Preface
Two questions that preoccupied me in the seventies motivated me
to write this book. (1) What is the relation between a statement (sentence,
proposition) and reality? (2) What is the fundamental character of logic?
Part I develops an answer to (1) and Part II (forthcoming) develops an
answer to (2). The Introduction and Chapter 1 give an outline of these
answers.
As I worked on the book, however, more and more of my life’s
experience teaching and researching questions of philosophy of logic, philosophy of mathematics and philosophy of language became relevant to the
subjects that I was discussing. So, the outcome is not just an attempt to answer questions (1) and (2), but is also, and perhaps primarily, a formulation
of a working philosophical viewpoint.
My viewpoint is essentially realist and metaphysical, influenced to
a very large extent by works of Plato, Aristotle, Frege, Russell, Gödel and
Hardy, among others. My aim in the book is to develop some aspects of
this viewpoint. Evidently, since I touch upon many different issues, it is
not possible to pursue them all in detail, and I will have to do that in other
writings. Nevertheless, I think that my answers to questions (1) and (2) are
developed in sufficient detail to serve as a basis for discussion.
In “Mathematical Proof” Hardy classifies philosophies as sympathetic and unsympathetic, tenable and untenable. I hope that some readers
will find the kind of view that I develop here as sympathetic as I do, and I
hope to show that it is at least as tenable as other views currently in vogue.
Although there is an overall structure to the book, on which I
will comment in the concluding remarks, most chapters are written in such
a way that they can be read independently. In fact, as can be seen in
the bibliography, versions of several of the chapters (from both parts) have
appeared in print. After reading the Introduction and Chapter 1, which give
a general sketch of position, the reader should feel free to read or browse
through chapters whose subject-matter may be appealing to him or her,
without worrying about earlier chapters. Each chapter has its own main
argument, which may be seen more clearly if on a first reading the end notes
are left out. The notes, some quite long, contain references, quotations,
amplifications, digressions, etc. and are not necessary for following the
main text. When a note is essential to the main text I indicate this by an
explicit reference of the form “see note such and such”.
11
I wrote the first version of the book in 1987-88, partly during a
sabbatical and partly in the context of a graduate seminar on philosophy
of logic. I am grateful to PUC-Rio for the sabbatical and I am equally
grateful to the students in that seminar, particularly to Arno Viero, for
many incisive and stimulating discussions of the texts coming hot out of
the computer. In 1989 I circulated a version of the typescript (then titled
“The Laws of Truth”) among some friends and colleagues. I am especially
indebted to John Corcoran and Charles Marks for detailed comments on
that typescript. In the intervening years I have presented versions of many
of the chapters at talks, conferences, colloquia, etc., and have had important
feedback from the audiences. I would like to thank Abel Casanave, André
Porto, Carlos di Prisco, Göran Sundholm, Itala D’Ottaviano, Jairo José
da Silva, Luiz Carlos Pereira, Luiz Henrique Lopes dos Santos, Michael
Wrigley, Raul Landim, Rodrigo Bacellar, Scott Soames, Sergio Fernandes
and Xavier Caicedo – and there are more. I would also like to thank CNPq
for a research fellowship during the years 1985-1993.
Writing a book is a trying experience and I appreciate the support I
received from my wife Siri and from my children Francisco, Barbara, Andrea
and Victor. I dedicate the book to them and to the memory of my mother,
Giovanna Vasta Bonino.
February 2001
Oswaldo Chateaubriand
Rio de Janeiro
12
Introduction
Logic, Ontology, and Epistemology
In its present day practice logic appears as a multiplicity of formal
systems conceptualized linguistically and mathematically. A logic, and a
formal system more generally, is conceived as a language consisting of a
syntax and a semantics. The syntax includes everything that can be treated
as a combinatorial of symbols, without regard for any content that these
symbols may have – i.e., without regard for what the symbols symbolize.
The setting up of the language (the grammar) is a main aspect of
syntax, but syntax is not restricted to the grammar. Proof is also treated
syntactically as consisting of operations (i.e., rules of inference) performed
over strings of symbols of certain categories such as formulas or sentences.
Considering a totality of applications of these operations one defines the
notions of logical deduction, logical consistency and logical theorem, which
together with certain notions of definition (abbreviative definitions, recursive definitions) are the main syntactic notions of logic.
The semantics of a logical language is based on the notion of interpretation (or of structure). This is mainly a set-theoretic notion that
involves a universe of discourse – a non-empty set – and a denotation function that assigns denotations to various symbols relative to the universe
of discourse. One can then introduce the notions of satisfaction and truth
relative to an interpretation. Considering a totality of interpretations one
defines the notions of logical consequence, satisfiability, and logical truth,
as well as a semantic notion of definition as individuation, which are the
main semantic notions of logic.
The systematic study of these notions and of their interconnections
belongs to proof theory, model theory and recursion theory, which are the
central areas of logic and are basically branches of mathematics. As a
science logic is supposed to be the combination of these theories and not
propositional and predicate logic as such. This was an important shift in
the conception of logic. For Frege and for Russell, for example, logic was
propositional and predicate logic; and it was a science.
Although the mathematical development of logic can fairly be said
to have begun with Aristotle, the modern project of an algebraization of
logic originates with Leibniz. Leibniz’s universal characteristic was to be
a language for pure thought combining an arithmetized theory of concepts
13
with an algebraic deductive calculus containing the fundamental principles
of all sciences1 .
Leibniz’s idea of an algebraic logical calculus came to fruition in the
nineteenth century, primarily through the work of Boole, whose The Mathematical Analysis of Logic set the basis for a systematic algebraic treatment
of logic along the lines envisaged by Leibniz. It was Frege, however, who
first axiomatized and formalized logic in what is essentially its modern form
in Begriffsschrift – and later in The Basic Laws of Arithmetic. These axiomatizations and formalizations were very fertile and are still the basis of
contemporary logical theory and practice. Frege formulated his logic as a
“conceptual script”, and following Leibniz referred to it as a “formula language” and as a “language for pure thought”, but he did not think of it as
an uninterpreted syntax. His logic was a specific theory with a definite content and scope, which left nothing to interpretation in the usual semantic
sense2 .
A significant factor in the development of the current linguistic
view of logic were the paradoxes of logic and set theory, which seemed to
undermine the Platonic elements in the conception of logic and set theory
that we find in Frege and in Cantor. It was partly as a reaction to the paradoxes that Hilbert emphasized the syntactic features of logic, though he did
not really consider logical syntax to be uninterpreted either. It was a “formula game”, but a formula game that expressed laws of thinking reflected
in language3 . Hilbert’s formalism and its consolidation as a philosophy of
mathematics was an important influence in the initial development of the
linguistic conception of logic. A relevant factor in the later transformation
of Hilbert’s view into a more straightforward treatment of logic as a theory
of formal languages was the emphasis on first order logic and the relativistic
conclusions that Skolem drew from the Löwenheim-Skolem theorem4 .
The main philosophical influences in shaping the modern linguistic
view of logic were Wittgenstein and the logical positivists, although Russell
also played a decisive role. His eliminative theory of classes, based on his
eliminative theory of descriptions, and his tendency to talk of propositional
functions as open sentences, gave room to the interpretation of Principia
Mathematica as a nominalistic theory of logic and mathematics which explained away the abstract content of these sciences in linguistic terms.
But it was Tarski’s work on truth, and his semantic conception of
truth, that led to the consolidation of the linguistic and mathematical view
of logic in its present form. Model theory is essentially Tarski’s creation,
and it completely transformed the character of modern logic. Skolem’s
philosophical relativism became a mathematical relativism apparently with-
14
out philosophical commitments. And Hilbert’s formalism, undermined by
Gödel’s incompleteness theorems, became a relativistic syntactic theory of
formal systems. The absolute conception of logic that one finds in Frege,
Russell, and even in Hilbert, gave way to a relativistic conception of logic
centered on model theory and proof theory as theories of formal systems.
Tarski claimed that semantics is part of the morphology of language and that the semantic conception of truth is neutral in relation to
metaphysical issues5 . This made his work acceptable to philosophers of
quite different persuasion, many of whom reject an ontological conception
of logic. It was Tarski’s formal analysis of truth that in a way managed
to separate the formula language from its content, however conceived, and
which gave a certain autonomy to syntax. One can hold the syntax to be
uninterpreted because it is interpreted by the semantics. This leads to the
idea that the syntax comes first and that it can stand on its own as a purely
symbolic manipulation – which is what I mean by autonomy. It was also
largely due to Tarski’s work that the problem of truth became primarily
a problem concerning a classification of sentences, which fits in quite well
with the idea that a central task of logic is the classification of sentences
into logical truths and others. This is an important aspect of the linguistic
view of logic, in any of its multiple forms, and is very much part of the
standard view of logic. It is common to Carnap and Quine, for instance, in
spite of their differences.
The basic positivistic view was that it is the meaning of the logical
words that accounts for logical truth. Logical truths were supposed to
be analytic, and analyticity was taken to be a feature of language alone.
Quine opposes this sort of linguistic view of logic, and attacks Carnap for
it, but nevertheless he holds language to be constitutive of logic. His view
is that logical truth, and logic, depend on grammar and truth rather than
on meaning. In part this is a way to avoid an ontological interpretation of
logic, for Quine agrees with the logical positivists that the idea that logic is
a science that formulates very general laws about the structure of reality is
nonsense6 . In part, however, the appeal to grammar and truth is Quine’s
way of avoiding the notions of meaning and analyticity, and what makes
this possible is Tarski’s work on truth.
In the Tarskian view of logic the notion of meaning largely disappears and is replaced by satisfaction and truth. Some interpretations of
Tarski’s semantic conception of truth see it as reducing truth to denotation, which is how truth in an interpretation is usually defined, but this is
not quite what Tarski himself does. In any case meaning is gone, and the
content of the logical notions must be recovered either through denotation
or through grammar. Even when truth is analyzed in terms of denotation,
15
however, the logical symbols typically do not denote, but their content is
explained as the role that they play in the truth or falsity of sentences – an
idea which goes back to Russell in “On Denoting”. Which leaves us with
grammar and truth, as suggested by Quine.
Quine’s animosity toward meaning and related notions is not shared
by other philosophers who also hold some form of linguistic view of logic.
In particular, there is now an important current of linguistic interpretations
of logic, primarily associated with proof theory and intuitionistic logic, that
denies the central place of truth in logic and goes back to the notion of
meaning as basic. According to this view, the meaning of the logical constants derives from our deductive practices. Meaning in general is supposed
to be intrinsically connected with proof7 .
Although there are different opinions about the relation between
logic, language and mathematics, the categorization of logical notions in
terms of syntax and semantics and the conception of logic as a language,
or languages, is basically taken for granted. Whether one agrees with the
details of one view or another, there is a fairly general consensus that the
conceptualization of logic as a language is the right conceptualization.
I do not deny that linguistic concepts have played an important
role in the development of logic, but in my view the fundamental character
of logic is metaphysical, not linguistic. On the one hand I see it as an
ontological theory that is part of a theory of the most general and universal
features of reality; of being qua being, as Aristotle said. On the other
hand I see it as an epistemological theory that is part of a general theory of
knowledge. A large part of the aim of this book is to develop and defend such
a metaphysical approach to logic, which involves also a critical evaluation
of the basic claims and assumptions of the linguistic approach. I shall now
outline some issues and ideas in a preliminary way.
Logic has always been centrally concerned with truth, and truth
has been traditionally conceived as an expression of what is real. The
strict connection between truth and being was emphasized by Plato and
by Aristotle8 . It was also Frege’s view of truth, though he gave it a new
twist by postulating two objects, the True and the False, to which true
statements and false statements refer. The connection between logic and
truth also goes back to Plato and Aristotle, but Frege stated it in a very
specific and distinctive way: logic formulates the laws of truth9 .
If logic has as at least part of its task the investigation of laws of
truth, and if truth is an expression of reality, then it would seem that an aim
of logic is the investigation of laws of being. The grounds of this ontological
aspect of logic were explicitly laid down by Aristotle in the Metaphysics,
16
where some of the basic laws of logic were held to be among “the most certain principles of all things”. The law of contradiction in particular, which is
still considered to be the most fundamental law of logic, was stated by Aristotle as “the most certain” of all such principles – Metaphysics 1005b10−35.
Evidently, the formulation of such principles depends on a categorization of
reality, and the traditional categorization of reality that has been central to
logic is the categorization in terms of objects and properties, or particulars
and forms, deriving from Plato. It is through this categorization and the notion of application (or instantiation, or participation) that the principles of
logic are formulated by Aristotle as ontological principles, whether directly
or in terms of truth. The principle of contradiction, for example, is formulated both as the principle that the same property cannot apply and not
apply (at the same time, in the same respect) to the same subject, and as
the principle that contradictory propositions cannot be true – Metaphysics
1005b18 and 1011b13. Although the connection between these formulations
of the principle of contradiction is quite close, because any proposition is
supposed to attribute a property or relation to some subject(s), they reflect
two rather different analyses of logic which are the basis for the division of
modern logic into propositional logic and predicate logic.
Propositional logic is generally considered to be a part of predicate logic, but I think that there is a fundamental difference between the
two. Propositional logic can be seen as a very broad theory of truth relations between propositions, quite independently of any analysis of the
logical structure of propositions and of their linguistic expression10 . As a
general theory of truth relations between propositions it is natural to say
that propositional logic formulates laws of truth. The principle of contradiction in its second formulation can be taken as the fundamental principle
of propositional logic.
Predicate logic has two distinct aspects. On the one hand it can
be seen as a general theory of properties and objects based on some specific
logical properties and operations. Its laws are laws of being in a sense
that is very close to Aristotle’s. The principle of contradiction in its first
formulation is one of the fundamental laws of predicate logic. In this sense
predicate logic is not a theory of logical truth, or of logical implication, but
a theory of reality. Although one can say that the laws of predicate logic
are truths of logic, it doesn’t follow that predicate logic is a theory of logical
truth in the sense of a classification of sentences or propositions.
The distinction between formulating laws of logic as laws of being and characterizing logical truth and logical implication brings out the
other aspect of predicate logic. This involves a concern with propositional
structure and its relation to reality. The connection between predicate logic
17
and propositional logic derives from the analysis of the logical structure of
propositions.
The modern theory of logical form is a theory of propositional
structure in terms of the categories of objects and properties and of specifically logical properties and operations. Through this structuring one can
bring together the logical analysis of the general features of reality with the
logical analysis of the truth relations between propositions.
One of the most fundamental truth relations between propositions
is the relation of material implication, which is basically a relation of factual
truth preservation. A proposition materially implies another if it is not the
case that it is true and the other is not true. Logical implication is a stronger
relation than material implication because it derives from the logical character of the propositions and not merely from their truth values. That is,
logical implication is an expression of logical necessity rather than a factual
relation between propositions. But truth preservation is a necessary feature
of any implication relation, and the usual analysis of logical implication (or
logical consequence) in terms of interpretations is a way of reducing logical
implication to material implication plus a totality of interpretations. But
even if one accepts this analysis of logical implication, it depends on certain
metaphysical assumptions.
An interpretation can either be conceived as a way in which the
world could be or as an aspect of how the world is. In order for the reduction of logical implication to material implication to work, one must either
assume that reality could contain infinitely many objects or that it does contain infinitely many objects, depending on which approach one takes. The
second approach normally involves the assumption that reality contains infinitely many mathematical objects – as well as infinitely many properties
(or sets), in fact. If this is indeed a feature of reality, it is presumably a
necessary feature of it, and it can only be accounted for in those terms. It
would be rather odd to have the analysis of logical implication depend on
whether or not, as a matter of fact, there are infinitely many objects and
properties in the world11 .
It seems to me, therefore, that the combination of propositional
and predicate logic should be seen as an investigation of the general structure of reality, including possible and/or necessary features of it, an account
of propositional structure, an account of truth relations between propositions, and an account of properties and operations that are specifically logical properties and operations. This is essentially what I mean by logic as
an ontological theory.
18
I take epistemology to be primarily concerned with the conditions
of knowledge and justification, especially in connection with our theories
about reality, and I take a central aspect of this to be a concern with proof
and justification. The investigation of the general structure of proofs, deductions, arguments, reasonings, definitions, etc., and of their correctness,
is considered to be another traditional aspect of logic. This epistemological
aspect of logic deals primarily with the analysis of logical deduction; that
is, with the general principles of inference that can be used to establish a
conclusion given certain initial assumptions.
The idea that proofs and definitions are syntactic seems to me a
particularly perverse idea. Philosophers may not agree about many things,
but on one thing in epistemology there is rather general agreement; namely,
that truth is a necessary condition for knowledge. But, by the same token,
the preservation of truth is a necessary condition for proof. If we start with
truths and arrive at a conclusion that is not true, then we haven’t got a
proof. But if truth is not a syntactic notion, then neither is proof.
This simple argument shows that the epistemological aspects of
logic depend on its ontological aspects at least through the dependence on
the notion of truth. A basic ontological constraint on the notion of logical
deduction is the preservation of truth. The laws of deduction may be said
to be laws of truth in the sense that they are laws for reaching truth and
for truth-preservation, and it is precisely this that one tries to guarantee
in their formulation. Therefore, our logical insights on truth are equally
fundamental for the development of a theory of proof.
That proof is an epistemological notion essentially connected to
the quest for knowledge, justification and truth is so obvious that it hardly
needs arguing. Yet, it has been so hammered into our heads that a proof
is essentially a sequence of syntactic transformations that these claims can
arouse considerable antagonism. One must have an account of proof and
deduction that is epistemologically significant and that gives plausibility
to the basic principles of proof that are accepted. It is clear that by its
very nature no purely syntactic account can do this – what it can do is
to codify such an account. I agree that the syntactic analysis of proof
and the analysis of effectiveness in the theory of recursive functions are an
important contribution to a general theory of proof and definition, but their
significance does not derive from purely syntactic considerations.
The truth preservation that is involved in logical and mathematical proofs is not simply a factual connection between the premises and
conclusion of a proof, but a necessary feature of proofs. A proof establishes
intrinsic connections between properties – as can be seen quite clearly from
19
mathematical proofs – and the analysis of truth preservation through logical form is an analysis of intrinsic (necessary) connections between logical
properties. This is what distinguishes the notions of proof and logical deduction, in a strict sense, from the more general notion of justification; for
justification in general does not require establishing intrinsic connections
between properties or guaranteeing truth preservation.
Nevertheless, in investigating the laws of logic we are also involved
with questions of justification in this broader sense. How can we justify
that these are laws of logic?12 How can we justify certain general principles
of definition? Given the generality of logic, these epistemological questions
get inextricably mixed up with logic itself. So much so that it seems almost
impossible to discuss a fundamental principle of logic such as the law of
contradiction without presupposing it. It is not clear, therefore, to what
extent one can separate the epistemological discussion of logic from logic as
an ontological theory13 .
There is a strong tendency to separate logic from ontological considerations, however, and the syntactic conception of proof and deduction
is partly an expression of this. The advent of natural deduction systems
encouraged the idea that deduction can be used to justify the laws of logic
independently of any considerations about truth and reality – and, therefore, that the notion of proof is the most fundamental logical notion. But
aside from purely syntactic considerations, an appeal to natural deduction
systems seems to me to show nothing about the priority of proof in logic.
Suppose that our system of natural deduction includes something
like the principle of reductio ad absurdum in the classical sense; namely, that
if from the assumption that S (or that ¬S), joined to some other assumptions we may have, we can deduce a contradiction, then from those other
assumptions alone we can deduce ¬S(or S). This, or something essentially
equivalent, is normally a part of natural deduction systems for classical
logic.
Now the problem is this: how does one justify reductio ad absurdum as a principle of proof? I know of only one good justification (for
classical logic), and that appeals ineliminably to truth-preservation and to
the principles of contradiction and of excluded middle. The justification is
the following. If we get a contradiction from S (or from ¬S), then, since a
contradiction cannot be true, by the principle of contradiction, S (or ¬S)
cannot be true either – because deductions must preserve truth. So, by
the principle of excluded middle, ¬S(or S) must be true. (Of course, this
is all relative to our other assumptions.) To which considerations can one
20
appeal in order to justify the principle of reductio ad absurdum that are not
considerations about truth (and reality)?
Should one appeal to rationality? What is it that makes it irrational to accept a contradiction? In practice, even the staunchest rationalists admit that we must live with contradictory theories. In any case,
however, unless deductions are truth-preserving we cannot infer anything
about S (or ¬S) from the fact that a contradiction was deduced from it.
Just as we can blame the contradiction on other hypotheses we may have
used rather than on S, we can blame the contradiction on our deduction; and
what we would be blaming it for is for not being truth-preserving. Reductio
ad absurdum presupposes that any combination of deductive principles in a
deduction is truth-preserving.
Should one appeal to deductive practices?14 What accounts for
these practices? Are they truth-preserving? If the principles of deductive
logic are based on actual deductive practices, then how (or why) does one
disqualify common practices of deduction that are not truth-preserving?
The practice of affirmation of the consequent, for instance, is very well established and can actually be quite useful in many specific circumstances. It
seems to me that no theory of deduction, based on meaning, deductive practices and the like, can guarantee truth-preservation. But truth-preservation
is an ontological constraint on deductions; for if we discover that an accepted principle of deduction is not truth-preserving, we disqualify it as a
principle of logic.
Naturally, there are complex issues involved here which I do not
mean to resolve now. What I am pointing out is that for classical logic
it is not a simple matter to account for proof independently of ontological
considerations, and that this is so precisely because classical logic is based
on considerations about truth and reality rather clearly reflected in the
principle of reductio ad absurdum.
A more promising approach may be to appeal to intuitionistic logic.
It is often said that intuitionistic logic is a logic of knowledge rather than a
logic of truth. In fact, the intuitionistic connectives are usually characterized
in terms of proofs (or mental constructions) and of assertibility conditions
– one can assert a disjunction (α ∨ β), for instance, if one has a proof of α
or one has a proof of β.
I don’t actually agree with this view of intuitionistic logic, for
it seems to me that intuitionistic logic deals with reality just as much as
classical logic, but that it deals with a structuring of reality that is not
subjectively transcendental in the same way as in classical logic. What
Brouwer maintains is “that truth is only in reality, i.e. in the present and
21
past experiences of consciousness”15 . Brouwer’s notions of consciousness,
mind, experience, truth, etc., are not really identifiable with the mental
content of a given subject, however. Consciousness is treated as a sort of
primaeval being rather than as the conscious activity of a subject; mind
derives from an initial structuring of consciousness, and the subject as well
as the object are aspects of mind. Moreover, it is not at all clear that
Brouwer’s constructions can be identified with proofs in an ordinary sense
– and certainly not in a syntactic sense.
So rather than a switch from a logic of truth to a logic of knowledge,
it is more natural to say that intuitionistic logic involves a switch from a
classical conception of reality to a different conception of reality, where truth
is in experience. Of course, this notion of truth may be more intimately
connected to knowledge than the classical notion, and I do agree that the
subject plays a more central role in intuitionistic logic than in classical logic,
but this is not to say that one is switching to a logic of knowledge or of proof
in an ordinary sense of these terms.
In any case, reductio ad absurdum and the principle of contradiction raise difficult questions for intuitionistic logic as well. Reductio ad
absurdum appears only in the non-parenthetic form, and since one doesn’t
have the principle of excluded middle, or of double negation (¬¬α ⇒ α),
one cannot get the parenthetic version. In fact, reductio ad absurdum is
built into the very notion of intuitionistic negation. The usual explanation
is (roughly) that one can assert a negation ¬α if one has a proof (or construction) that would deduce a contradiction from any hypothetical proof
of α. It is partly this interpretation of negation that leads to the rejection of the principles of double negation and of excluded middle. But if
¬α is true, i.e., if I can experience it, then the hypothetical constructions
of α can have no reality; i.e., cannot be experienced. So how can I reason
about them? This is Parmenides’ old point about non-being making its
appearance in the intuitionistic conception of reality. As Parmenides before
him, Griss concluded that one should not speak of what is not, nor reason
about it, and was led to develop a negationless intuitionistic mathematics
and logic in which negative statements are either interpreted directly or
are replaced by positive, generally stronger counterparts. The main point
of Griss’ criticism was not merely technical, however; it was rather the result of a mystical and idealistic view of reality which had many points in
common with Brouwer’s16 .
This criticism of negation raises serious problems for the interpretation of intuitionistic logic and mathematics in terms of proof. How does
one justify negation? What is a contradiction supposed to be? This is what
Heyting, the founder of intuitionistic logic, says: “I think that contradic-
22
tion must be taken as a primitive notion. It seems very difficult to reduce
it to simpler notions, and it is always easy to recognize a contradiction as
such”17 . I don’t agree with the last point, but in any case this doesn’t say
what contradictions are or what the principle of contradiction is about. Is it
about the present and past experiences of consciousness? Does it say that in
some sense there are no contradictions in the structuring of consciousness?
This seems to involve an appeal to truth and reality just as in classical logic.
Does it say that the subject cannot experience contradictions? How come?
So I don’t think that intuitionistic logic, interpreted in a way that
would respect the metaphysical idealistic content of Brouwer’s views, gives
an epistemological approach to proof that is clearly independent of ontological considerations. In fact, Brouwer’s remark that “logic . . . cannot deduce
truths that would not be accessible in another way as well” seems to me
another way of saying that deduction is subordinate to truth and reality (in
his sense).
Of course, one can leave aside Brouwer’s own metaphysical views
and interpret intuitionistic logic from a different perspective. This is Dummett’s and Prawitz’ approach through a theory of meaning, linguistic behavior and deductive practices18 . But, as for classical logic, a serious difficulty
for this approach is the question of truth-preservation of deductions and
how this relates to meaning as use and to deductive practices. Martin-Löf,
who also takes intuitionistic logic as primary, concludes that the notion of
validity of a proof, or correctness of a proof, must be an absolute (transcendental) notion. As a logic of knowledge in his sense intuitionistic logic
would be an ontological theory of proof19 .
In my view any reasonable epistemological development of the notions of proof and deduction must have a strong ontological component, but
I see this in terms of a classical realistic theory of truth. Thus, I emphasize
classical logic rather than intuitionistic logic. Nevertheless, I have sympathy
for Brouwer’s approach and I think that in some respects it is not that far
from Frege’s. One important characteristic that their approaches to ontological and epistemological matters have in common is a certain subsidiary
role they attribute to language – though Frege emphasizes notation and
Brouwer does not.
A rather different approach to the question of the relation between
proof and ontology in logic is due to Quine, who claims that if we know that
a system of deduction is sound (i.e., is truth-preserving) and complete (i.e.,
can deduce all logical consequences), then we can discard the appeal to
ontology in favor of a syntactic notion of deducibility. If we assume an
ontological analysis we can justify a syntactic analysis, and therefore we
23
can get rid of the ontological analysis. It’s a case of throwing away your
ladder after you have climbed it.
Quine offers this argument in various places to show that ontology can be eliminated from logic in favor of deducibility, or of proof
procedures20 . Of course, the soundness and completeness theorems hold for
propositional and first order logic, but completeness does not hold for other
logics such as second order logic. Well, so much the worse for second order
logic, says Quine – who doesn’t like second order logic anyway21 . But second order logic cannot be so easily discarded, because, among other things,
it is an integral part of mathematical practice22 .
What Quine suggests about second order logic is that we should
replace it by a first order counterpart such as some system of set theory.
But, on the one hand, this doesn’t settle the question of mathematical
practice because that practice involves second order logic in a way that is
not simply reducible to first order set theory. And, on the other hand, the
claim that it is the existence of proof procedures that characterizes logic
begs the question; Quine does not deny the ontological character of second
order logic, but disqualifies it as logic precisely because it cannot be reduced
to purely syntactic proof procedures.
It is essentially the restriction of logic to first order logic, beginning in the twenties with Skolem and Hilbert23 and strongly defended by
Quine, that has encouraged the idea of proof as syntactic. The appeal to
the completeness theorem then suggests this rather formalistic approach of
Quine’s to the question of ontology in logic. His argument depends on the
idea, characteristic of linguistic conceptions, that what matters in logic are
the classifications we make. It suggests that any appeal to ontology, or to
epistemology for that matter, is basically irrelevant aside from generating
a certain classification of sentences. If we can show that this classification
can be recovered by purely syntactic means, it follows that we can get rid of
whatever ontological or epistemological assumptions we used to begin with.
What Quine doesn’t do, but in my view should do for the argument to show the priority of proof over ontological considerations, even for
first order logic, is to discuss the notion of proof, or deduction, aside from
a purely formal characterization of proof procedures. What makes this notion philosophically central to logic? It is quite remarkable that in Quine’s
extensive work on logic we find no attempt to discuss the notion of proof
philosophically.
Quine’s elimination of ontology in favor of proof procedures is more
in the nature of a reduction to grammar than a reduction to proof in any
significant sense, and his main view of logic is indeed in terms of grammar
24
and truth rather than proof. As he puts it in Philosophy of Logic: “Logic
is, in the jargon of mechanics, the resultant of two components: grammar
and truth”24 . To the extent that Quine holds truth to be a central concern
of logic, he agrees with Frege. But why should logic depend on (or result
from) grammar? Because the notion of logical truth, which for Quine is the
main notion of logic, is held to be characterized by grammar and truth.
Let’s interpret grammar as logical form – or logical grammar.
(Quine’s formulation to which I referred above is supposed to be a generalization of this.) A sentence is a logical truth if all sentences with the
same logical form are true; or, alternatively, a sentence is a logical truth if all
its substitution-instances are true. If this is not some sort of cosmic coincidence, then there must be something about the logical form that guarantees
it. The standard idea is that it is the interpretation of the logical symbols
that guarantees it. In doing interpretations that’s the one thing that is kept
fixed; one is not allowed to fiddle with the interpretation of the logical symbols. This already suggests that the grammar is not so meaningless after
all. Why not interpret those symbols in any way we want? Because they
are the logical symbols characteristic of logical form. But what do they
stand for? It would seem natural that they would stand for logical notions,
or logical properties, or logical operations of some sort, but in practice they
don’t stand for anything. They are logical notions, in some sense, but their
content seems to be to be translated away into other expressions of them.
In order to define logical truth substitutionally Quine assumes
a language that is sufficiently strong to include the truths of first order
arithmetic. His substitutional definitions of logical truth are then justified
through the fact that any interpretation for first order logic can be reflected
in such a language – which again depends on an appeal to the completeness
theorem and to a form of the Löwenheim-Skolem theorem.
Quine’s grammatical characterization of logical truth is supposed
not only to avoid an ontological interpretation of logic, but also to avoid
any appeal to possibility, necessity, and meaning in connection with logic.
The sense in which ontology, necessity, possibility, and meaning are avoided
by these grammatical definitions is not altogether clear, however. In the
case of interpretations we know that if we are dealing with actual interpretations rather than possible interpretations, we depend on there being
enough reality to give us all the interpretations we need. Similarly, if we are
dealing with sentences which are actually true rather than possibly true,
we also depend on there being enough things in reality to disqualify various
sentences as logical truths – for example, a sentence that asserts that there
are at most n objects, for sufficiently large n. If we assume that reality
contains the objects and structures of mathematics, then there isn’t any
25
problem, and something like this is generally assumed. Quine’s assumption
that his language contains the truths of first order arithmetic is a version
of precisely that assumption.
And Quine must appeal to arithmetical truth in an ontological
sense because the attempt to replace the ontological notion of mathematical
truth by a structural (syntactic) notion of mathematical truth characterized
in terms of proof did not work. What are the basic properties of truth? The
laws of contradiction and of excluded middle. And what are their deductive
counterparts for formal systems? Consistency and completeness of the formal system. Consistency for a formal system means that contradictions are
not provable in that system; completeness for a formal system means that of
a pair of contradictory sentences of the system at least one is provable in it.
Therefore, if we have a consistent and complete formal system, the notion of
theorem of the system may be considered a reasonable substitute for the notion of truth; at least in the sense that it has two very fundamental features
of truth. The problem is that any formal system which includes a modicum
of arithmetic – i.e., that can represent the primitive recursive functions –
is either inconsistent or incomplete; which is the content of Gödel’s first
incompleteness theorem25 .
Quine claims that his assumption is very modest, but from an ontological point of view he is assuming that reality has a rather complex
infinite structure which on any reasonable view should be a necessary feature of it. So in effect, even for first order logic, his argument depends on
assumptions which are naturally taken to be about the necessary structure
of reality rather than about grammar and about truth in the sense of a mere
classification of sentences.
Logic is supposed to be universal in some sense, yet not an ontological theory. It is actually suggested that this derives precisely from
the universality of logic. In terms of interpretations the suggestion is that
since logic must hold in every interpretation, and interpretations come in all
shapes and sizes, it cannot be about anything in particular. Another way of
putting the thesis is to say that logic has no existential commitments; that
it doesn’t imply or presuppose the existence of anything26 .
Everybody would agree, of course, that logic shouldn’t imply the
existence of non-logical entities such as tables and chairs – something like
this was one of the main problems with Russell’s axiom of infinity27 . But the
question is, or should be, whether logic says something about the structure
of reality, and in particular whether there are specifically logical entities of
some sort. In my view there are.
26
If we assume the kind of categorization of reality which Frege used,
and which still underlies standard logical practice, logic treats of objects,
properties (concepts) of objects, relations among objects, properties of properties of objects, relations among properties of objects and objects, etc.
That is, one has a hierarchy of levels beginning with objects (level 0), continuing with properties and relations of these objects (level 1), and so on,
indefinitely. Are there among these entities some that have the character of
universality that one would expect of logical entities?
Leaving aside the question as to whether there are logical objects
at level 0, suppose that we start at level 128 . There seem to be only two kinds
of relations at this level that are universal in the required sense; namely,
Identity and Diversity relations – I will capitalize logical properties. There
are actually infinitely many of these; for example, there are pairwise Diversity relations for all arities (i.e., number of arguments) greater or equal than
229 .
But at the second level is where things begin to get really interesting. As Frege saw it, the first order quantifiers (quantifying over level
0) appear at level 2 as properties of properties of level 0 objects. Frege
emphasized that existence is not a property of objects but of concepts –
his terminology for properties. ‘There are philosophers’ asserts something
about the property of being a philosopher – namely, that it doesn’t have an
empty extension – and not about the individuals who are philosophers30 .
Therefore, the use of quantification over level 0 objects “generates”, in a
manner of speaking, a large number of logical properties of level 2. Subordination between level 1 properties, for example (Aristotle’s ‘All A is B’);
Exclusion (‘No A is B’); etc. And as Frege showed in The Foundations of
Arithmetic, the Cardinality relations between level 1 properties (e.g., the
level 1 property F applies to the same number of level 0 objects as the level
1 property G), also appear at level 2. And there are many, many more. In
fact, there are infinitely many logical properties at all levels.
This view of logical properties also suggests an approach to logical
truth that is independent of linguistic forms and that shows the connection
between the laws of logic and the logical truths. Is it a logical truth that
every object is self-identical? Well, yes, in the sense that it is a law of logic
that for any object x, x = x. This can be understood as asserting Reflexivity
– i.e., a certain level 2 property expressed by the universal quantifier – of
the level 1 Identity relation. The logical truth ‘∀xx = x’ is an expression of
this feature of Identity and, therefore, an expression of the logical law.
But is it the case that specific sentences of the form ‘a = a’ are
logical truths? I would say that sentences of the form ‘a = a’ are not
27
logical truths in the sense of being true in virtue of logical or grammatical
form, because logic does not guarantee that the expression substituting ‘a’
has denotation, and therefore does not guarantee that a sentence of that
form is true. If one holds, with Frege, that sentences containing expressions
which do not denote are neither true nor false, then it is pretty obvious that
grammar is not a source of logical truth – because it isn’t a source of truth.
Since in fact this holds for any sentences containing non-logical expressions,
be they names or predicates31 , the only guaranteed logical truths will be
sentences such as ‘∀xx = x’ consisting exclusively of logical notions. What
accounts for their truth, however, are the logical notions (or properties),
not grammar.
Sentences can only be considered to be logically true in something like a grammatical sense relative to certain assumptions concerning
the nature of the logical expressions and the denotation of the non-logical
expressions that they contain. This is what one does in the usual semantic
characterization of logical truth by limiting the totality of interpretations to
those in which the logical symbols denote (or get translated as) the “right”
things and the non-logical symbols always denote something. What this
suggests, however, is that it is the ontological analysis, rather than the
grammar, that accounts for logical truth. To assume that the non-logical
“constants” must always denote is tantamount to declaring that one wants
to deal with reality, not with grammar32 . Why not then formulate the law
of identity by saying that every object is (necessarily) self-identical rather
than talk about grammar?
In addition, by taking logical truth to be a feature of sentences
one is assuming that each sentence has a definite logical form. But as far
as ordinary sentences are concerned none of these assumptions are justified.
What one really does is to work with a language of pure forms, the logical languages, conceived syntactically. It seems more natural to conceive
of these languages as directly expressing the logical properties than as expressing sentential forms. That is, the usual so-called logical grammars are
really theories of logical properties, and the linguistic forms represent these
properties. The real forms are the properties themselves33 .
The combination of grammar and truth only seems to work because
one makes enough ad hoc assumptions to guarantee that the notion of logical
truth will come out right – i.e., in agreement with the basic ontological
content of the laws of logic. If truth is an expression of reality, logical truth
should also be an expression of reality – it should express certain necessary
features of reality. The idea that logical truth has to do with logical form
is natural enough, but the idea that logical form is a grammatical feature
of sentences (or is syntactic) seems to me completely unnatural.
28
Take the sentence ‘Theaetetus is sitting’, for instance. Its logical
form is represented by, say, ‘Fa’. In my view it is not the expression ‘Fa’ that
can be considered to be a logical form, but the logical property Application
(or Instantiation) of a certain type; this property is expressed linguistically
by the juxtaposition of a property letter ‘F ’ to an object letter ‘a’. I agree
that the expression ‘Fa’ represents the logical form, but this is not a purely
syntactic feature of it. To analyze the sentence ‘Theaetetus is sitting’ as
having the logical form ‘Fa’ is to analyze it as expressing the instantiation
of a property by an object, with the specific association of the property
‘sitting’ to ‘F ’ and the object Theaetetus to ‘a’ – where ‘F ’ and ‘a’ are
thought of as non-logical constants. We may also say that we are analyzing
the form of the sentence ‘Theaetetus is sitting’ as corresponding to a certain
type of state of affairs consisting of the instantiation of a property in an
object.
States of affairs are quite independent of language, and that Identity is Reflexive can be considered to be a logical state of affairs; a matter
of logical necessity one might say, or a logical law. But for each object
its self-identity is also a matter of logical necessity, though it is not a logical law that, say, Hesperus is self-identical. The existence of Hesperus is
a contingent matter, and so is the existence of the state of affairs of Hesperus’ self-identity, even though it involves a logically necessary feature of
Hesperus.
As Frege said, the laws of logic are not merely laws of what happens
but of what is, in the sense that they express fundamental features of the
structure of reality. The relation of the law of identity to the logical truth
that Hesperus is self-identical seems to me analogous to the relation of a
law of physics to a specific consequence of that law (a physical truth, say)
concerning Hesperus. Just as it is not the primary business of physics to
classify sentences into physical truths and others, it is not the primary
business of logic to classify sentences into logical truths and others. These
classifications are derivative from the laws of these sciences. It is precisely
because I take the laws of logic to express fundamental features of reality
that I see logic as a science, or as a theory, rather than as a language.
As other sciences logic came to maturity by the application of
the axiomatic method developed initially by Aristotle. And it was with
Frege’s first axiomatization in Begriffsschrift that logic was born as a mature
science. Not really by rejecting Aristotle’s logical project but by improving
on it34 . The axiomatic method is an epistemological method, and Frege’s
axiomatization of logic was just as much part of his epistemology of logic
as Newton’s axiomatization of mechanics was part of his epistemology of
physics. Theory-making is an epistemological affair. Frege did not base his
29
theory of logic primarily on an analysis of language but on ontological and
epistemological considerations. Yet his idea of logic as formulating the laws
of truth was also meant to delimit the scope of logic vis-à-vis both ontology
and epistemology.
Frege’s second axiomatization in The Basic Laws of Arithmetic
was a rather direct attempt to express the laws of logic as laws of truth,
with the truth values the True and the False playing a very central role. But
given the peculiar character of these objects, they seem to be a merely formal
expedient. In his last publications on logic Frege no longer explicitly appeals
to truth values as objects, although the conception of logic as formulating
the laws of truth is still there and is expressed in much the same terms
that he used in the introduction to The Basic Laws. In “Thoughts” Frege
argues rather strongly against a view of truth in terms of correspondence,
maintaining that it is both obscure and question begging, and concludes
that truth is sui generis and undefinable. It pertains to logic to spell out
the contents of this notion in the laws of truth. It must be assumed that for
him this spelling out is an axiomatic spelling out akin to Newton’s axiomatic
spelling out of the contents of the notion of motion. We don’t ask anymore
what motion is, in a direct definitory sense, but what its laws are. Similarly,
I see Frege’s suggestion in “Thoughts” as a suggestion that one shouldn’t
ask what truth is in a direct definitory sense, but what its laws are; and
that this is the task for logic. This is not a linguistic (or syntactic) view of
logic, though, for truth for Frege is in reality, not in words or in thoughts35 .
The paradoxes discredited Frege’s system while keeping certain aspects of his formalism more or less above the fray. Eventually this formalism
became somewhat independent of the philosophical dispute and gave way
to the notion of formal language. There is an important ambiguity in this
notion, however, for one must distinguish the use of a special notation in
formulating one’s theories about reality from the notion of formal language
in the sense in which mathematical logic is said to be a theory of formal languages. Frege’s conceptual notation was an example of the former, though
it also inspired the latter. Frege’s system of notation is linguistic and has
the characteristics and limitations of language generally. Formal languages,
on the other hand, are not languages at all but abstract mathematical structures, which can even be conceived as generalized arithmetics36 , that are correlated in various ways with other mathematical structures. From a Fregean
point of view, they can also be seen as complex higher order properties37 .
The result of mixing up these two altogether different things is that logic as
a theory, in Frege’s sense, gets conceptualized as a formal language, which is
both an abstract mathematical structure that can be correlated with other
mathematical structures, and something linguistic. As a logical language
30
it has some kind of definite content, yet it is not a theory of anything; it
is rather like a schema that can be used in the formulation of theories38 .
Logic as theory is then the theory of one or another formal language, or of
many of them.
This leads to a view of logic as an autonomous mathematical discipline that studies formal systems. The philosophical content of logic is
not necessarily denied on this view, but it is conceptualized in terms of
philosophical implications of the mathematical theory to be studied in the
philosophy of logic. Although there is a certain amount of truth in this
view, I think that it puts the cart before the horse, for as I see it logic is
philosophy studied and developed mathematically. Physics did not cease to
be physics by being mathematized; similarly, logic did not cease to be metaphysics by being mathematized. The sense in which mathematical logic is
an autonomous mathematical discipline seems to me exactly the same sense
in which mathematical physics is an autonomous mathematical discipline.
This is not to deny the importance of this mathematical development; on the
contrary, I consider it the most significant ontological and epistemological
advance in modern times.
The view of logic as metaphysics (in a sense that includes both
ontology and epistemology) is not unusual and has been a major point of
dispute throughout the century. Opponents of classical logic often maintain
that it should be rejected precisely because they see it as a metaphysical
(realistic) theory. Curiously enough, however, many defenders of classical
logic chose to fight on the grounds that it is not – a notable exception was
Gödel, but he was definitely part of a small minority39 .
31
Notes
1. In “Towards a Universal Characteristic” Leibniz says (Selections, pp. 22-23):
In order to establish the Characteristic which I was after – at least
in what pertains to the grammar of this wonderful language and
to a dictionary which would be adequate for the most numerous
and most recurrent cases – in order to establish, in other words, the
characteristic numbers for all ideas, nothing less is required than the
founding of a mathematical-philosophical course of study according
to a new method, which I can offer, and which involves no greater
difficulties than any other procedure not too far removed from familiar concepts and the usual method of writing. Also it would not
require more work than is now already expended on lectures and
encyclopedias. I believe that a few selected persons may be able to
do the whole thing in five years, and that they will in any case after
only two years arrive at a mastery of the doctrines most needed in
practical life, namely, the propositions of morals and metaphysics,
according to an infallible method of calculation.
Once the characteristic numbers are established for most concepts,
mankind will then possess a new instrument which will enhance the
capabilities of the mind to a far greater extent than optical instruments strengthen the eyes, and will supersede the telescope and
microscope to the same extent that reason is superior to eyesight.
In his introduction Wiener comments (pp. xxvi-xxvii):
There are two parts to Leibniz’s universal characteristic: one is the
system of primitive characters that stand for the irreducible simple
concepts, the alphabet of the universal script; the other is the calculus of reasoning (calculus ratiocinator) which contains the rules of
reducing all composite ideas to the simple ones and of combining the
simple characters into composite ones. The former supplies the ultimate premises of all the sciences; the latter, the rules for combining
concepts and propositions. Thus Leibniz’s universal characteristic
was both a metaphysical system and an instrument of demonstration. It was not a novelty to conceive of logic as both a part of
philosophy and an instrument. Boethius in the sixth century argued for this dual function of logic, comparing it to the eye which is
both a part of the body and an aid to its orientation.
2. The initial system is in Begriffsschrift, subtitled a formula language, modeled
upon that of arithmetic, for pure thought. Frege’s thought underwent a marked
development from 1879, when he published Begriffsschrift, to 1893, when he published the first volume of The Basic Laws of Arithmetic. An overview of some
aspects of this development is given in Chapter 8.
32
3. Commenting on his formal system for logic and arithmetic in ”The Foundations
of Mathematics”, Hilbert remarks (p. 475):
The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For
this formula game is carried out according to certain definite rules,
in which the technique of our thinking is expressed. These rules form
a closed system that can be discovered and definitively stated. The
fundamental idea of my proof theory is none other than to describe
the activity of our understanding, to make a protocol of the rules
according to which our thinking actually proceeds. Thinking, it so
happens, parallels speaking and writing: we form statements and
place them one behind another.
This suggests that Hilbert viewed the laws of logic – or at least some laws of logic –
as expressing laws of thought in a rather literal sense. An interesting examination
of this question is Hallet “Hilbert’s Axiomatic Method and the Laws of Thought”.
4. See Skolem “Sur la Portée du Théorème de Löwenheim-Skolem”. In its basic formulation this theorem says that if a first order theory (formulated in a
denumerable language) has an infinite model, then it has a denumerable model.
5. See “The Concept of Truth in Formalized Languages”, pp. 265, 273, and “The
Semantic Conception of Truth”, pp. 361-362. A critical evaluation of Tarski’s
views is given in Chapter 7.
6. In Philosophy of Logic (p. 96) Quine argues that the idea that “logic [is] a
compendium of the broadest traits of reality . . . [is] unsound; or all sound, signifying nothing.” Quine attacks what he calls the “linguistic doctrine of logical
truth” (that would make logic depend on language alone) in Chapter 7 of Philosophy of Logic and in his earlier paper “Carnap and Logical Truth”. My use of
‘linguistic view of logic’ is broader than Quine’s and includes his own view as a
main example.
7. See Prawitz “Proofs and the Meaning and Completeness of the Logical Constants”, p. 26.
8. We find an expression of this view in Cratylus 385b, where Socrates asks: “Then
that speech that says things as they are is true, and that which says them as they
are not is false?” It is extensively discussed in the Sophist. See also Aristotle’s
discussion of truth in Metaphysics Γ and Θ10.
9. Frege opens his paper “Thoughts” with the following well-known words:
33
Just as ‘beautiful’ points the way for aesthetics and ‘good’ for ethics,
so do words like ‘true’ for logic. All sciences have truth as their goal;
but logic is also concerned with it in a quite different way: logic has
much the same relation to truth as physics has to weight or heat. To
discover truths is the task of all sciences; it falls to logic to discern
the laws of truth. The word ‘law’ is used in two senses. When
we speak of moral or civil laws we mean prescriptions, which ought
to be obeyed but with which actual occurrences are not always in
conformity. Laws of nature are general features of what happens
in nature, and occurrences in nature are always in accordance with
them. It is rather in this sense that I speak of laws of truth. Here
of course it is not a matter of what happens but of what is.
This view of logic seems to have been developed by Frege independently of the
specific treatment of truth in “On Sense and Reference”, where the truth values
the True and the False are introduced. It is presented in some detail in the
introduction to the first volume of The Basic Laws of Arithmetic and in two
manuscripts titled “Logic” in his Posthumous Writings. In The Basic Laws (pp.
xiv-xv) Frege comments in more detail on the two senses of ‘law’:
Our conception of the laws of logic is necessarily decisive for our
treatment of the science of logic, and that conception in turn is
connected with our understanding of the word “true”. It will be
granted by all at the outset that the laws of logic ought to be guiding
principles for thought in the attainment of truth, yet this is only too
easily forgotten, and here what is fatal is the double meaning of
the word “law”. In one sense a law asserts what is; in the other
it prescribes what ought to be. Only in the latter sense can the
laws of logic be called ‘laws of thought’: so far as they stipulate the
way in which one ought to think. Any law asserting what is, can
be conceived as prescribing that one ought to think in conformity
with it, and is thus a law of thought in that sense. This holds for
laws of geometry and physics no less than for laws of logic. The
latter have a special title to the name “laws of thought” only if we
mean to assert that they are the most general laws, which prescribe
universally the way in which one ought to think if one is to think at
all. But the expression “law of thought” seduces us into supposing
that these laws govern thinking in the same way as laws of nature
govern events in the external world. In that case they can be nothing
but laws of psychology: for thinking is a mental process.
10. An initial clarification of this claim is given in the part on propositional logic
of Chapter 6, while a full discussion is contained in Chapter 16. The basic idea
is that (classical) propositional logic can be formulated as a theory of the logical
predicates (or properties) ‘is True’ and ‘is False’.
34
11. A recent examination of philosophical issues concerning logical implication,
which has generated a fair amount of discussion, is in Etchemendy The Concept
of Logical Consequence.
12. This raises some difficult issues already discussed by Aristotle in Metaphysics
Γ in connection with the principle of contradiction – see Dancy Sense and Contradiction: A Study in Aristotle for a detailed examination. Frege also discusses
the question in the introduction to The Basic Laws of Arithmetic, where he says
(p. xvii):
The question why and with what right we acknowledge a law of logic
to be true, logic can answer only by reducing it to another law of
logic. Where that is not possible, logic can give no answer. If we step
away from logic, we may say: we are compelled to make judgments
by our own nature and by external circumstances; and if we do
so, we cannot reject this law – of Identity, for example; we must
acknowledge it unless we wish to reduce our thought to confusion
and finally renounce all judgment whatever. I shall neither dispute
nor support this view; I shall merely remark that what we have
here is not a logical consequence. What is given is not a reason for
something’s being true, but for our taking it to be true. Not only
that: this impossibility of our rejecting the law in question hinders
us not at all in supposing beings who do reject it; where it hinders
us is in supposing that these beings are right in so doing, it hinders
us in having doubts whether we or they are right. At least this is
true of myself. If other persons presume to acknowledge and doubt
a law in the same breath, it seems to me an attempt to jump out of
one’s own skin against which I can do no more than urgently warn
them. Anyone who has once acknowledged a law of truth has by the
same token acknowledged a law that prescribes the way in which one
ought to judge, no matter where, or when, or by whom the judgment
is made.
13. Although one can draw a distinction between logic and epistemology in terms
of the notion of deduction in relation to the more general notion of justification,
this distinction becomes somewhat blurred by the “metamathematical” results
obtained in model theory, proof theory and recursion theory. This is where logic as
an epistemological theory came of age, and results such as Gödel’s incompleteness
theorem had a profound impact on the axiomatic method and on epistemology
more generally.
14. The view that the justification of the principles of logic depends on our deductive practices is defended by Goodman in Fact, Fiction, and Forecast, pp. 62-63.
35
The justification of logic through deduction, and natural deduction systems, is
nowadays mostly associated with intuitionistic logic rather than with classical
logic.
15. “Consciousness, Philosophy, and Mathematics”, p. 1243. He says:
From the above report, especially from the rejection of the hypothesis of plurality of mind, follows that truth is only in reality, i.e. in
the present and past experiences of consciousness.
And in the next paragraph he comments about logic:
Further there is a system of general rules called logic enabling the
subject to deduce from systems of word complexes conveying truths,
other word complexes generally conveying truths as well. Causal behaviour of the subject (isolated as well as cooperative) is affected by
logic. And again object individuals behave accordingly. This does
not mean that the additional word complexes in question convey
truths before these truths have been experienced, nor that these
truths always can be experienced. In other words, logic is not a reliable instrument to discover truths and cannot deduce truths which
would not be accessible in another way as well.
Brouwer did not like logic much because he conceived it linguistically. Language
for him is an inessential component of thought, and the idea that the turning of
a linguistic handle could generate truths was horrifying to him. That’s why he
never developed an intuitionistic logic.
16. See Griss “Mathématiques, Mystique et Philosophie”.
17. Intuitionism: An Introduction, p. 102. This is in answer to the question: “Is
it not necessary to clarify the notion of a contradiction?” Heyting briefly discusses
Griss’ objections and has a section (8.2) on negationless mathematics.
About his axiom X (¬p ⇒ (p ⇒ q)), that from a contradiction anything
follows, Heyting remarks (p. 106):
Axiom X may not seem intuitively clear. As a matter of fact, it adds
to the precision of the definition of implication. You remember that
p ⇒ q can be asserted if and only if we possess a construction which,
joined to the construction p, would prove q. Now suppose that ⊢ ¬p,
that is, we have deduced a contradiction from the supposition that
p were carried out. Then, in a sense, this can be considered as
a construction, which, joined to a proof of p (which cannot exist)
leads to a proof of q. I shall interpret the implication in this wider
sense.
36
18. See Prawitz “Meaning and Proofs: On the Conflict Between Classical and Intuitionistic Logic” and “Proofs and the Meaning and Completeness of the Logical
Constants”, and Dummett “The Philosophical Basis of Intuitionistic Logic”.
19. This is essentially the conclusion of his paper “Truth of a Proposition, Evidence of a Judgment, Validity of a Proof”. In pp. 418-419 he says:
As should be clear from what I have just said, this notion of validity
or conclusiveness or correctness of a proof is a very fundamental
notion. Indeed, it is the most fundamental one of all, the one of all
the notions that I have discussed which has no other notion before
it, because to say that a proof is valid or conclusive or correct, as
should be clear from the formulations that I have used, is nothing
but saying, either that it is a proof with an emphatic is, or, better,
that it is a real proof and not a deceptive proof: it is a real proof
or it is a true proof . . . So the notion of validity or conclusiveness or
correctness as applied to proof is nothing but the notion of truth or
reality with its opposite falsehood or appearance . . . applied to the
particular acts and objects with which we are concerned in logic,
namely, acts of knowing and objects of knowledge.
20. This is what he says in “Logic and the Reification of Universals” (p. 116):
Even in the theory of validity it happens that the appeal to truth
values of statements and extensions of predicates can be finally eliminated. For truth-functional validity can be redefined by the familiar
tabular method of computation, and validity in quantification theory can be redefined simply by appeal to the rules of proof (since
Gödel has proved them complete). Here is a good example of the
elimination of ontological presuppositions, in one specific domain.
This theme is elaborated further in Philosophy of Logic, pp. 56-58.
21. Quine’s objections to second order logic are in Philosophy of Logic, pp. 68-72.
At the end (p. 72) he uses the fact that there is no complete proof procedure for
set-theory and second order logic as the reason to place them outside the scope
of logic:
We are out beyond the reach of complete proof procedures, and in a
domain even of competing doctrines. It is no defect of the structural
versions of logical truth that they exclude genuine set theory from
the field of logic.
Broadening the totality of interpretations for second order logic to interpretations of a certain kind, Henkin proved a completeness theorem for this
logic. But by generating compactness – i.e., that if the sentence α is a logical consequence of a set of sentences Γ, possibly infinite, then α is a logical consequence
37
of a finite subset of Γ – this seems to falsify the notion of logical consequence
of second order logic. Quine’s arguments against second order logic are critically
discussed by Boolos in “On Second-order Logic”.
22. There is a detailed examination of this question in Shapiro “Second-order
Languages and Mathematical Practice”.
23. An interesting historical discussion of this question is in Moore “The Emergence of First-order Logic”.
24. Philosophy of Logic, p. 60. His grammatical characterization of logical truth
is introduced as follows (p. 58):
Now the further idea suggests itself of defining logical truth more abstractly, by appealing not specifically to the negation, conjunction,
and quantification that figure in our particular object language, but
to whatever grammatical constructions one’s object language may
contain. A logical truth is, on this approach, a sentence whose grammatical structure is such that all sentences with that structure are
true.
Sentences have the same grammatical structure when they are interconvertible by lexical substitutions. Our new definition of logical
truth, then, can also be put thus: a logical truth is a sentence that
cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same
grammatical categories, the resulting sentence is true.
A critical discussion of Quine’s views on logic is contained in the logic chapters beginning with Chapter 15 – a version of Chapter 15 has been published as “Logical
Forms”.
25. The philosophical (and mathematical) project of replacing the notion of mathematical truth by the notion of theorem of a first order formal system was an
important aspect of Hilbert’s project for the foundations of mathematics. This
project has characteristics that are similar to those of other philosophical projects
that attempt to justify transcendental notions in terms of a given that is subjectively clear – phenomenalism, for instance – and was definitely refuted, at least
in its original formulation, by Gödel’s theorems.
26. I have recently found a very explicit formulation of this view in Hale “Is
Platonism Epistemologically Bankrupt?”. He says (p.92):
The notion that logic is distinguished from other disciplines by its
complete lack of existential commitment is so familiar and so widely
38
accepted that it is, perhaps, hard to see how it could – relevantly
and soberly – be challenged.
27. This axiom is part of the system of Principia Mathematica and its rejection
as a logical axiom is entirely justified. The problem was not only the assertion of
existence of non-logical entities, but the assertion that there are infinitely many
such entities.
28. The assumption that there are specifically logical objects at level 0 raises
some difficult questions which I will not discuss now. Frege held that extensions
are logical objects, though in a somewhat different sense of ‘logical object’, and
this was an important aspect of his logical account of arithmetic. Even aside
from this assumption, however, Frege’s analysis of the arithmetical properties as
logical properties did succeed in establishing a close connection between logic
and arithmetic. Whether arithmetic – and mathematics more generally – can be
considered to be a part of logic, in some reasonable sense, seems to me to be
still an open question. Boolos and Heck have done very interesting work in this
connection – see the papers in Demopolous Frege‘s Philosophy of Mathematics.
29. We could also say that there are no logical entities at level 1 and that the
Identity and Diversity relations among level 0 objects appear at level 3. This
follows if we organize the hierarchy “definitionally” using the Leibnizian definition
of Identity adopted by Frege – i.e., that a = b if a and b have the same level 1
properties.
30. One can hold, however, that Existence is a property of objects, in which
case it would appear at level 1. It would also appear at all higher levels as a
property of properties distinguishable from the property expressed by existential
quantification.
31. This claim is justified in Chapter 11.
32. In Philosophy of Logic Quine does the same thing in a different way by leaving
names out of his logical grammar and by assuming that he is only dealing with
predicates for which that sort of failure cannot occur. To each denotationless
name then corresponds a predicate with an empty extension.
33. This view is developed in detail in the logic chapters beginning with Chapter
15.
34. To appreciate the continuity between Frege’s project and that of Aristotle see
Lear Aristotle: The Desire to Understand, Chapter 6.
39
35. This is quite clear when Frege contrasts his conception of logic to the psychologistic conception in the introduction to The Basic Laws (p. xvi):
I understand by ‘laws of logic’ not psychological laws of takings-tobe-true, but laws of truth. If it is true that I am writing this in
my chamber on the 13th of July, 1893, while the wind howls outof-doors, then it remains true even if all men should subsequently
take it to be false. If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not
psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is
because of this that they have authority for our thought if it would
attain to truth. They do not bear the relation to thought that the
laws of grammar bear to language; they do not make explicit the
nature of our human thinking and change as it changes.
36. See Kleene Introduction to Metamathematics §50.
37. This was Frege’s point about Hilbert’s formal axiomatization of geometry. In
“The Foundations of Geometry”, p. 374, he argues that what Hilbert’s axiom
system defines is a second level concept.
38. But Frege encouraged this sort of interpretation to some extent by referring
to certain formulas of his concept script as “empty schemata”. See “The Aim of
“Conceptual Notation””, p. 97 [7].
39. Like Frege, Gödel was influenced by Leibniz, and he opens his paper “Russell’s
Mathematical Logic” with the following words:
Mathematical logic, which is nothing else but a precise and complete formulation of formal logic, has two quite different aspects.
On the one hand, it is a section of Mathematics treating of classes,
relations, combinations of symbols, etc., instead of numbers, functions, geometric figures, etc. On the other hand, it is a science prior
to all others, which contains the ideas and principles underlying all
sciences. It was in this second sense that Mathematical Logic was
conceived by Leibniz in his Characteristica universalis, of which it
would have formed a central part. But it was almost two centuries
after his death before the idea of a logical calculus really sufficient
for the kind of reasoning occurring in the exact sciences was put into
effect (in some form at least, if not the one Leibniz had in mind)
by Frege and Peano. Frege was chiefly interested in the analysis of
thought and used his calculus in the first place for deriving arith-
40
metic from pure logic. Peano, on the other hand, was more interested
in its applications within mathematics and created an elegant and
flexible symbolism, which permits expressing the most complicated
mathematical theorems in a perfectly precise and often very concise
manner by single formulas.
A recent defence of metaphysical realism in logic is developed by Bealer
in Quality and Concept, and my own defence in this book shares his main aims (pp.
1-2). As opposed to Bealer, however, my ideas are heavily influenced by Frege and
have a substantial Fregean orientation – in spite of my rejection of many specific
Fregean theses. This makes my approach rather different from Bealer’s, both in
the general development and in details – although there are points in common.
41
PART I
TRUTH AND DESCRIPTION
.
46
Chapter 1
Truth, Description, and Identification
A natural way of looking at the connection of a statement to reality
is as a description: a true statement describes an aspect of the world, a
false statement does not. Just as we describe objects, properties of objects,
relations among objects, etc., we also describe states of affairs, situations,
circumstances and the like, and these are typically described by statements.
The truth of a statement lies precisely in this description of reality. The
truth of the statement that I am in my study right now lies in my being in
my study right now; i.e., in a certain state of affairs. The statement that I
am in my living-room right now, on the other hand, does not describe an
aspect of the world and is not true.
This view of truth goes back at least to Plato, and is the backbone
of all realist accounts of truth. Though not generally formulated in terms of
description and states of affairs, the fundamental point of a realist account
of truth is that truth lies in reality, or is an expression of reality. This seems
natural enough, but there are difficulties in elaborating the view precisely.
Some of these difficulties are related to the notion of falsity.
I said that a true statement describes a state of affairs and that
a false statement does not1 . Yet it would seem that both true and false
statements describe the world and that the difference between them is that
whereas true statements describe the world correctly, false statements describe the world incorrectly. If true statements describe by describing states
of affairs, in what sense do false statements describe? What do they describe? This is a version of Plato’s problem of non-being2 .
One can approach this problem by postulating possible states of
affairs: true statements describe actual states of affairs; false statements describe possible states of affairs that are not actual. This is analogous to the
view that definite descriptions such as ‘the present king of France’ describe
possible individuals which may happen not to exist (as in this particular
case). And the problems are similar in both cases. In particular, just as a
definite description may describe an “impossible” object (‘the square circle’,
‘the largest prime’, etc.), a statement may describe an “impossible” state
of affairs; that 2 + 2 = 5, for example. This means that for this sort of
approach to work uniformly one must allow impossible entities as well as
possible entities3 .
47
It is more natural to say that a false statement describes the world
in the sense that it states something of an identifying character about the
world, and not in the sense that there is a possible state of affairs that
is described by it. Both true and false statements describe the world to
the extent that they state something of an identifying character about the
world, but true statements identify something whereas false statements do
not. This is indeed parallel to what happens with definite descriptions. The
definite description ‘the elephant in my study right now’ has an identifying
character but does not identify anything because there is no elephant in my
study right now. Similarly, the statement that I am in my living-room right
now has an identifying character but does not identify anything because I’m
not in my living-room right now; there is no such state of affairs.
The feature of description that I am emphasizing is to be identifying, roughly in the sense of involving identity conditions that can be
fulfilled by at most one thing – object, state of affairs, property, etc. There
are identity conditions for something being an elephant in my study right
now, and although there could be many elephants in my study right now,
there can be at most one thing that is the elephant in my study right now.
Similarly for the statement that I am in my living-room right now; there are
identity conditions for my being in my living-room right now, and although
these conditions could be fulfilled in many different ways, there can be at
most one state of affairs of my being in my living-room right now4 .
One respect in which there seem to be asymmetries between definite descriptions and statements is in relation to negation, and many traditional issues about statements concern negation. In fact, since any false
statement can be turned into a true statement by means of negation, and
any true negative statement has a false counterpart, questions about falsity
go hand in hand with questions about negation.
Negation gives rise to other aspects of the problem of non-being.
What is it to deny existence? Is one denying existence of something?5 To
deny that there is an elephant in my study right now it is natural to appeal
to the property of being an elephant in my study right now and to take
the denial of existence as a denial that this property is instantiated. For
the statement that I am in my living-room right now we can take a similar
approach. We can say that this statement is false because the relation ‘is in’
does not relate (or is not instantiated in) me and my living-room, in that
order, right now. Or in other words, to say that the state of affairs of my
being in my living-room right now does not exist is to say that I don’t have
the relation ‘is in’ to my living-room right now.
48
But the statement that I am not in my living-room right now is
true, so according to a view of truth as description it must identify a state
of affairs. Which state of affairs is this? It may seem natural that the state
of affairs that would be involved in the truth of this statement were the
state of affairs of my being in my study right now, but it is quite clear that
the statement that I am not in my living-room right now cannot be said to
identify this state of affairs. If, on the other hand, we say that the state of
affairs identified by this statement is the state of affairs of my not being in
my living-room right now, then we may wonder what sort of state of affairs
(aspect of reality) is that? There are so many places in which I am not
right now; are we to say that in each case there is a state of affairs of my
not being there?
Well, why not? I can say that the state of affairs identified consists
of my having the relation ‘is not in’ to my living-room right now. This might
seem odd if we think of not being somewhere as a sort of lack, but there is no
clear sense in which having the relation ‘is in’ to something is more real than
having the relation ‘is not in’ to something. Not being in my living-room is
just as much a real relation to the living-room as being in the living-room.
Similarly, to say that nothing instantiates the property of being an elephant
in my study right now is to say that everything instantiates the property ‘is
not an elephant in my study right now’. This is essentially Plato’s idea, it
seems to me.
Plato identified very strongly truth with reality; what is true is
what is, what is real. His formulations vary a bit, but they always suggest
this strict correlation between truth and being. Sometimes the suggestion
is of an identification of being with truth, sometimes it is more along the
lines of an agreement. This is what led to his detailed examination of the
problem of falsity. Is falsity an attribution of reality to what is not real?
Would falsity be possible then? His conclusion was that the non-being that
is involved in falsity is not an expression of what is not real, in a literal
sense, and is not such as to make falsity impossible or nonsensical.
Plato’s examples of a true and a false statement are ‘Theaetetus is
sitting’ and ‘Theaetetus is flying’. In both cases there is a subject (Theaetetus), and real properties (sitting, flying) are involved. The first statement
is true because it states of Theaetetus things that are (sitting) as they are
– Theaetetus is sitting. The second statement is false because it states of
Theaetetus things that are (flying) as they are not – Theaetetus is not flying. The non-being that is involved in the falsity of the statement may be
thought of as a lack of fit (or combination) of Theaetetus with the property flying. This lack of fit is itself something, however, because it can be
expressed by saying that Theaetetus is not flying, and ‘not flying’ is a real
49
property. Thus, the falsity of the statement that Theaetetus is flying is
accounted for by the truth of the statement that Theaetetus is not flying.
Plato’s solution presupposes that the describing done by statements is due to the connections between the various terms (singular and
general) that make up the statement with aspects of reality. The statement
that Theaetetus is sitting describes reality in the sense that it says of what
is (Theaetetus, sitting) that it is thus and so. The thus and so is the participation of Theaetetus in the form Sitting, which is indeed an aspect of reality
– and a feature of Theaetetus. Thus, the statement describes what is as it
is and is true. The statement that Theaetetus is flying also describes reality
in the sense that it says of what is (Theaetetus, flying) that it is thus and
so. The thus and so is the participation of Theaetetus in the form Flying,
which in this case is not an aspect of reality. Thus, the statement describes
what is as it is not and is false. But the non-participation of Theaetetus
in the form Flying is not a lack, a nothing; it is, rather, a participation in
the form Other-than-Flying, which is a combination of the form Otherness
with the form Flying. This is an aspect of reality as well, and a feature of
Theaetetus; thus what accounts for the falsity of the statement is a feature
of reality6 .
In terms of states of affairs we can say that the truth of the statement that Theaetetus is sitting lies in the state of affairs that is the participation of Theaetetus in Sitting, and that the falsity of the statement that
Theaetetus is flying lies in the state of affairs that is the participation of
Theaetetus in Other-than-Flying. Using the terminology of properties one
can say that the first state of affairs consists of the combination of Theaetetus with the property ‘sitting’, and the second state of affairs consists of the
combination of Theaetetus with the property ‘not flying’.
Aside from the question as to whether this is a correct interpretation of Plato, at least in general lines, this approach to the problem of
falsity and negation seems to me basically sound. It can be questioned on
several grounds, however. To begin with, there is the appeal to an ontology of forms, or properties, and the problems generally associated to that.
Second, there is the question of the legitimacy of such properties as ‘not
flying’. Third, there is a question as to whether this sort of approach works
in general for statements of more complex structure. I shall comment on
some of these points later, but first there is still the question of how one is
to express the relation between the false statement and reality in terms of
states of affairs. Although one can say that the first statement identifies the
state of affairs of Theaetetus’ participation in Sitting, or having the property ‘sitting’, one cannot properly say that the second statement identifies
50
the state of affairs of Theaetetus’ participation in Other-than-Flying, or of
having the property ‘not flying’.
In “The Philosophy of Logical Atomism” Russell took the line
that a true statement points to reality and that a false statement points
away from reality. In Russell’s view, it is the state of affairs (or fact) of
Theaetetus’ sitting that makes the statement that Theaetetus is sitting
true; the statement points to that state of affairs and is thereby true. Again
it might seem natural to say that the very same state of affairs is what
makes the statement that Theaetetus is flying false, and that it would also
make false an indefinite number of other statements, all of which would
point away from reality – for example, the statements that Theaetetus is
running, or going up the stairs, etc. But what about the statement that
Theaetetus is not flying? Which state of affairs makes it true? As with
my formulation in terms of description we cannot very well say that the
statement that Theaetetus is not flying points to the fact that Theaetetus
is sitting.
So, like Plato, Russell is led to the view that what is expressed in
the negative statement is not a lack of reality but an aspect of reality; he
postulates the negative fact of Theaetetus’ not flying. That is, he rejects
the idea that it is a relation, or lack of relation, to the fact of Theaetetus’
sitting that makes the statement that Theaetetus is flying false. His view is
that each statement points to or away from either a positive or a negative
fact. The statement that Theaetetus is sitting points to the positive fact
of Theaetetus’ sitting; the statement that Theaetetus is not sitting points
away from this very same fact. The statement that Theaetetus is not flying
points to the negative fact of Theaetetus’ not flying, and the statement that
Theaetetus is flying points away from this negative fact.
Russell’s preoccupation was exactly the same as Plato’s; to analyze
falsity in terms of reality. One could actually interpret the negative fact of
Theaetetus’ not flying as the combination of Theaetetus with the property
‘not flying’. Russell does not do this, and gives no actual analysis of the
structure of negative facts, but in this respect the solutions are quite similar
– and Russell got just as much flack about his negative facts as Plato did
about his Other-than-X combinations.
With respect to the question we are discussing, however, the problem is rather with the pointing away from a fact. What is it to point away
from a fact? Or to point away from anything, for that matter? Statements
seem to be directed at reality, and to attempt to describe it in some way;
to say that false statements point away from reality seems to say nothing
at all. How does a statement point away from reality? Could we say that
51
the description ‘the elephant in my study right now’ points away from reality? If I mistake a rock in the distance for an elephant and state ‘that’s
an elephant’, am I pointing away from reality? Unless one gives a more
specific content to the notion of pointing away, this appears to be a merely
terminological solution to the problem.
It may seem more natural to understand Plato’s idea as a formulation of a correspondence view of truth. One can say that the true statement
describes reality as it is because the statement matches the state of reality,
whereas the false statement describes reality as it is not because it does not
match the state of reality. Plato’s analysis of falsity would still hold and
would show that this does not involve an appeal to unreality. One can raise
questions about the nature of the matching, however.
The general idea of the correspondence theory of truth is that a
statement is true when it corresponds to (or agrees with) reality, and false
when it doesn’t. It tries to preserve the original conception of truth as being by shifting the burden of explanation to the notion of correspondence.
But, as Frege and others have argued, it is not at all clear what sort of
correspondence is at issue here. A correspondence between what and what?
Between what the statement says and how the world is, is the natural suggestion. If this is a relation between two somethings, though, then we are
back to the problem of specifying what in reality corresponds (or not) to the
content (or description) of the statement, what is this content, and what is
the appropriate notion of correspondence7 .
I think that the view of truth as correspondence is quite compatible with the view of truth as identification, and that Frege’s idea that truth
is a form of denotation, which is a form of identification, is a natural interpretation of the notion of correspondence if one thinks of statements as
descriptions – as Frege does in “On Sense and Reference”. For Frege both
true and false statements denote, and what they denote are the truth values
the True and the False – which are logical objects. I see a close connection
as well between Frege’s ideas and Plato’s.
For Frege, the statement that Theaetetus is flying also involves
an object, Theaetetus, a concept (or property), flying, and a lack of fit
between the object and the concept – the object does not fall under the
concept. This lack of fit between object and concept is what Frege refers to
as the circumstance that the statement is false and objectifies in the False.
Similarly, the fit between Theaetetus and sitting, i.e., the circumstance
that Theaetetus falls under the concept sitting, is objectified by Frege in
the True. Both true and false statements refer to reality, but in both cases
their determination as true and false by reality is interpreted “globally” by
52
Frege, with all true statements denoting the True and all false statements
denoting the False.
This global objectification is rather odd, and again has the ring of
a merely terminological solution, but Frege held that statements, whether
true or false, could not refer to isolated aspects of reality; he argued that
their connection to reality must be, so to speak, in toto. Nevertheless, Frege
held that it is possible to discern in the case of each true or false statement
which specific features of reality are involved in its truth or falsity. This is a
feature of judgement, which connects the sense expressed by the statement
(the thought) and its truth value.
As I see it, therefore, Frege’s analysis of truth and falsity is not that
far from Plato’s and Russell’s. And I think that one can accept Frege’s denotational approach to the connection between statements and reality without
committing oneself to truth values; i.e., that one can combine Frege’s view
of truth as denotation with Russell’s view of truth in terms of facts or states
of affairs.
This is not to say that there aren’t fundamental differences in their
views. For one thing, there is Frege’s argument that it is not possible to
treat the specific features of reality that account for the truth of a statement
as an ontological unit; as something like a Russellian fact, for instance. I
shall comment on this argument and its ramifications a little later. And
Russell had arguments to the effect that it is not possible to treat truth as
a form of denotation. He held that if a true statement were to denote a
fact, then a false statement would denote nothing and be nonsensical – a
mere noise. This is a version of Plato’s problem of non-being again, and is
one of the main reasons for Russell’s conceptualization of truth and falsity
as a pointing to or away from facts. Russell’s view that if truth were a
form of denotation false statements would be nonsensical reflects another
major disagreement between Frege and Russell concerning the content of
statements.
For Frege, a statement expresses an objective thought, which is
its sense. A sense, in general, is what contains a mode of presentation.
An object can be presented in many ways by means of its properties and
its relations to other objects, and each way of presenting it is a sense of
it. But Frege also held that a sense need not be a sense of anything, and
illustrates this with names and descriptions such as ‘Odysseus’ and ‘the least
rapidly convergent series’. A sentence that contains non-denoting terms
may express a sense but is neither true nor false. The explanation for this is
that the sense expressed by a sentence (the thought expressed by it) depends
only on the senses expressed by its parts and not on their denotations. Since
53
senses are themselves part of reality for Frege, even the objective content of
statements that are neither true nor false is explained entirely by appeal to
reality.
Russell rejects Frege’s distinction between sense and denotation
and his treatment of sentences containing non-denoting terms is rather different. But if one looks at it in terms of properties, it is also an ontological
solution. The statement that the present king of France is bald is false,
according to Russell, precisely because no particular uniquely instantiates
the property of being presently a king of France. This means that the statement has a subject, but that this subject is not an alleged present king of
France, for there is no such thing, but the property of being presently a king
of France. If the subject were whatever is named by the description ‘the
present king of France’, then there would be no subject and the statement
would be nonsensical. But for Russell it isn’t nonsensical; it is clearly false.
Of course, this does not settle what to do with statements containing non-denoting names rather than descriptions. How does one analyse the
statement that Pegasus does not exist, or the statement that Pegasus is flying? Assuming that these statements say something (are not nonsensical),
about which most people agree, the question is whether these statements
are true, or false, or neither.
Russell thought that the first statement is true and the second
false. But what does the first statement say? If we hold that it is denying
existence of Pegasus, then we seem to get back to the problem of non-being;
what are we denying existence of? On the other hand, it doesn’t look like
there is a property about which we are asserting that it is not uniquely
instantiated. Russell’s suggestion was that it might not look like it, but, in
fact, there is such a property. A proper name stands for or abbreviates a
definite description, and what we are denying is that the property (or combination of properties) involved in the description is uniquely instantiated.
Thus, if ‘Pegasus’ abbreviates ‘the winged horse captured by Bellerophon’,
the statement that Pegasus does not exist denies that the property of being
a winged horse captured by Bellerophon is uniquely instantiated. Therefore,
the statement is true. And precisely because the property is not uniquely
instantiated Russell holds the statement that Pegasus is flying to be false.
But there are difficulties with this approach. The idea that proper
names abbreviate definite descriptions has undesirable consequences and it
seems very unlikely that it fits in with actual linguistic practice8 . For one
thing, since the thesis applies to all names, not just non-denoting names,
it implies that when I say that Russell was a logician, I am not talking
about Russell but about some property or properties (that may or may not
54
individuate Russell). This seems very implausible. In fact, this approach
of Russell’s basically discards the idea that proper names and definite descriptions can be used to talk about things; i.e., it eliminates the denoting
function of such terms.
Another difficulty with Russell’s approach concerns the appropriate choice of the description that is abbreviated by each name, even in
specific contexts. What if we didn’t have a description to substitute for
‘Pegasus’ ? Quine, who follows Russell in this matter, takes a short-cut to
the problem by introducing a property of pegasizing – expressed by the predicate ‘pegasizes’, or ‘is-Pegasus’9 . To deny that Pegasus exists is to deny
that some one thing pegasizes. And he would also hold that the statement
that Pegasus is flying is false because nothing pegasizes.
Although Quine’s introduction of the ad hoc predicate ‘pegasizes’
seems to give a uniform way out of the problem – or, at least, a way of
avoiding the problem when no other way can be found – it is not at all
clear what this predicate is supposed to be. What are its conditions of
applicability? One cannot very well say that it applies to something if
and only if that thing is Pegasus. Without an account of proper names
it is questionable whether the predicate ‘pegasizes’ (or ‘is-Pegasus’) is well
defined, or can be said to express a property, or to have an extension, or
to be true of or false of anything. Thus, this artificial technique of Quine’s
doesn’t seem to really clear up the problems that arise in connection with
Russell’s abbreviative conception10 .
It seems more natural to me to hold that the statement that Pegasus is flying is neither true nor false and that the statement that Pegasus
does not exist is ambiguous – treating ‘exists’ as a predicate. This is based
on Frege’s view that statements containing non-denoting terms, whether
names or definite descriptions, are neither true nor false. The ambiguity
that I claim for the second statement has to do with the negation that appears in it. If this is predicate negation (not-exist), then the statement is
neither true nor false, for the same reason as before. If this is statement
negation (it is not the case that, it is not true that), then the statement
is true; because the statement that Pegasus exists is neither true nor false,
and, hence, not the case, or not true. Thus, Russell’s intuition that the
statement that Pegasus does not exist is true is justified in this sense. And
his intuition that the statement that Pegasus is flying is false can also be
justified in a similar way as an intuition of the truth that it is not the case
that Pegasus is flying.
But why isn’t the statement that it is not the case that Pegasus is
flying neither true nor false if it contains the non-denoting name ‘Pegasus’ ?
55
Because what is involved in this statement is the content of the statement
that Pegasus is flying, and this content is an aspect of reality – and an
appropriate component of states of affairs. In other words, the statement
that it is not the case that Pegasus is flying identifies the state of affairs of
the description that Pegasus is flying not identifying a state of affairs. Thus,
not only states of affairs but also the content of descriptions will be treated
as a legitimate ontological category. And, evidently, just as I will have to
develop a suitable account of the aspects of reality that are identified by
true statements, I will have to develop a suitable account of descriptioncontents – or, in Frege’s terminology, senses. I.e., although I agree with
Russell that there are properties involved in the describing that statements
do, these seem to me more closely connected to Frege’s distinction between
sense and reference.
But is there any need for an account of states of affairs as some sort
of ontological units? Why appeal to a state of affairs in this sense if we can
give the truth conditions of statements in terms of objects, properties and
relations? The semantic analysis of the notion of truth deriving from Tarski
can be seen as giving an account of the truth of a statement in terms of
truth conditions arising from an analysis of the structure (or logical form) of
the statement. Even interpreted realistically it seems to avoid the problem
of identifying specific aspects of reality (such as states of affairs) that would
account for the truth of a statement. One can still say that a statement
is a description, but even in the case of a true statement what it describes
would not be identified as some sort of ontological unit. A true statement
describes reality in the sense that its truth derives from reality being thus
and so, as specified by the truth conditions, but it does not describe reality
in the sense of identifying something.
It is not clear, however, that this sort of account is a way of dispensing with the idea that a statement is a description which when true
identifies a state of affairs. In effect, on the realist reading, what the account does is to analyse the truth of each particular statement by means of
a more elaborate description which depends on the logical structure of the
statement and on an ontological analysis of what corresponds in reality to
the various parts of the statement. This is very helpful, and is something
that would have to be done no matter what (in a realist approach), but it
may be seen as giving an analysis of the structure of states of affairs rather
than as eliminating them11 .
In spite of their differences I see Frege and Russell as following
Plato’s idea of explaining significance, truth, and falsity in terms of an
abstract ontology of non-sensible entities, which is what largely characterizes
the Platonic approach. Moreover, in all three accounts truth and falsity are
56
directly related to reality. The account of truth that I will develop combines
several of the ideas I presented above. I agree with Plato’s analysis of
falsity and negation, and I’ll show that it can be suitably generalized to all
statements in a way that also accounts for Frege’s and Russell’s postulation
of a connection between false statements and reality. I agree with Russell’s
ontology of facts, though I analyse the structure of those facts (or states of
affairs) in a way that is inspired by Frege and Plato as well as by Russell.
I agree with Frege that truth is a form of denotation, or identification, and
I agree with the distinction between sense and reference, though I disagree
with the postulation of truth values as the denotation of statements.
I also disagree with Frege’s view of senses as objects, for it seems
more natural to me to take senses to be properties of a certain sort rather
than objects. They are properties which have the logical feature that if they
apply to anything, they apply only to that thing – that’s part of the idea
of the identifying character of a description that I mentioned earlier. This
makes Frege’s analysis of the content of sentences (i.e., of thoughts) a natural
development in relation to a Platonic ontology of forms. Not only truth and
falsity but the content of discourse generally depends on forms – or concepts,
or properties. This is quite compatible with Frege’s general outlook, because
although he treats senses as objects, they are logical objects of a rather
Platonic character.
My view, therefore, is that what is involved in truth is identification
and that the connection that statements have to reality is similar to the
connection that definite descriptions have to reality. A statement is true
when it identifies a state of affairs that is a combination of properties and
objects or properties and properties; it is not true when it does not identify
a state of affairs. It is false when its predicate negation identifies a state of
affairs; otherwise, it is neither true nor false. Some of the general advantages
that I see in formulating the realist conception of truth as identification (or
denotation12 ) of states of affairs are the following.
To begin with, it does seem to be a way of capturing the idea of
truth as being. If when I use an expression – a name, say – I am referring
to its denotation, if any, then when I use a sentence I am referring to its
denotation, if any. So if I say that I am in my study right now, I am
referring to my being in my study right now, and the truth of my statement
lies in the existence of this state of affairs. On the other hand, the falsity
of the statement that I am in my living-room right now lies in that I am
not in my living-room right now. Although the statement does not refer to
a state of affairs, it doesn’t follow that it is not a description of reality – it
says something of an identifying character about various aspects of reality to
which its terms refer. It so happens that reality is not like that, as evidenced
57
by the truth of its predicate negation which does refer to a state of affairs.
Therefore, in this formulation, what is true is what describes reality as it
is, and what is false is what describes reality as it is not.
Second, the close parallelism between the way in which sentences
function and the way in which definite descriptions function seems very
striking to me. If I describe a natural number as the largest even prime,
this definite description denotes something because the property of being
even, the property of being prime, and the relation greater than among
natural numbers, combine in such a way that there is a unique number that
is the largest even prime. On the other hand, the property of being odd,
the property of being prime, and the relation greater than among natural
numbers, do not combine in such a way that there is a unique number that
is the largest odd prime. Whereas the first description identifies an object,
the second identifies nothing. In the case of the first description, as in the
case of the first statement in the previous paragraph, certain properties,
relations, and objects, combine in such a way that there is something – an
object, or a state of affairs – which is identified by the definite description
or by the statement13 .
Finally, to assimilate statements to definite descriptions, and to
hold that the relation of statements to reality is similar to the relation of
definite descriptions to reality, seems to me quite compatible with the basic
intuition that underlies the correspondence view of truth. The obvious
relational features of identification and the dependence of identification on
reality, show truth as a fundamentally relational notion. Although it might
be somewhat misleading to say that what is true is what is and what is
false is what is not, this emphasizes the dependence of truth and falsity on
reality and suggests that the sense in which truth and falsity are properties
of statements is derivative from their connection to reality – which I take
to be the fundamental content of the correspondence view of truth.
One specific problem for the view of truth as identification, or
denotation, is Frege’s peculiar conclusion that on such a view all true statements must denote the same thing and all false statements must denote the
same thing. He argued that since co-referential terms are intersubstitutable
preserving truth value, it would be possible to gradually change a statement
by means of substitutions so that one would reach statements whose only
common feature with the original one would be sameness of truth or falsity.
In order to uphold the view of truth as denotation this led him to postulate
the object the True as the denotation of all true statements, and the object
the False as the denotation of all false statements. Frege concluded that in
its connection with reality all that is specific to a statement is obliterated
58
and that we are left merely with the circumstance of its truth or with the
circumstance of its falsity.
Frege’s arguments for truth values are not decisive, but there have
been several attempts to improve on them and to show that Frege’s conclusion is unavoidable on a denotational view of truth. Of these the most
interesting is suggested by Gödel in “Russell’s Mathematical Logic”. Simplifying a bit, Gödel’s idea is that a sentence of the form ‘a is F’ means
the same thing (or says the same thing) as a sentence of the form ‘a is that
thing that is F and is (identical to) a’; i.e., as the identity ‘a = ιx(F x and
x = a)’. Assuming that this sameness of meaning guarantees sameness of
denotation, and using Frege’s substitutivity principle, one can show, for example, that any two true sentences of the forms ‘a is F’ and ‘b is G’ denote
the same thing. Although this much does not actually establish Frege’s conclusion, it would certainly refute the view of truth as denotation of states
of affairs, because no reasonable account of states of affairs would make the
state of affairs of my being in my study right now the same as the state of
affairs of Socrates’ dying in 399 B.C. For all practical purposes this would
make Frege’s the only sensible option for truth as denotation.
It is quite essential therefore to understand precisely what is involved in Gödel’s argument and what may be wrong with it, if anything.
And this is not as easy as it may seem, because Gödel’s principle seems
quite plausible, as can be seen by trying it in a few specific cases. We get,
for example, that the statement that Socrates died in 399 B.C. means the
same thing as the statement that Socrates is that individual who died in
399 B.C. and is (identical to) Socrates. Similarly, the statement that 17
is a prime number means the same thing as the statement that 17 is that
number which is prime and is (identical to) 1714 .
Whether the formulation of truth as identification of states of affairs can be developed satisfactorily cannot be judged in advance because it
depends not only on the details of the treatment of negation and of the other
connectives and quantifiers, but on the treatment of states of affairs, senses,
descriptions, names, etc. I shall now make a few preliminary remarks about
the more specific development of some of the ideas.
The most natural view of states of affairs is intensional, and I
have been talking of a state of affairs as a combination of a property with
objects and/or other properties. From an ontological point of view states
of affairs may be taken to be one of three basic categories, together with
properties and objects, in an analysis of the structure of reality. Reality may
be conceived as structured hierarchically in a sequence of levels with objects
at the bottom level 0 and states of affairs and properties appearing at all
59
higher levels – states of affairs may also be viewed as higher level objects.
This is the structuring of reality first introduced by Frege (without states of
affairs) and developed more systematically (but in a rather different way)
by Russell in the theory of types.
Frege’s hierarchy of levels is important for my purposes because,
among other things, it gives a natural account of logical forms as properties15 .
As Frege saw it, the first order quantifiers (quantifying over level 0) appear
at level 2 as properties of properties of level 0 objects. Frege emphasized
that existence is not a property of objects but of properties. Thus ‘there are
philosophers’ asserts something about the property of being a philosopher
– namely, that it doesn’t have an empty extension, – and not about the individuals who are philosophers16 . Therefore, the use of quantification over
level 0 objects generates, in a manner of speaking, a large number of logical
properties of level 2. Subordination between level 1 properties, for example
(Aristotle’s ‘All A is B’); Exclusion (‘No A is B’); etc. In fact, there are
infinitely many logical properties at all levels higher than 017 .
The most fundamental logical property is for Frege the property
of instantiation (what he called ‘falling under a concept’18 ), and a natural
extension of Frege’s idea of treating the quantifiers as logical properties
makes it possible to treat all statements as having subject-predicate form –
i.e., every statement will express the instantiation of a property in one or
more subjects which can be either objects or properties or states of affairs.
This means that predicate negation will be everywhere defined and that
Plato’s treatment of falsity in terms of predicate negation can be generalized
to all statements, including statements involving quantification. Since a
state of affairs is a combination of a property with objects and/or other
properties, and will appear at the same level in the hierarchy as the initial
property, there will be a natural correlation between statements and states
of affairs that will account for their truth and falsity.
Senses can be treated as a subcategory of the category of properties; they are identifying properties in the sense mentioned earlier that it is
a logical feature of a sense that if it applies to anything it applies only to
that thing. One can say that a property that applies to exactly one thing –
be it an object, a property, or a state of affairs – is an individuating property. Not all individuating properties are senses, however, nor are all senses
individuating. The property of being a person in this room is individuating
in that it applies only to me, but it is not a sense. The property of being a
series that converges less rapidly than any other series, on the other hand,
is a sense that does not individuate anything; which agrees with Frege’s
view19 . Propositions are senses which purport to individuate states of
60
affairs – this is an analogue of Frege’s thoughts, for propositions in this
sense are non-linguistic modes of presentation.
All senses are non-linguistic, though they can be expressed linguistically in various ways. I shall say that a sense which individuates an entity
(property, object, or state of affairs) denotes that entity, but for linguistic
expressions I shall mostly talk of denotation in relation to their use – rather
than in relation to the expression as such. Nevertheless, one can talk ambiguously of the denotation of a name, or definite description, or predicate,
or sentence, and for certain purposes it is convenient to do so.
Since a proposition is true when, and only when, it denotes a state
of affairs, and a sense denotes an entity when it is instantiated in that entity,
as a relation truth is a special case of the fundamental logical relation of
instantiation.
There are many objections to talk of properties, states of affairs,
senses, propositions, statements, etc. The general tendency in logic is to
talk of truth in connection with sentences and to treat of denotation settheoretically as a relation between names, predicates, etc., and objects, sets
of objects, relations among objects, etc. Although this approach does not
seem to me the most natural it provides a common ground for discussion
and I shall use it in the next few chapters to illustrate the discussion of
various issues.
On an extensional interpretation we can think of states of affairs as
certain finite sequences of sets, relations, objects, etc. Thus, a pair < A, a >,
where A is a set and a is an element of A, is a certain kind of state of affairs;
a triple < R, a, b >, where R is a binary relation and < a, b > is an element
of R, is another kind of state of affairs; a pair < C, A >, where C is a set of
sets and A is an element of C, is another kind of state of affairs; and so on.
Specific examples of the various kinds of states of affairs listed
above are the following:
(1e ) <set of philosophers, Quine>
(2e ) <set of pairs of persons such that the first taught the second, Frege,
Carnap>,
(3e ) <set of non-empty sets of objects, set of philosophers>.
These states of affairs are denoted (identified) by the sentences, among
others:
(1) Quine is a philosopher,
(2) Frege taught Carnap,
and
(3) There are philosophers,
respectively.
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The usual semantic conditions for the truth of sentences can be
seen as conditions for sentences denoting states of affairs; therefore, all
semantic conditions are conditions for denotation. For example, the conditions for (2) denoting a state of affairs are that ‘Frege’ and ‘Carnap’ denote
objects, that ‘taught’ denote a set of ordered pairs, and that this set include
the ordered pair of the denotations of ‘Frege’ and ‘Carnap’, in this order.
Among the sentences that do not denote states of affairs one can
distinguish those which contain terms that lack denotation, e.g.,
(4) Sherlock Holmes taught Carnap;
from those whose terms denote but that do not satisfy the other semantic
conditions, e.g.,
(5) Carnap taught Frege.
With Frege, I will say that (5) is false whereas (4) is neither true nor false.
The circumstance that a sentence does not denote a state of affairs
is itself a certain kind of state of affairs, although one that has a linguistic
component; namely, the sentence itself. As I pointed out before, this is
relevant to the treatment of negation.
One should distinguish negation as a predicate operator from negation as a sentential operator. Consider the sentence
(6) Quine is a dentist,
for instance. This sentence is false because even though ‘Quine’ denotes an
object (Quine) and ‘is a dentist’ denotes a set of objects (the set of dentists), Quine does not belong to the set of dentists and therefore there isn’t
a corresponding state of affairs denoted by (6). On the other hand, the
sentence
(7) Quine is not a dentist,
where ‘not’ is a predicate operator, does denote a state of affairs; namely,
the state of affairs
(7e ) <set of non-dentists, Quine>20 .
And the sentence
(8) It is not the case that Quine is a dentist,
where ‘it is not the case that’ is a sentential operator, also denotes a state
of affairs, but an altogether different kind of state of affairs that has the
sentence (6) as a component; something like
(8e ) <set of sentences that do not denote, ‘Quine is a dentist’>.
The distinction between (7) and (8) is exactly the same as the
distinctions of scope that are made in Russell’s theory of descriptions. In
this particular case both (7) and (8) are true, but if we take
(9) Sherlock Holmes is not a dentist,
and
62
(10) It is not the case that Sherlock Holmes is a dentist,
with the same interpretations as before, then we have a difference in truth
value – (10) is true whereas (9) is neither true nor false. And, just as in
Russell’s theory of descriptions, as the logical structure of the sentences
gets more complex there are more and more different interpretations (or
alternative readings) of its logical form.
In fact, this question of structural ambiguity arises even at the
level of sentences that have no connectives or quantifiers. For example, the
primary interpretation of sentence (2) above is probably as denoting the
state of affairs (2e ). But it can also be interpreted as denoting any of the
following states of affairs:
(2/12e ) <set of teachers of Carnap, Frege>,
(2/13e ) <set of students of Frege, Carnap>,
(2/14e ) <set of binary relations among persons that include the pair <Frege,
Carnap>, set of ordered pairs of persons such that the first taught the
second>.
At a certain level of analysis there are nine different interpretations of the
logical structure of (2), of which the following correspond to the four given
above:
(11) [Rxy](a,b),
(12) [Rxb](a),
(13) [Rax](b),
(14) [Zab]([Rxy]).
The notation should be read as follows. In each case we have a
predicate of a certain logical type – the part within square brackets on the
left – followed by an appropriate number of arguments which are listed
within parentheses on the right. Lower case letters such as ‘x’ and ‘y’ are
variables over objects, and capital letters such as ‘Z’ are second order variables over relations among objects and over sets of objects21 . The other five
interpretations referred to above are:
(15) [Zxb]([Rxy],a),
(16) [Zax]([Rxy],b),
(17) [Zxy]([Rxy],a,b),
(18) [Zx]([Rxb],a),
(19) [Zx]([Rax],b).
In a case such as that of sentence (2), where all terms occurring in
the sentence denote something, it follows that if under any of these different
interpretations of its logical structure the sentence denotes a state of affairs,
then it denotes a state of affairs under all the interpretations – though these
states of affairs may all be different, as is the case with (2). Hence, as long
as one works under a convention that all singular and general terms denote
63
something, and that one is only concerned with the question of whether or
not sentences are true (i.e., denote some state of affairs), it is not necessary
to distinguish the different interpretations of the logical structure22 .
If the situation is like that of ordinary language, on the other
hand, where singular and general terms may fail to denote, then we are not
even guaranteed sameness of truth value with the different interpretations.
Complex predicates such as ‘x reasons like Sherlock Holmes’ or ‘x looks
like a unicorn’ may very well denote sets of objects (or properties) without
the singular term ‘Sherlock Holmes’ or the general term ‘unicorn’ denoting
anything. This means that the sentence
(20) John reasons like Sherlock Holmes,
may be true (or false) if interpreted as having the logical structure (12), i.e.,
(20/12)[x reasons like Sherlock Holmes](John),
but will be truth-valueless if interpreted as having either of the structures
(11) or (13); i.e.,
(20/11)[x reasons like y](John, Sherlock Holmes),
(20/13)[John reasons like x](Sherlock Holmes)23 .
These distinctions of logical structure generalize to sentences involving quantifiers and connectives as well. Take the sentence
(21) All men are mortal,
for example. Its logical structure is usually expressed as
(22)∀x(x is a man ⇒ x is mortal),
but this can be interpreted in different ways, among which:
(23)[∀x(Zx ⇒ W x)]([x is a man], [x is mortal])
and
(24)[∀xZx]([x is a man ⇒ x is mortal]).
According to the first interpretation we have a relation of inclusion between
two sets, and the state of affairs denoted by (23) is
(23e ) <Inclusion, set of men, set of mortals>.
According to the second interpretation we have the universality of a set,
and the state of affairs denoted by (24) is
(24e ) <singleton of the universal set (of objects), set of objects that are not
men or are mortal>.
In (23) ‘⇒’ is combined with the universal quantifier to form a logical predicate; in (24) ‘⇒’ is used as a predicate operator combining the predicates
‘x is a man’ and ‘x is mortal’.
In terms of properties we can say that whereas in (23) ‘[∀x(Zx ⇒
W x)]’ denotes a (level 2) logical property Subordination, and that the state
of affairs is the combination of this property with the properties of being a
man and of being mortal, in (24) ‘[∀xZx]’ denotes a (level 2) logical property
64
Universality, and that the state of affairs is the combination of this property
with the property of not being a man or being mortal. I.e.,
(23i ) <Subordination, being a man, being mortal>
and
(24i ) <Universality, not being a man or being mortal>24 .
With this analysis we can see that a sentence of pure logic such as
(25)∀xx = x
denotes a logical state of affairs. (25) can be analysed as
(26)[∀xZxx]([x = y]),
which denotes the state of affairs
(26i ) <Reflexivity, Identity>,
where Reflexivity is a level 2 logical property and Identity is a level 1 logical
property. Extensionally, of course, Reflexivity will be the set of all reflexive
relations (over a given domain) and Identity will be the identity relation for
that domain. Evidently this depends on treating quantification as a form
of predication and analysing the structure of (25) as a predicate applied to
an argument. But this is a natural consequence of Frege’s idea that the
quantifiers are higher level properties. In fact, it is easy to generalize this
sort of analysis to all sentences of first order (and higher order) logic so that
each sentence in standard notation has a number of different interpretations
(or readings) in terms of subject-predicate structure each of which either
denotes or does not denote a state of affairs. From which it follows that we
can define falsity in general as truth of the predicate negation. Thus
(27)[∀x(Zx ⇒ W x)]([x is a man], [x is a dentist])
is false because
(28)[¬∀x(Zx ⇒ W x)]([x is a man], [x is a dentist])
is true – i.e., denotes the state of affairs
(28e ) <Non-inclusion, set of men, set of dentists>.
This subject-predicate analysis is the main tool I use in the account
of truth, descriptions, senses, propositions, etc. to be developed in the following chapters. The basic intuition behind it, which goes back to Plato
and to Aristotle, is that for any statement (assertion, declarative sentence,
proposition) it is proper to ask what it states (asserts, predicates) about
what. Although this seems to go directly against the analysis of quantification in modern logic beginning with Frege, my claim is that it does not.
For as I suggested above, and will argue in more detail in Chapter 8, my
analysis is essentially a generalization of Frege’s analysis of quantification
in terms of higher level properties.
Another central aspect of my approach is that I consider the exclusion of non-denoting terms and truth-valueless sentences to be not only
unwarranted, but to obscure and confuse many important points and dis-
65
tinctions relevant to the account of truth, descriptions, senses, etc. So,
in my discussions, both formal and informal, I allow non-denoting proper
names, definite descriptions, predicates, and sentences (which are thereby
truth-valueless). Here I am in disagreement with Frege, for although he
allows for non-denoting terms and truth-valueless sentences in his informal
considerations about ordinary language, he held this to be an imperfection
of ordinary language that should be avoided in the formulation of a logical
language.
The notation and distinctions introduced in the last few pages
are used in chapters 3 and 4, and are presented and developed formally in
Chapter 6 – which also includes an elaboration of distinctions and notions
introduced in those earlier chapters. If in doubt about the notation, the
reader may want to refer to the beginning pages of Chapter 6.
66
Notes
1. I shall use ‘state of affairs’ as a generic term for the aspects of the world
described by true statements even though in many cases other terms (‘situation’,
‘circumstance’, ‘fact’, etc.) may be linguistically more appropriate. But while
discussing the views of authors who use an alternative terminology (e.g., Russell)
I shall often switch to their own terminology.
2. Plato’s most systematic discussion of these questions is in the Sophist beginning
in 236. The problem of falsity is neatly set up in 240d-241a:
STR. But, again, false opinion will be that which thinks the opposite
of reality, will it not?
THEAET. Yes.
STR. You mean, then, that false opinion thinks things which are
not?
THEAET. Necessarily.
STR. Does it think that things which are not, are not, or that things
which are not at all, in some sense are?
THEAT. It must think that things which are not in some sense are
– that is, if anyone is ever to think falsely at all, even in a slight
degree.
STR. And does it not also think that things which certainly are, are
not at all?
THEAET. Yes.
STR. And this too is falsehood?
THEAET. Yes, it is.
STR. And therefore a statement will likewise be considered false, if
it declares that things which are, are not, or that things which are
not, are.
THEAET. In what other way could a statement be made false?
STR. Virtually in no other way; but the sophist will not assent to
this. Or how can any reasonable man assent to it, when the expressions we just agreed upon were previously agreed to be inexpressible,
unspeakable, irrational, and inconceivable? Do we understand his
meaning, Theaetetus?
THEAET. Of course we understand that he will say that we are contradicting our recent statements, since we dare to say that falsehood
exists in opinions and words; for he will say that we are thus forced
repeatedly to attribute being to not-being, although we agreed a
while ago that nothing could be more impossible than that.
A recent discussion of the development of Plato’s thought on these issues is Denyer
Language, Thought and Falsehood in Ancient Greek Philosophy.
67
3. Russell takes up these problems in “On Denoting” and Quine in “On What
There is”. The temptation to postulate possible entities of one sort or another is
very strong, and even when it seems that this sort of approach is dead and buried
it comes back with renewed strength, as with the notion of possible world.
4. The word ‘description’ can be misleading in this connection. Ordinarily we
describe something by stating properties of that thing that characterize it in some
way (which may depend on the kind of thing in question and on context) but
that do not necessarily identify it. If I have a set of dinning-room chairs, I may
describe one of them in such a way that applies equally well to the others and yet
say that this is a description of that particular chair. On the other hand, I may
identify something without describing it in the ordinary sense. If there is a chair
in my study, and only one, the definite description ‘the chair in my study’ is not
really a description of that chair in the ordinary sense, although I may properly
say that I am identifying something as the chair in my study. The addition of
the qualification ‘definite’ in ‘definite description’ is meant to indicate that we
are dealing with a description of a definite thing. Thus, definite descriptions may
be included in a general category of identifiers, and what I am suggesting is that
statements should also be included in this category.
5. This is how Quine formulates the problem of non-being at the beginning of
“On What There is” (pp. 1-2):
It would appear . . . that in any ontological dispute the proponent of
the negative side suffers the disadvantage of not being able to admit
that his opponent disagrees with him. This is the old Platonic riddle
of nonbeing. Nonbeing must in some sense be, otherwise what is it
that there is not? This tangled doctrine might be nicknamed Plato’s
beard; historically it has proved tough, frequently dulling the edge
of Occam’s razor.
It is some such line of thought that leads philosophers like McX
to impute being where they might otherwise be quite content to
recognize that there is nothing. Thus, take Pegasus. If Pegasus were
not, McX argues, we should not be talking about anything when we
use the word; therefore it would be nonsense to say even that Pegasus
is not. Thinking to show thus that the denial of Pegasus cannot be
coherently maintained, he concludes that Pegasus is.
6. I don’t mean to be giving a textual account of this difficult and controversial
matter here, but merely my understanding of Plato’s view. In Plato’s Theory of
Knowledge Cornford translates the passage in Sophist 263a-d as follows:
STR. I will make a statement to you, then, putting together a thing
with an action by means of a name and a verb. You are to tell me
what the statement is about.
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THEAET. I will do my best.
STR. ‘Theaetetus sits’ – not a lengthy statement, is it?
THEAET. No, of very modest length.
STR. Now it is for you to say what it is about – to whom it belongs.
THEAET. Clearly about me: it belongs to me.
STR. Now take another.
THEAET. Namely – ?
STR. ‘Theaetetus (whom I am talking to at this moment) flies’.
THEAET. That too can only be described as belonging to me and
about me.
STR. And moreover we agree that any statement must have a certain
character.
THEAET. Yes.
STR. Then what sort of character can we assign to each of these?
THEAET. One is false, the other true.
STR. And the true one states about you the things that are as they
are.
THEAET. Certainly.
STR. Whereas the false statement states about you things different
from the things that are.
THEAET. Yes.
STR. And accordingly states things that are-not as being.
THEAET. No doubt.
STR. Yes, but things that exist, different from things that exist in
your case. For we said that in the case of everything there are many
things that are and also many that are not.
THEAET. Quite so
STR. So the second statement I made about you . . . must be about
something.
THEAET. Yes.
STR. And if it is not about you it is not about anything else.
THEAET. Certainly.
STR. And if it were about nothing, it would not be a statement at
all; for we pointed out that there could not be a statement that was
a statement about nothing.
THEAET. Quite true.
STR. So what is stated about you, but so that what is different is
stated as the same or what is not as what is – a combination of verbs
and names answering to that description finally seems to be really
and truly a false statement.
THEAET. Perfectly true.
(Op.
Although I don’t agree with Cornford’s interpretation of the passage
Cit., pp. 312-317), he also discusses it in terms of states of affairs.
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Another related discussion is Wiggins’ in “Sentence Meaning, Negation, and
Plato’s Problem of Non-Being”.
7. The correspondence view of truth is often traced back to Plato and to Aristotle’s
formulation in Metaphysics Γ 7 (1011b 26): “To say of what is that it is not, or
of what is not that it is, is false, while to say of what is that it is, and of what
is not that it is not, is true”. It is not clear, however, that this is anything but
a reformulation of Plato’s view of truth as being. It is quite close to Plato’s
formulation in Cratylus 385b: “Then that speech which says things as they are
is true, and that which says them as they are not is false?”. What Aristotle’s
formulation does is to eliminate a certain ambiguity in the expressions ‘says things
as they are’ and ‘says things as they are not’. For false statements, Plato also
makes this explicit in the passage I quoted in note 2 when he says that a statement
is false “if it declares that things which are, are not, or that things which are not,
are.” Naturally, the counterpart of this for true statements would be that a true
statement declares that things which are, are, or that things which are not, are
not; which is quite close to Aristotle’s formulation. And as long as one agrees
that this sort of terminology makes sense, it shows that the problem of non-being
arises just as much for truth as for falsity, since it seems equally problematic to
attribute non-being to non-being as to attribute being to non-being.
Aristotle has another formulation at the beginning of Metaphysics Θ 10,
in terms of combination and separation, which seems a more direct statement of
a correspondence view of truth. He says:
The terms ‘being’ and ‘non-being’ are employed firstly with reference
to the categories, and secondly with reference to the potency or
actuality of these or their non-potency or non-actuality, and thirdly
in the sense of true and false. This depends, on the side of the
objects, on their being combined or separated, so that he who thinks
the separated to be separated and the combined to be combined has
the truth, while he whose thought is in a state contrary to that of
the objects is in error.
This suggests a correspondence view of truth in terms of the state of thought
matching (or not matching) the state of the objects. Truth, however, is clearly
subordinate to reality for Aristotle, for, as he goes on to say: “It is not because
we think truly that you are pale, that you are pale, but because you are pale we
who say this have the truth.” Thus, I think that Aristotle is again formulating
Plato’s view of truth as being; truth is what expresses what is as it is.
I have no objection to calling Plato’s and Aristotle’s view of truth a
correspondence view of truth, but the word ‘correspondence’ is open to many
different interpretations and must be given a more definite content. I discuss this
issue in a more general setting in Chapter 12.
8. See Kripke Naming and Necessity for an important examination of this issue.
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9. In “On What There Is”, pp. 7-8:
In order thus to subsume a one-word name or alleged name such as
‘Pegasus’ under Russell’s theory of description, we must, of course,
be able first to translate the word into a description. But this is
no real restriction. If the notion of Pegasus had been so obscure
or so basic a one that no pat translation into a descriptive phrase
had offered itself along familiar lines, we could still have availed
ourselves of the following artificial and trivial-seeming device: we
could have appealed to the ex hypothesi unanalysable, irreducible
attribute of being Pegasus, adopting, for its expression, the verb ‘isPegasus’, or ‘pegasizes’. The noun ‘Pegasus’ itself could then be
treated as derivative, and identified after all with a description: ‘the
thing that is-Pegasus’, ‘the thing that pegasizes’.
10. Quine treats the problem as a sort of abstract logical problem, for which
his solution is reasonable, but Russell’s abbreviative conception was based on a
fundamental epistemological distinction between knowledge by acquaintance and
knowledge by description – see “On Denoting”, pp. 41-42 and 55-56. This is a
distinction that Russell maintained (with some variations) throughout his later
philosophical work, but it is doubtful that he would have agreed in any of the
versions that we are acquainted with a property ‘pegasizes’.
I am not claiming that there are no such properties as ‘is-Russell’ or
‘is-Quine’, but only that since ‘Pegasus’ does not denote one cannot specify the
property in terms of identity to the bearer of the name.
11. In “The Semantic Conception of Truth” Tarski raises a number of questions
about both realism and correspondence. He sees as one of the merits of Aristotle’s
formulation in Metaphysics Γ that it avoids the notion of correspondence. I am
not suggesting that Tarski’s own view of truth should be interpreted realistically;
on the contrary, Tarski wanted to avoid this sort of issue.
12. It is usual to talk of the relation between a description and what it identifies, when it does identify something, as denotation, and I shall use this word
in connection with statements as well. One can talk of reference, or designation,
or use various other words as well to express the same relation. There are important differences between these words, but to try to keep them apart would
complicate the discussion of various authors who use them to express roughly the
same notion. The fundamental point for me is that what is involved in denotation is the fulfillment of identity conditions; hence my use of ‘identification’ or
‘individuation’.
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13. I am taking the term ‘combines’ from Aristotle in the passage from Metaphysics Θ10 quoted in note 7 above. Later in the chapter (1051b 33) Aristotle
expresses himself in a way that is suggestive of the view of state of affairs that I
indicate in the text: “As regards the ‘being’ that answers to truth and the ‘nonbeing’ that answers to falsity, . . . there is truth if the subject and attribute are
really combined, and falsity if they are not combined”.
14. Gödel’s argument is discussed in detail in Chapter 4.
15. I use ‘property’ in the sense that includes relations – i.e., properties of any
number of terms. Frege used ‘concept’ and ‘function’. The remarks in the next
few paragraphs are rather compact and I mean them as a brief indication of where
I’m heading.
16. This idea was first formulated explicitly by Frege in The Foundations of Arithmetic, p. 65 – and presumably somewhat earlier in his posthumously published
“Dialogue with Pünjer on Existence”. It is elaborated in “Function and Concept”
and in The Basic Laws of Arithmetic. One can hold, however, that Existence is a
logical property of objects, in which case it appears at level 1. It will also appear
at all higher levels as a property of properties distinguishable from the property
expressed by existential quantification.
17. Aside from Existence, I will assume that the logical properties that appear at
level 1 are the Identity relation and pairwise Diversity relations of all arities.
18. The view that instantiation is the fundamental logical relation is expressed
by Frege in “Comments on Sense and Meaning”, p. 118: “The fundamental
logical relation is that of an object’s falling under a concept: all relations between
concepts can be reduced to this.” I don’t quite agree with the last point though,
because I hold that there are non-extensional relations between properties.
19. Besides these senses, which we may call ‘singular senses’, there are also plural
senses which purport to identify pluralities (or totalities). We may say, in general,
that a sense is a property with the logical feature that if it applies to a plurality
of things, then it applies only to those things. Thus, the property of being the
authors of Principia Mathematica is a sense (of Russell and Whitehead), whereas
the property of being an author of Principia Mathematica is not a sense.
20. According to standard set theory there would be no such thing as the set
of non-dentists. One has to relativize sets to a given domain of discourse – or
distinguish types, which is what one does with properties. Whenever I use a set
or relation like this we may suppose such a relativization.
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21. This notation is an abbreviation for the notation introduced more formally in
Chapter 6. It is a form of abstraction notation (as in λ-notation) and the argument
variables should be displayed. Thus (11) abbreviates ‘[[Rxy](x, y)](a, b)’, where
the variables ‘x’ and ‘y’ are displayed. Since I am distinguishing variables and
constants by the notation, and I am abstracting simultaneously on all the variables
that occur free within the brackets, as long as the ordering of the variables is clear
relative to the argument places there will be no possibility of confusion. I will also
follow the usual convention of omitting superscripts when the arity of the second
order variables (or constants) is clear from context.
22. This fact has played an important role in the modern conceptions of logic
and of truth. It leads to the view, common in presentations of logic, that the
only connection between a sentence (or statement, or proposition) and reality is
the circumstance that it is true or the circumstance that it is false. Even though
one sets up fairly detailed connections between singular and general terms and
reality, when it comes to sentences all detail is lost – i.e., it is restricted to the
description of the truth conditions but not considered to be some complex feature
of reality. This is actually rather puzzling and is hard to explain convincingly to
students. What one normally says is that sentences denote truth values, T or F,
which is precisely Frege’s view. As opposed to Frege, however, the modern view
does not assign to truth values any clear status as entities. But if truth values are
not objects, in some sense, then why say that sentences denote truth values?
23. It is not only for sentences involving non-denoting terms that these distinctions
of logical structure are relevant. An interesting case is that of counterfactual
conditionals. In Methods of Logic pp. 14-15 Quine used the following examples
as an argument against the possibility of having a logical theory of counterfactuals:
(a) If Bizet and Verdi had been compatriots, Bizet would have been Italian;
(b) If Bizet and Verdi had been compatriots, Verdi would have been French.
His point being that we wouldn’t know how to adjudicate between the two.
But consider the following pair now:
(c) If Bizet had been a compatriot of Verdi, Bizet would have been Italian;
(d) If Verdi had been a compatriot of Bizet, Verdi would have been French.
It seems quite clear, on intuitive grounds, that (c) and (d) are true whereas (a)
and (b) are false. Part of the reason for this, as Quine himself points out, is that
our use of counterfactuals involves a certain projection: we suppose that things
have properties other than they actually have and draw conclusions. But how do
we make these projections?
In (c) we are supposing that Bizet has the property of being a compatriot
of Verdi, and from this it seems to follow clearly that since Verdi is Italian, and
compatriots have the same nationality, also Bizet would be Italian. In (d) we are
supposing that Verdi has the property of being a compatriot of Bizet, and from this
again it seems to follow intuitively that since Bizet is French, also Verdi would be
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French. In Quine’s examples (a) and (b), on the other hand, we are supposing that
Bizet and Verdi have the relation of being compatriots, from which nothing follows
concerning their individual nationalities, except for the fact that they would have
the same nationality – i.e., they would both be French, or both Italian, or both
Canadian, or whatever.
The reason Quine thinks that his examples are damaging to a logical
theory of counterfactuals is due to the fact that the standard analysis of the
logical structure of the antecedents of (a)-(d) gives them all the same structure;
something like ‘Cbv’. In terms of the alternative structures (11)-(19), however,
we see that whereas (a) and (b) do indeed have antecedents of form (11), the
form of the antecedents of (c) and (d) is (12). In the actual ordinary language
formulations these distinctions are made explicit by means of the word order. The
different forms of the antecedents help indicate what is involved in the projection.
24. I use bigger corner brackets for states of affairs interpreted intensionally to
indicate that these should not be thought of as ordered sequences in the settheoretic sense.
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Chapter 2
The True and the False
Frege’s basic distinction between sense and reference for singular
terms (proper names and definite descriptions) is that the sense of a singular
term contains a mode of presentation, whereas the reference of the singular
term is the object thus presented. Although each singular term has (or
expresses) a sense, it does not necessarily refer to anything; senses are (or
contain) manners of presentation which may or may not actually present
something1 .
After distinguishing sense and reference for singular terms, Frege
raises the question as to whether there is a similar distinction for declarative
sentences2 . He states that a sentence contains a thought and argues that
this thought cannot be the reference of the sentence, if any. The argument,
based on the principle
(R) The reference of a composite expression depends only on the reference
of the parts and not on their sense,
is essentially that whereas substitution of co-referential terms in a sentence
would not change its reference (if any), it may change the thought contained
in the sentence3 .
It is perfectly natural for Frege to argue like this, because one motivation for the distinction between sense and reference for singular terms
is that sentences such as
(1) The morning star = the morning star,
and
(2) The morning star = the evening star,
express different thoughts even though corresponding parts have the same
reference. But in a way Frege’s argument begs the question, because if sentences referred to thoughts, then principle (R) would be false for sentences.
For the argument to be conclusive Frege would need an independent justification for (R) that makes it plausible for reference in general.
Having concluded that the thought contained in a sentence cannot
be identified with its reference, Frege identifies the thought with the sense
of the sentence4 . He goes on to consider whether there may be something
else that can be identified as the reference of a sentence. It is natural to
suppose that there must be, because the notion of sense for singular terms,
as a mode of presentation, depends on the notion of reference for these
terms. That is, if one could not talk intelligibly of reference in relation to
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singular terms, then one could not talk intelligibly of modes of presentation,
because this means a mode of presentation of a reference, or of a presumed
reference. Frege’s notion of sense for singular terms is naturally interpreted
as a manner in which an object is identified in terms of its properties and its
relations to other objects. Since a sense need not be the sense of anything,
we can think more generally of a sense as a manner of identification, but
the question as to what sort of thing can be identified (or presented) by the
sense of a sign must be significant.
The analogous procedure for sentences would be to introduce their
sense as the mode of presentation of their reference, or presumed reference,
and so to look first for candidates for the role of referents of sentences. If
a thought, characterized as the objective content of a sentence (32n), is a
sense, then it contains a mode of presentation, and it must make sense to ask
what it is that is (or can be) presented by a thought. It seems inevitable to
me that if one looks for candidates one will hit upon states of affairs, facts,
or something like that. The similarity between a definite description, as a
way of presenting an object, and a sentence, as a way of presenting a state
of affairs, is so striking and obvious, at least for simple examples, that it
just can’t be missed.
There is no explicit indication that Frege considered states of affairs as candidates for the role of referents of sentences, but one of his
arguments suggests that he might have and that he did not see how states
of affairs could be treated as units to which sentences may refer. So what
he does is to argue his way from thoughts to the nature of the (possible)
referents.
Frege starts by arguing that the thought alone is not enough because if all that interested us about a sentence was its sense – i.e., the
thought, – then we wouldn’t have to consider whether a part of a sentence
has a reference; for all that matters for the sense of a sentence is the sense of
the parts. This is based on a principle for senses analogous to (R), namely:
(S) The sense of a composite expression depends only on the senses of the
component parts and not on their reference.
But, Frege argues, when we are concerned with the truth value of a sentence, then the reference and not just the sense of the component parts is
relevant. Whether or not Odysseus was set ashore at Ithaca while sound
asleep depends on the reference of ‘Odysseus’, for it is of that reference that
the predicate is affirmed (32-33). So, if we are interested in whether the
sentence ‘Odysseus was set ashore at Ithaca while sound asleep’ is true or
not, we cannot merely rest with the sense of this sentence. From this he
concludes “that the reference of a sentence may always be sought, whenever
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the reference of its components is involved; and that this is the case when
and only when we are inquiring after the truth value” (33).
At this point in the argument this conclusion is rather overstated,
however. What Frege has established so far is that the reference of the parts
of the sentence, and not only their sense, may be relevant for the truth
value of the sentence. From this nothing follows directly as to whether or
not sentences have reference. Frege says that “(t)he fact that we concern
ourselves at all about the reference of a part of a sentence indicates that
we generally recognize and expect a reference for the sentence itself” (33).
But this is not clear, because a sentence may involve reference to the world
without itself referring to some one thing – be it a truth value or a state
of affairs. That’s one reason I think that it is better to make the case for
reference first, and then consider how a thought presents a reference.
At any rate, Frege now goes on to his main conclusion: “We are
therefore driven into accepting the truth value of a sentence as constituting
its reference” (34). Given the earlier claim, with its “when and only when”,
this may not be as unreasonable as it seems, and Frege goes on to justify
his contention in three different ways.
First, he considers the objection that the truth value is not the
reference of the sentence but a property of it. This is perfectly compatible
with Frege’s earlier conclusions: the reference of the parts of the sentence
are relevant to its truth value but sentences have no reference. It is also
compatible with (some) sentences having a reference and the truth value
being a property of sentences related to this. But Frege offers an argument
to the effect that the truth value cannot be a property of sentences and that
it must be considered as their reference. I will discuss this argument later
on5 .
Second, he claims that with the truth values as reference for sentences the substitutivity principle (R) is satisfied when sentences occur as
component parts of more complex sentences. In fact, the rest of Frege’s paper – about half of the entire paper – is occupied with certain special cases of
this principle as applied to sentences. In particular, he argues that indirect
discourse does not provide a counter-example6 . Although these arguments
are quite essential to Frege’s case for truth values, they are not used directly
to establish that truth values must be the reference of sentences, if sentences
have reference, but as a test for his conclusion (36).
Frege thought that (R) makes truth values a compelling choice,
and his third argument is to suggest this. He asks: “What else but the
truth value could be found, that belongs quite generally to every sentence
if the reference of its components is relevant, and remains unchanged by
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substitutions of the kind in question?” (35)7 . Although this argument may
justify Frege’s choice to some extent, and can also be taken as a challenge
to find reasonable alternatives, it is not really convincing. For, as far as
this challenge is concerned, it is quite easy to argue intuitively that states
of affairs are a reasonable alternative. Let us consider what happens in one
example when we perform the substitutions that Frege had in mind.
We have seen that the sentence
(3) Frege taught Carnap,
can be taken as referring to the state of affairs of Frege teaching Carnap,
which we can represent as
(3i ) <teacher of, Frege, Carnap>,
or, in terms of sets, as
(3e ) <set of pairs of persons such that the first taught the second, Frege,
Carnap>.
Principle (R) allows us to substitute parts of (3) by co-referential expressions, so that, for example, we could substitute for ‘Frege’ the description
‘the greatest logician since Aristotle’, getting
(4) The greatest logician since Aristotle taught Carnap.
In place of ‘Aristotle’ we could now substitute ‘the author of the Nicomachean Ethics’, say, getting
(5) The greatest logician since the author of the Nicomachean Ethics taught
Carnap.
Continuing in this way we can produce extremely elaborate sentences containing any number of expressions referring to all sort of things, but it is
not at all clear that by doing this we can reach any sentence that refers to a
state of affairs that is different from the state of affairs (3i ), or (3e ), referred
to by (3), as Frege’s remark may lead us to expect. Thus, if one accepts
Frege’s principle (R), what one can clearly infer is that for each sentence
there are an indefinite number of other sentences that have the same reference, but not that the only thing that these sentences have in common is
their truth value.
The point of Frege’s question may perhaps be put as follows,
though. By substituting into a sentence it seems that one can make more
and more aspects of reality relevant to the reference of that sentence. The
reference of ‘Aristotle’ is certainly relevant to the reference of (4), which is
the same as the reference of (3). Moreover, it is not only the reference of
‘Aristotle’ that is relevant to the reference of (4), but the circumstance that
Frege was the greatest logician since Aristotle. By doing more substitutions
that preserve the reference of (3) there are more and more circumstances
that are relevant to the common reference of all these sentences. Since this
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can go on indefinitely, how do we isolate the circumstance that accounts for
the common reference of all these sentences? Is there such a circumstance?
By taking the truth values as a sort of global contribution of reality
Frege avoids this issue. But this means that “in the reference of the sentence
all that is specific is obliterated” (35), which seems to leave out of account
the obvious difference in the contribution of reality to the truth or falsity
of different sentences. Frege’s conclusion was that it is the judgement that
analyses the truth value through the thought and, as it were, divides it into
parts. The combination of thought and truth value for each sentence would
then be what distinguishes the relevant circumstance8 . Though Frege does
not develop this idea any further, I think that he is getting close to a notion
of state of affairs. The difference between the sentences (3), (4), and (5), is
that Frege is identified in different ways. If we think of the state of affairs
as (3i ), or (3e ), we can think of the same state of affairs being identified
in different ways. The different contributions of reality have to do with
the identifications of Frege – i.e., with different senses of Frege – which is
precisely the difference of one sentence to another9 .
Frege’s earlier considerations also raise a question for false sentences in relation to states of affairs. The reference of the parts of a sentence
are not only relevant for its truth but for its falsity. Yet, a false sentence
cannot refer to a state of affairs – in any reasonable sense of this term which
does not include possible or impossible states of affairs that are not actual.
But if false sentences don’t refer even though the reference of their parts
is relevant to their truth value, why should true sentences refer? Frege’s
argument that consideration of the reference of the parts of a sentence is
relevant to its truth value, and leads us to expect a reference for the sentence itself, is completely independent of whether the sentence is true or
false. So to the extent that it establishes that true sentences refer, it also
establishes that false sentences refer. In my view the argument does not
establish either conclusion, but if it is accepted, then it supports the choice
of truth values over states of affairs because truth and falsity are treated
symmetrically.
Truth values are objects for Frege, and there are only two of them:
the True and the False. When he introduces these objects, he characterizes
them as follows: “By the truth value of a sentence I understand the circumstance that it is true or false” (34)10 . But the circumstance that a sentence
is true or false seems to be a characteristic, or property of a sentence, not an
object. This is not Frege’s view, however, and is the point of his argument
that the relation between a sentence and its truth value is not like a relation
of a subject to a predicate – see note 5.
Frege claims that the sentence
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(6) The thought that 5 is a prime number is true
says nothing additional to what is said in the sentence
(7) 5 is a prime number.
He argues that the claim of truth contained in (6) arises from the form of
the declarative sentence and not from the predication ‘is true’, and that the
claim of truth contained in (6) is exactly the same as the claim of truth
contained in (7). Frege supports this point with the argument that when
an actor utters (6) or (7) he is only expressing a thought, not making an
assertion, and that the thought he expresses with (6) is exactly the same as
the thought he expresses with (7).
In my view this argument conflates two different questions. One
is whether (6) says the same thing (or expresses the same thought) as (7);
the other is whether the claim of truth arises from predication or from
assertion. I agree with Frege on the latter, and his considerations about
actors are relevant to this – in fact, in “On the Foundations of Geometry”
Frege draws this conclusion using again the example of the actor but without
making a specific point about the predicate ‘is true’11 . On the other hand,
I don’t agree with Frege on the former.
Take the sentence
(8) The sentence ‘5 is a prime number’ is true.
Clearly, (8) says something about a sentence, namely the sentence ‘5 is a
prime number’; it says of that sentence that it is true. (7) says something
about the number 5; it says of that number that it is prime. The subject
of discourse is different. I don’t turn (7) into an assertion by predicating ‘is
true’ of it, and the remark about the actor is relevant to this, but it doesn’t
follow either that the content of what I’m saying with (7) and with (8) is
the same, or that when I assert (8) I am making the same assertion that
I make when I assert (7). The same considerations apply to the relation
between (6) and (7); just as (8) says something about a sentence, (6) says
something about a thought.
One can also see that the content of (6) is different from the content
of (7) by considering a different example. Take the sentences
(9) The thought that Odysseus was set ashore at Ithaca while sound asleep
is true,
and
(10) Odysseus was set ashore at Ithaca while sound asleep.
Assuming that ‘Odysseus’ has no reference, (10) contains only a thought
and is neither true nor false. This is Frege’s view, with which I agree. But
it follows from this that (9) is false; for, otherwise, it wouldn’t be true to
say that (10), or the thought expressed by (10), is neither true nor false. It
follows also that the assertion of (9) is a different assertion than the assertion
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of (10), and that even when an actor utters (9) and (10) he is expressing
different thoughts. And, evidently, the same points can be made about the
sentence
(11) The sentence ‘Odysseus was set ashore at Ithaca while sound asleep’ is
true12 .
Since (7) is true, it follows that (6) is also true, but not that
it contains the same thought as (7) or that asserting it is the same as
asserting (7). The confuson arises from the fact that (6) and (7) are logical
consequences of each other, and, therefore, that whoever is prepared to
assert (6) should be prepared to assert (7), and viceversa. This shows
something about the (logical) character of the predicate ‘is true’ and is
completely independent of the fact that (7) is true, for it holds for any such
pair, including (9) and (10). But the contents are always different.
What is important to notice in connection with these examples
is that when we allow for statements that are neither true nor false, two
notions of logical equivalence that coincide for two-valued statements do not
coincide anymore. These notions are:
(12) Two forms φ and ψ are c-logically equivalent if and only if each is a
logical consequence of the other,
where, in terms of interpretations, φ is a logical consequence of ψ if there is
no interpretation in which φ is true and ψ is not true.
(13) Two forms φ and ψ are tv-logically equivalent if and only if they have
the same truth value in every interpretation13 .
Such pairs as (6)-(7) or (7)-(8) or (9)-(10) or (10)-(11) are always
c-logically equivalent but not tv-logically equivalent – and what I pointed
out before is that (9)-(10) and (10)-(11) are not materially equivalent.
But what follows from this about truth as reference and about the
predicates ‘is true’ and ‘is false’ ? In what sense is Frege arguing that truth
is not a property of a thought? Is he arguing that ‘is true’ and ‘is false’ are
not predicates?
On the face of it, it would be rather odd to argue that ‘is true’ and
‘is false’ cannot be predicated of thoughts, or of sentences. Obviously they
can, and Frege does it all the time. Moreover, if truth values are objects
to which thoughts refer, then it would seem that there will be predicates ‘x
refers to the True’ and ‘x refers to the False’, which can be abbreviated as
‘x is true’ and ‘x is false’ and which apply precisely to the true thoughts and
to the false thoughts, respectively. So what is Frege arguing when he argues
that truth is not a property of thoughts, or that the truth or falsity of a
thought or sentence should not be analyzed as a subject-predicate relation?
In “On Sense and Reference” Frege does not explicitly say that ‘is
true’ and ‘is false’ aren’t legitimate predicates, nor does he say that truth
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is not a property of thoughts. What he says is that the relation between a
thought and its truth value cannot be compared with the relation between
a subject and a predicate; from which it would follow also, I suppose, that
it cannot be compared with the relation between an object and a property.
But this does suggest that truth and falsity are not properties of thoughts,
and in “Thoughts” he is explicit about it. He argues that whenever an
object has a property, the thought that that object has that property would
have the property of truth; yet this property does not add anything to that
thought. And he concludes: “May we not be dealing here with something
which cannot be called a property in the ordinary sense at all?”14
I would put the point by saying that as a property truth is derived
from the connection between a thought and reality. Going back to the
example (6)-(7), it is one thing to say that (6) and (7) express the same
thought, or that (6) says nothing that isn’t said in (7), and quite another
to say that (6) doesn’t add truth to the thought expressed by (7). Part of
what Frege seems to be arguing is that truth is a feature of the connection
between the thought and reality and not a feature of the thought itself.
That’s why he insists that in judging and asserting one is passing from the
level of sense to the level of reference, and that truth arises at this level.
We can compare the case of sentences with that of names. Is there
something analogous to asserting for names? It seems to me that to use a
name referentially is similar to asserting a sentence. When we use a name
– to call someone, for example, or just to talk about something – we are at
the level of reference, and it is included in the use of a name that it has a
reference – though it may not, in fact, have it. Just as assertion involves a
claim of truth, using a name referentially involves a claim of existence.
A point similar to Frege’s can be made about the redundancy of
appealing to reference in the act of using a name. To call
(14) The referent of ‘John’ !,
for example, adds nothing to simply calling
(15) John!
And if the calling is not a real calling, as by an actor on the stage, then the
use of the expression ‘the referent of’ won’t turn it into a referring use.
Like asserting, referring is an act which involves a passage to the
level of reference. Assertion, for Frege, is the expression of a judgement,
which is itself the admission of the truth of a thought (34n)15 . In my view
this must hold for referring in general, and the referential use of a name
must involve an admission of existence. I.e., referential use involves a claim
to reference, which is also a judgement. One could say that a judgement is
a commitment to reality and that the claim of truth involved in assertion
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and the claim of reference involved in the referential use of names arise from
this16 .
Let me argue now in more detail that the property of truth of a
thought is derived from the connection between the thought and reality.
This means, in particular, that the truth or falsity of a sentence cannot be
determined by its sense, which seems to go against another Fregean principle
about sense and reference; namely, that sense determines reference.
Church formulates this principle by saying that the reference of an
expression is a function of the sense. With this I agree if it is interpreted as
the claim that if two expressions have the same sense, then they have the
same reference – or no reference, as the case may be. But Church goes on
to say that “given the sense, the existence and identity of the denotation
are thereby fixed, though they may not necessarily therefore be known to
every one who knows the sense”17 . This seems to me somewhat misleading,
because it can be interpreted as suggesting that it is the sense that fixes the
existence and identity of the reference. I think that the sense contributes
to the fixing by giving something like the identity criteria, but that the
connection is fixed by reality.
When one talks about something determining something one tends
to think of this in terms of functional dependence in a mathematical sense,
but the fact that there is a functional correlation between two classes of
entities does not mean that the existence and identity of the values of the
function is determined by the function – the function can simply be an
expression of this determination, as with functions in extension. We can, for
example, correlate to each building the first person, if any, who ever walked
around it three times, but this does not mean that the existence and identity
of this person is determined by the building or by the functional rule. On the
contrary, we would say that it is the existence and identity of the correlation
– the function in extension – that is determined by what happens in reality;
i.e., by people walking (or not walking) around buildings. The function in
extension is derivative from those circumstances. What Frege’s argument
may be taken to show is that the sense in which ‘is true’ and ‘is false’ are
predicates of sentences (or of thoughts) is similarly derivative from what
happens in reality.
What we get from the mathematical formulation in terms of functional correlation is that to any sense there cannot correspond more than
one reference, from which it follows that there is a (partially defined) function in extension from senses to references. But this doesn’t tell us anything
about the character of that function, nor does it tell us anything about determination in any stronger sense of determination. One can also formulate
principle (R) in functional terms, saying that the reference of a complex
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expression is a function of the references of its parts, but it certainly does
not follow that the references of the parts determine the reference of the
whole. In fact, for a sentence, even the sense of the sentence together with
the references of the parts cannot determine the truth value of the sentence.
Why not? It is certainly a contingent matter that Frege taught
Carnap. The sense of that sentence does not depend on that contingency,
however, and neither do the references of ‘Frege’, ‘Carnap’ and ‘taught’ –
taking the latter to refer to a relational property rather than to a set of
ordered pairs. So even though the truth value of the sentence ‘Frege taught
Carnap’ depends on the sense expressed by this sentence, and depends on
the referents of its parts, which are not necessarily determined by their sense,
this is a functional dependence, not a determination. What determines the
truth of the sentence is the circumstance that Frege taught Carnap, to
which, through its sense, the sentence refers. I would say that it is the
state of affairs of Frege teaching Carnap, whereas Frege simply talks about
the circumstance of the sentence being true, and conceptualizes this as the
object the True.
I think, therefore, that the sense in which truth is not a property
of thoughts is that truth is not a feature of thoughts simpliciter, but that it
derives from the relation of the thought to reality. Another way of putting
the point that the truth of a thought is derivative from the connection between the thought and reality is to say that truth is ineliminably relational.
Since one of the terms of this relation (reality) is considered to be fixed,
one may get the idea that it is a property of the thought simpliciter that
it is true. This happens with many properties. For example, one could
say that to be indoors is a property of people, but this is clearly derived
from a relation. It is not a feature of a person simpliciter to be indoors,
although I do say something about a person when I say that that person
is indoors. Similarly, I do say something about a thought when I say that
it is true, but it is not a feature of the thought simpliciter to be true. I
am indoors right now because I am in a certain relation to my home right
now; a thought is true because it is in a certain relation to reality – i.e., its
identifying conditions are fulfilled in reality. Frege’s idea that the truth of
a thought consists in referring to the True seems to me to be one way of
expressing the ineliminable relational nature of truth.
This goes against what Frege says in “Thoughts”, however, because
there he argues that the main problem with the conception of truth as
correspondence is that it takes truth to be a relation. He begins with the
observation that to take truth to be a relation “goes against the use of
the word ‘true’, which is not a relative term and contains no indication
of anything else to which something is to correspond.” Although this is
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correct, it doesn’t follow that ‘true’ is not derived from a relation, and
Frege’s subsequent argument is mainly an argument against the definability
of truth as a relation between a thought and something else18 . It seems to
me, therefore, that the claim that truth is essentially relational is compatible
with the claim that truth is not a relation.
Whether this may throw any light on Frege’s claim that truth is not
a property of thoughts remains to be seen. For the time being I will simply
draw some preliminary conclusions from the discussion in this chapter.
First, although I agree with Frege’s view that truth is a form of
denotation, I do not think that Frege has established that truth must be
analyzed denotationally. In fact, I don’t think that this can be established
in any conclusive sense by means of direct argumentation. One must argue
for the fruitfulness and richness of the view, as Frege himself does in the
introduction to The Basic Laws of Arithmetic19 .
Second, I don’t think that Frege establishes that truth is not a
property of thoughts (or that ‘true’ is not a predicate), although contrary
to his stated opinion I think that he makes a good case for truth being
relational.
Third, I think that the argument that a sentence S says the same
thing or expresses the same thought as a sentence of the form ‘the thought
that S is true’ is a basic fallacy. Such sentences are c-logically equivalent,
but if one admits that there are sentences which are neither true nor false,
then the fallacy is shown clearly by examples like (9) and (10) where the two
sentences are not materially equivalent. If one restricts oneself to sentences
that are either true or false, then the pairs are materially equivalent, but it is
still clear that the two sentences do not say the same thing and do not have
the same content – though, naturally, this depends on what one means by
‘content’20 . The issues involved here are important because Frege’s claim,
or related claims, have played a central role in discussions of truth in the
twentieth century21 .
Finally, although I think that Frege gives some reasons for taking
truth values as the reference of sentences, his arguments do not seem to
me to establish that on a referential conception of truth this is either the
only or the most natural or the best choice. The nature of these objects
is left completely unexplained, even in a rough intuitive sense, and the
introduction of the True as the circumstance that a sentence is true, and of
the False as the circumstance that a sentence is false seems to presuppose
the notions of truth and falsity. If truth is not a property of sentences (or
thoughts), then what are these circumstances of being true and of being
false?
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Notes
1. “On Sense and Reference”, pp. 26-28 – throughout this chapter page references
to this paper will be simply indicated by numbers within parentheses.
It is questionable whether ‘reference’ is an appropriate translation of
Frege’s ‘Bedeutung’, and the translators of his Collected Papers, Phostumous
Writings, and Philosophical and Mathematical Correspondence have used ‘meaning’ instead. Although this may be the better translation, I find it somewhat confusing given the standard philosophical use of the word ‘meaning’. In this chapter
I will mostly use ‘reference’, and in general I will use ‘denotation’, ‘reference’,
‘designation’, and other similar words more or less interchangeably, depending on
which text or author I am discussing.
2. In “On Sense and Reference” Frege does not discuss the notions of sense and
reference in relation to general terms, but he does in his posthumously published
“Comments on Sense and Meaning”.
3. This is what he says (32):
Let us assume for the time being that the sentence has reference.
If we now replace one word of the sentence by another having the
same reference, but a different sense, this can have no bearing upon
the reference of the sentence. Yet we can see that in such a case
the thought changes; since, e.g., the thought in the sentence ‘The
morning star is a body illuminated by the Sun’ differs from that in
the sentence ‘The evening star is a body illuminated by the Sun’.
Anybody who did not know that the evening star is the morning
star might hold the one thought to be true, the other false.
4. Frege continues (32):
The thought, accordingly, cannot be the reference of the sentence,
but must rather be considered as the sense. What is the position
now with regard to the reference? Have we a right even to inquire
about it? Is it possible that a sentence as a whole has only a sense,
but no reference?
5. The argument is the following (34-35):
One might be tempted to regard the relation of the thought to the
True not as that of sense to reference, but rather as that of subject
to predicate. One can, indeed, say: ‘The thought, that 5 is a prime
number, is true’. But closer examination shows that nothing more
has been said than in the simple sentence ‘5 is a prime number’.
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The truth claim arises in each case from the form of the declarative
sentence, and when the latter lacks its usual force, e.g., in the mouth
of an actor upon the stage, even the sentence ‘The thought that 5
is a prime number is true’ contains only a thought, and indeed the
same thought as the simple ‘5 is a prime number’. It follows that
the relation of the thought to the True may not be compared with
that of subject to predicate. Subject and predicate (understood in
the logical sense) are indeed elements of thought; they stand on the
same level for knowledge. By combining subject and predicate, one
reaches only a thought, never passes from sense to reference, never
from a thought to its truth value. One moves at the same level but
never advances from one level to the next. A truth value cannot be
a part of a thought, any more than, say, the Sun can, for it is not a
sense but an object.
6. Indirect discourse is the most obvious counter-example to (R) applied to contexts involving sentences, for one cannot generally substitute a sentence S ′ for a
sentence S in a context of the form ‘x said that S’ preserving truth value when
S and S ′ have the same truth value. Frege argues that this is only an apparent
counter-example by distinguishing the customary and the indirect reference of an
expression, and claims that whereas the customary reference of a sentence is its
truth value, the indirect reference of the sentence is its (customary) sense (28).
Thus, in ‘He said that 2 + 2 = 4’ the reference of ‘2 + 2 = 4’ is not its truth value
but its ordinary sense, and the only substitutions that are pertinent to (R) in this
context are substitutions of sentences with the same sense as ‘2 + 2 = 4’. In the
introduction to The Basic Laws of Arithmetic (p. x) Frege claims that only by
taking the truth values as the reference of sentences “can indirect discourse be
correctly understood.”
7. Many commentators have indeed agreed with Frege on this point. Dummett,
for instance, says that “it is difficult to think of anything else of which we have
a guarantee that it will not change.” Frege: Philosophy of Language, p. 182. A
critical discussion of Frege’s argument, which makes some of the points I make below, is in Barwise and Perry “Semantic Innocence and Uncompromising Situations
(pp. 393-395) – and also in Situations and Attitudes (pp. 20-26). As opposed to
them, however, I want to preserve and develop Frege’s theory of senses (Chapter
11), and I accept principle (R) (which they reject).
8. The passage is the following (35-36):
If now the truth value of a sentence is its reference, then on the one
hand all true sentences have the same reference and so, on the other
hand, do all false sentences. From this we see that in the reference
of the sentence all that is specific is obliterated. We can never be
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concerned only with the reference of a sentence; but again the mere
thought alone yields no knowledge, but only the thought together
with its reference, i.e. its truth value. Judgments can be regarded as
advances from a thought to a truth value. Naturally this cannot be a
definition. Judgment is something quite peculiar and incomparable.
One might also say that judgments are distinctions of parts within
truth values. Such distinction occurs by a return to the thought. To
every sense belonging to a truth value there would correspond its
own manner of analysis. However, I have here used the word ‘part’
in a special sense. I have in fact transferred the relation between the
parts and the whole of the sentence to its reference, by calling the
reference of a word part of the reference of the sentence, if the word
itself is part of the sentence. This way of speaking can certainly
be attacked, because the whole reference and one part of it do not
suffice to determine the remainder, and because the word ‘part’ is
already used in another sense of bodies. A special term would need
to be invented.
9. Of course, if one could show that from any true sentence one can pass to any
other true sentence by means of substitutions in accordance with (R), then Frege’s
conclusion would be unavoidable. Church, Davidson, and Gödel tried to show this
and I discuss their arguments in Chapter 4.
10. The full passage is the following:
By the truth value of a sentence I understand the circumstance that
it is true or false. There are no further truth values. For brevity I
call the one the True, the other the False. Every declarative sentence
concerned with the reference of its words is therefore to be regarded
as a proper name, and its reference, if it has one, is either the True
or the False. These two objects are recognized, if only implicitly,
by everybody who judges something to be true – and so even by a
skeptic. The designation of the truth values as objects may appear
to be an arbitrary fancy or perhaps a mere play upon words, from
which no profound consequences could be drawn. What I mean
by an object can be more exactly discussed only in conexion with
concept and relation. I will reserve this for another article. But
so much should already be clear, that in every judgment [Note: A
judgment, for me is not the mere comprehension of a thought, but
the admission of its truth.], no matter how trivial, the step from the
level of thoughts to the level of reference (the objective) has already
been taken.
11. He says (Op.Cit., p. 371):
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Take the proposition “Two is a prime number.” Linguistically we
distinguish here between a subject, “two,” and a predicative constituent, “is a prime number.” One usually associates an assertive
force with the latter. However, this is not necessary. When an actor
on the stage utters assertoric propositions, surely he does not really
assert anything, nor is he responsible for the truth of what he utters.
Let us therefore remove its assertive force from the predicative part,
since it does not necessarily belong to it!
12. The same point can also be made for
(9′ ) That Odysseus was set ashore at Ithaca while sound asleep is true,
instead of (9). In other words, it is not because one adds ‘the thought’ that one
is referring to the thought.
13. Where sameness of truth value in an interpretation means both true or both
false or both truth valueless.
14. The same theme is also developed in the posthumously published “My basic
logical Insights”. The passage in “Thoughts” is the following (pp. 61-62):
All the same it is something worth thinking about that we cannot
recognize a property of a thing without at the same time finding
the thought this thing has this property to be true. So with every
property of a thing there is tied up a property of a thought, namely
truth. It is also worth noticing that the sentence ‘I smell the scent
of violets’ has just the same content as the sentence ‘It is true that I
smell the scent of violets’. So it seems, then, that nothing is added to
the thought by ascribing to it the property truth. And yet is it not
a great result when a scientist after much hesitation and laborious
reserches can finally say ‘My conjecture is true’ ? The meaning of
the word ‘true’ seems to be altogether sui generis. May we not
be dealing here with something which cannot be called a property
in the ordinary sense at all? In spite of this doubt I will begin by
expressing myself in accordance with ordinary usage, as if truth were
a property, until some more appropriate way of speaking is found.
In this paper and in the other two papers of the Logical Investigations Frege does
not explicitly appeal to truth values as objects anymore, but in his “Notes for
Ludwig Darmstaedter”, which dates from the same period, he does refer to the
truth values (p. 255):
It is not, however, only parts of sentences that have meaning; even
a whole sentence, whose sense is a thought, has one. All sentences
that express a true thought have the same meaning, and all sentences
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that express a false thought have the same meaning (the True and
the False).
15. In “Thoughts” (pp. 62-63) Frege says:
. . . it is possible to express a thought without laying it down as true.
The two things are so closely joined in an assertoric sentence that it
is easy to overlook their separability. Consequently we distinguish:
(1) the grasp of a thought – thinking,
(2) the acknowledgement of the truth of a thought – the act of judgment,
(3) the manifestation of this judgment – assertion.
We have already performed the first act when we form a propositional question.
An advance in science usually takes place in this way: first a thought is grasped,
and thus may perhaps be expressed in a propositional question; after appropriate
investigations, this thought is finally recognized to be true. We express acknowledgement of truth in the form of an assertoric sentence. We do not need the word
‘true’ for this. And even when we do use it the properly assertoric force does
not lie in it, but in the assertoric sentence-form; and where this form loses its
assertoric force the word true cannot put it back again.
16. At the end of the paragraph quoted in the previous note, which continues
with remarks about actors and poetry, Frege says:
Therefore the question still arises, even about what is presented in
the assertoric sentence-form, whether it really contains an assertion.
And this question must be answered in the negative if the requisite
seriousness is lacking.
What kind of seriousness is Frege talking about here? It is not that the actor
or the poet aren’t serious in an ordinary sense, but that what they are saying
involves no commitment to truth; and a commitment to truth is a commitment
to the world. If I assert to you that my wife went to the neighbor’s, you may act
on it and look for her there. Not so with what an actor says on the stage.
17. Introduction to Mathematical Logic I, p. 9. Church says that his notion
of function is the notion of function in extension (pp. 15-16). In a function in
extension everything is fixed because it is a set of ordered pairs, but as I shall argue
below it is quite misleading to say that a function in extension fixes anything, or
that the sense fixes the reference due to a functional correlation between senses
and references. Earlier in the book (p. 6), when he introduces the notion of
sense, Church says: “Of the sense we say that it determines the denotation, or is
a concept of the denotation.”
18. These arguments are in pp. 59-60 of “Thoughts”, and Frege’s claim is really
that any attempt to define truth will beg the question:
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But could we not maintain that there is truth when there is correspondence in a certain respect? But which respect? For in that case
what ought we to do so as to decide whether something is true? We
should have to inquire whether it is true that an idea and a reality,
say, correspond in the specified respect. And then we should be
confronted by a question of the same kind, and the game could begin again. So the attempted explanation of truth as correspondence
breaks down. And any other attempt to define truth also breaks
down. For in a definition certain characteristics would have to be
specified. And in application to any particular case the question
could always arise whether it were true that the characteristics were
present. So we should be going round in a circle. So it seems likely
that the content of the word ‘true’ is sui generis and indefinable.
This argument is essentially a version of Zeno’s Dichotomy paradox – which says
that one can’t move because in order to get anywhere one must first get to the
halfway point. My claim is that even if it is accepted it doesn’t follow that truth
is not relational.
19. He says (Op. Cit., p. x):
How much simpler and sharper everything becomes by the introduction of truth values, only detailed acquaintance with this book can
show. These advantages alone put a great weight in the balance in
favor of my own conception, which indeed may seem strange at first
sight.
20. If two sentences being a logical consequence of each other is taken as a criterion
for these sentences expressing the same thought, then Frege’s conclusion follows
directly. He formulates such a criterion in a letter to Husserl (Philosophical and
Mathematical Correspondence, pp. 70-71):
It seems to me that an objective criterion is necessary for recognizing
a thought again as the same, for without it logical analysis is impossible. Now it seems to me that the only possible means of deciding
whether proposition A expresses the same thought as proposition B
is the following, and here I assume that neither of the two propositions contains a logically self-evident component part in its sense.
If both the assumption that the content of A is false and that of B
true and the assumption that the content of A is true and that of
B false lead to a logical contradiction, and if this can be established
without knowing whether the content of A and B is true or false,
and without requiring other than logical laws for this purpose, then
nothing can belong to the content of A as far as it is capable of
being judged true or false, which does not also belong to the content
of B; for there would be no reason at all for any such surplus in
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the content of B, and according to the presupposition above, such a
surplus would not be logically self-evident either. In the same way,
given our supposition, nothing can belong to the content of B, as
far as it is capable of being judged true or false, except what also
belongs to the content of A. Thus what is capable of being judged
true or false in the contents of A and B is identical, and this alone
is of concern to logic, and this is what I call the thought expressed
by both A and B.
The thoughts expressed by (9) and (10) satisfy the conditions that Frege lays
out here, and yet cannot be the same thought. Frege does not allow thoughts
that are neither true nor false in his logical system, but to restrict the criterion to
thoughts that are either true or false seems to me a considerable (and unjustifiable)
weakening.
In Begriffsschrift (§3) Frege gives a related criterion for two judgements
having the same conceptual content; namely that “the consequences derivable from
the first, when it is combined with certain other judgments, always follow also from
the second, when it is combined with these same judgments”. The interpretation
of this depends on whether one takes ‘derivable’ to be relative to some specific
rules and axioms, or takes it in the general sense of logical consequence.
21. The so-called “redundancy theory of truth”, defended by Ramsey in “Facts
and Propositions”, is often traced back to Frege’s views. Also Tarski’s semantic
conception of truth derives from the idea that such pairs as (6) and (7) – or,
rather, (7) and (8) – are materially equivalent. I argue in Chapter 7 that this idea
undermines Tarski’s account as a general semantic account of truth.
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Chapter 3
Use, Mention, and Russell’s Theory of Descriptions
A typical way of identifying an object is by means of some properties of the object that uniquely characterize it in a given context. The
properties used to identify an object in this way are often simply an aid to
identification. We may have no interest whatever in the fact that the object
has these properties aside from the fact that they help to identify it. This
is very common. Suppose, for example, that I want you to bring me my
copy of Word and Object from my study. I remember that I left the book
on top of the printer, and I say: Bring me the book that’s on top of the
printer in my study. I am not at all interested in the printer, or in the fact
that my copy of Word and Object is on it, except insofar that it helps me
to identify the book I want you to bring me. I’m interested in getting the
book, and I’m talking about it, and I use the fact that the book is on the
printer at that time to identify it for you. Intuitively, there is a reasonably
clear distinction between what we are talking about, what we want, what
we are interested in, etc., and the means we use to talk about it or to get
to it.
This distinction is related to the use-mention distinction, but the
latter is usually drawn in a very narrow way. One says, basically, that in
order to mention an object one can use a name or description of it, but
that this name or description is not mentioned, only used. If one wants
to mention the name or description, or linguistic expression generally, one
needs some device – quotation marks, for instance – to obtain a name or
description of it. It seems to me, however, that this is a rather superficial
analysis of use and mention and that the standard story about quotes is not
very convincing1 .
What is it that I use, after all, when I say ‘the book that’s on top
of the printer in my study’ ? Although I do use a linguistic expression, I also
use my mouth and my tongue. Why concentrate on the former rather than
on the latter? Not because it is a bunch of sounds, or marks, but because
it is an expression of something; namely, of a property, or characteristic, or
feature of the book at that particular moment. It is not important whether I
write down my instruction or utter it, nor is it important whether I use that
specific form of expression; the description ‘the book on the printer in my
study’ would have done just as well. It is not even important whether I use
this language or that, as long as we both know it, or whether I use a mixture
of language and gestures: ‘the book on the printer in there’, pointing to my
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study. In all cases what’s important is the property expressed: being the
book that’s on top of the printer in my study. This is a complex property,
involving the relations ‘is on top of’ and ‘is in’, the property ‘is a book’,
the printer, my study, and some logical properties and relations. Why not
say that I am using those things, as well as the linguistic expression – and
my tongue, my mouth, my larynx, etc. – to identify or refer to my copy of
Word and Object?
In fact, why should quotes name linguistic expressions? It is a
well established convention in logic that quotes are used to name the linguistic expression within them, and that they are a purely syntactic device,
but it is very hard to actually make sense of this. Is the quotation expression a name of the spatio-temporal (material) inscription within? Of the
inscription-type (in some sense)? Of the linguistic expression as a syntactic
expression of a specific language? Of the linguistic expression as a semantic
(i.e., meaningful, referential, etc.) expression of a specific language? Is the
linguistic expression tied up to a specific context of use? Take ‘Socrates’, for
instance. There were, and are, many people called that. When I say that a
philosopher called ‘Socrates’ was Plato’s teacher, and that a soccer player
called ‘Socrates’ was in Brazil’s national team in 1982, I may be actually
referring to a name as a linguistic expression independently of the specific
semantic content of that particular name – except for being a name, which
is really a semantic category rather than a syntactic category. But when I
say that ‘Socrates’ refers to the same thing as ‘the teacher of Plato’, I cannot take ‘Socrates’ to be a linguistic expression divorced from a context of
use. Although this last problem is partly a problem of ambiguity of names,
I think that the general problem has to do with the interpretation of the
quotes, not with the names and other expressions that are placed within
them.
It is undoubtedly true that what we place within quotes are linguistic expressions, or symbolic expressions of some sort, but the quotes are
meant to display this symbolic expression and not necessarily to refer to it
in a direct way. We can do this displaying in many different ways – using
colon, or a separate line, or a special font – and in none of these forms of
displaying it is at all clear that we are naming the symbolic expression that
is displayed, though we are referring to it in a broad sense of ‘referring’. In
some cases we may indeed intend to name it, but in most cases we display
the symbolic expression as an expression of something else.
Quotes are highly ambiguous and context dependent even as used
in philosophical and logical writing. I can perfectly well say that ‘human’
is an English word; that ‘human’ is a property (many philosophers refer to
properties in this way and I did earlier on); that ‘human’ is part of ‘Schu-
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mann’; and so on. We can use quotes in these different ways, and if not
clear from context alone, we normally indicate what we mean by tacking on
qualifications such as: the name ‘. . .’, the sentence ‘. . .’, the expression ‘. . .’,
the word ‘. . .’, the sequence of letters ‘. . .’, the term ‘. . .’, the phrase ‘. . .’,
the utterance ‘. . .’, the inscription ‘. . .’, the property ‘. . .’, the notion ‘. . .’,
the operation ‘. . .’, the concept ‘. . .’, etc. We also qualify these qualifications
by combining them in various ways and by adding references to specific languages, sciences, religions, sports, etc. To make the convention that quotes
are used to name the linguistic expression within them, in some explicit
and unambiguous sense of the terms ‘linguistic expression’ and ‘name’, is to
paint oneself into a corner, because it is practically impossible to stick to the
convention consistently except (perhaps) in very restricted contexts. And
that’s not because we aren’t clever enough, but because that’s not what
quotes are used for.
So, going back to my copy of Word and Object, the linguistic expression ‘the book that’s on top of the printer in my study’ that I used is
not the only thing that’s involved in my identification of the book. It isn’t
even the main thing, just as my mouth, and tongue, and larynx aren’t the
main thing. I can properly say that the main thing I used is the property of
being the book that’s on top of the printer in my study, and that in using
this property I have used the printer, my study, and the various properties
and relations that I mentioned earlier.
To identify something we describe it in a certain way, and what we
use to do this are properties of that thing that may involve various objects
and other properties and relations. The linguistic expressions and gestures
that we use are an expression of that, and we shouldn’t assume that when
we refer to our description, by quotation or by other means, we are referring
to the linguistic expression in order to name it.
Not only definite descriptions but also declarative sentences are
used to identify. If I say that I am in my study right now, I am identifying a
certain state of affairs. If I say that my copy of Word and Object is on top of
the printer, I am identifying another state of affairs. These states of affairs,
and my descriptions of them, involve objects, properties, and relations, and
it is important to distinguish what’s part of the state of affairs itself and
what’s part of the description I use to identify it.
Consider the sentence:
(1) Quine speaks Portuguese.
This sentence identifies a state of affairs that consists of a certain individual,
Quine, having a certain property, speaking Portuguese, and it is reasonable
to think of the state of affairs as having these two aspects (or components)
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that are identified by my use of the name ‘Quine’ and of the expression
‘speaks Portuguese’.
In communicating with other people, however, we have to take
into account the fact that we don’t share all information. What’s more,
we share different information with different people. So, if I want to identify the state of affairs, or circumstance, of Quine’s speaking Portuguese to
some other person, and I want to be informative relative to that person’s
knowledge and momentary interest, I may have to use different descriptions
that identify that same state of affairs. I might say, for example,
(2) The man you saw browsing in the bookstore yesterday speaks Portuguese,
or,
(3) The philosopher who lectured at MIT last Thursday speaks Portuguese,
or any number of other sentences.
What Frege’s principle (R) of substitutivity of reference does,
among other things, is to assure us that by using these alternative descriptions of Quine I am still identifying the same state of affairs. I don’t
change the subject, so to speak. My descriptions are geared to the information that the other person has through my knowledge of it. So I try
to pick a description which uses some properties that I believe identify the
object for that person. The point, however, is that these properties may be
connected only in a casual way to what I’m trying to identify. They may
play a merely auxiliary role. So in (2) I use the bookstore, and the fact
that Quine was browsing in it yesterday, and in (3) I use MIT, and the fact
that Quine lectured there last Thursday, without thereby making that the
subject of conversation. Parts of my descriptions, of course, mention the
bookstore, the lecture, MIT, yesterday, etc., but these things are not part
of the subject-matter. A friend may have asked me whether I know anyone
who speaks Portuguese, and since I know that he doesn’t know Quine by
name, and I want to be more informative than a simple ‘Yes’ answer, I
identify Quine for him by means of one or another description to identify
the circumstance that Quine speaks Portuguese.
What is used to identify Quine with those descriptions is not
merely the linguistic expressions ‘the bookstore’, ‘MIT’, etc., but the things
themselves, through those expressions of them. We use things and properties to identify other things and properties that we want to mention in this
more general sense of use and mention. This happens all the time, and it’s
quite irritating and disconcerting when someone turns something we are using to describe something, or to make a point, into the subject of discourse.
Thus, if you say ‘the man over there drinking champagne stole my wallet’,
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and the other person replies ‘yes, champagne is a wonderful drink; let’s get
some’, you feel at a loss.
Of course, there are also circumstances in which I do want to use
a description that is pertinent to the subject-matter in a more direct way.
You may claim that no author of a major philosophical work in English
speaks Portuguese, and I may counter by saying that the author of Word
and Object does. In this case my description of Quine as the author of Word
and Object is quite relevant to the subject-matter. But in this case the state
of affairs that I am describing is more complex. I’m saying that the author
of Word and Object speaks Portuguese and (implicitly, presumably using
your knowledge) that Word and Object is a major philosophical work in
English. The state of affairs described involves Quine’s speaking Portuguese,
being the author of Word and Object, and Word and Object being a major
philosophical work in English. But in other circumstances my use of the
description ‘the author of Word and Object’ may have only an auxiliary
role and I may just be referring to the circumstance of Quine’s speaking
Portuguese and identifying him as the author of Word and Object.
I would summarize my view of (1)-(3) roughly as follows. By
asserting (1) I mean to refer to a state of affairs by describing it. My
assertion is about Quine, and I am saying of him that he has the property
of speaking Portuguese – i.e., I am asserting a property of an object. In
my linguistic description of the state of affairs I use the name ‘Quine’ to
identify Quine and the linguistic expression ‘speaks Portuguese’ to identify
the property of speaking Portuguese. I would say the same thing about
(2) and (3), except that there I am identifying Quine in different ways
using various properties and relations that Quine has to other objects, and
using different linguistic expressions of this. These properties, relations, and
objects are not part of the state of affairs to which I mean to refer, however,
but are only part of my identification of it. The linguistic expressions ‘the
man you saw browsing in the bookstore yesterday’ and ‘the philosopher who
lectured at MIT last Thursday’ in the context of (2) and (3) refer to Quine
by expressing properties that identify him in a certain way.
With these remarks as introduction let me now turn to theories of
definite descriptions, especially Russell’s theory of descriptions. As Quine
said about modal logic, I think that Russell’s theory of descriptions was
conceived in sin: the sin of confusing use and mention.
The basic idea of Russell’s theory is that (definite) descriptions,
appearances to the contrary, do not name or designate things, and that sentences involving descriptions have a logical structure that is different from
their grammatical structure. Thus, the sentence
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(4) The author of Word and Object speaks Portuguese,
which appears to be of subject-predicate form, is not really so from the
logical point of view. According to Russell, the description
(5) the author of Word and Object
does not name an individual, and therefore (4) does not predicate of that
individual the property of speaking Portuguese. Nevertheless (4) is meaningful and true, and its proper logical analysis is:
(6) ∃x(x is an author of W &O & ∀y(y is an author of W &O ⇒ y = x)&x
speaks Portuguese).
In general, given any sentential context containing a description
(7) . . . ιxF x . . .,
Russell’s theory says that this context is to be analysed as
(8) ∃x(F x & ∀y(F y ⇒ y = x) & . . . x . . .),
which is also formulated as
(9) ∃x(∀y(F y ⇔ y = x) & . . . x . . .),
or as
(10) ∃!xFx & ∀x(F x ⇒ . . . x . . .),
where ‘∃!x’ is read ‘there is a unique x’. If the context in which the description occurs has logical structure, then there are many different ways to
eliminate the description and one has to introduce the notion of scope of
the description – this is very important but it will not be relevant for my
immediate considerations.
Now why do I say that Russell’s theory confuses use and mention?
Because when we apply Russell’s theory to such contexts the properties
used in our identifying descriptions and the property that we are predicating become the subject-matter of our discourse2 . Thus, on a natural
reading of (6) the property of being an author of Word and Object and the
property of speaking Portuguese are the subjects of our assertion; we are
talking about them. If we apply Russell’s theory to (2) and (3), the properties of browsing in the bookstore yesterday and of giving a lecture at MIT
last Thursday also become our subject-matter together with the property
of speaking Portuguese. And the answer you got when you were concerned
about your wallet being stolen is entirely logical according to Russell’s theory. You said:
(11) ∃x(x is a man & x is over there & x is drinking champagne & ∀y((y
is a man & y is over there & y is drinking champagne) ⇒ y = x) & x
stole my wallet),
and there is no logical reason why your interlocutor should think that the
stealing of the wallet is of more concern to you than the drinking of the
champagne or the being over there. The distinctions that I have been trying to emphasize simply vanish in Russell’s analysis of these sentences. (Of
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course, if one knows where (11) came from, then one can make the distinction, but not thanks to Russell’s analysis.)
In fact, I think that there are confusions of use and mention in all
theories of description – at least in those that are normally used in logic
and philosophy, which are variations on Russell’s theory and on Frege’s3 .
Frege’s view is that descriptions are used to denote (or name)
objects, and he would analyse (1)-(4) pretty much as I did as involving
different ways of identifying (or presenting) Quine, who is the subject of
these sentences. The problem comes when we take descriptions that do
not denote anything because the property used in the description does not
apply to a unique object. Frege thought that a sentence containing such
descriptions is neither true nor false, but he saw this as an unwanted feature
of ordinary language. In a well designed language for logic, mathematics, or
science, there should be no non-denoting expressions, and he sought to avoid
them in his own formulations of logic by characterizing a formal counterpart
of descriptions in such a way that a denotation is always assured4 .
The way this is usually done in specific formal languages that include a description operator is through a convention that for properties F
which do not apply to a unique thing, the description ‘ιxF x’ denotes some
arbitrarily specified object. In number theory the object may be 0, and in
set theory it can be taken to be the empty set or the set of objects that
have the property F. The description could also be taken to be undefined,
however, as for natural languages, but then one must come to terms with
sentences that are neither true nor false, which is the main reason why this
is generally avoided.
Russell’s theory can be applied equally well to natural languages
and to formal languages, which is one of its advantages over Frege’s. But
from a purely formal point of view Frege’s technical solution has the advantage of not requiring distinctions of scope, which make Russell’s theory
rather messy in practice.
The question of scope has nothing to do, really, with the description operator, but is rather a matter of the logical structure of the sentence
or formula in which the description occurs. Thus, if we take a negative
sentence of the form
(12) ¬Ga,
there are two interpretations of its logical structure. One has negation as a
predicate operator
(13) [¬Gx](a),
and the other has negation as a propositional operator
(14) ¬(Ga).
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If what we’ve got instead of ‘a’ is a description ‘ιxF x’, and we apply Russell’s theory, we get
(15) ∃x(F x & ∀y(F y ⇒ y = x) & ¬Gx)
and
(16) ¬∃x(F x & ∀y(F y ⇒ y = x) & Gx),
respectively, which are not logically equivalent and will have different truth
values if F does not apply to a unique object – (15) will be false and (16)
true. If descriptions always denote, however, then what we get following
Frege is
(17) [¬Gx](ιxFx)
and
(18) ¬(G(ιxFx)),
which do have the same truth value.
As the sentences get more complex they are more ambiguous as
to logical structure, and if we distinguish the different structures precisely
we get all the distinctions of scope that are necessary for Russell’s theory.
If we have two negations, for example, then we will have to distinguish the
three forms
(19) [¬¬Gx](ιxF x),
(20) ¬([¬Gx](ιxF x)),
and
(21) ¬(¬(G(ιxF x))).
My view is that both Frege and Russell were wrong in their analyses of sentences involving descriptions, but that a more adequate theory can
be obtained that combines some aspects of each of their theories. Let me
begin to argue for this in terms of a simple case that is quite central to all
theories of description.
Consider statements of the form:
(22) a is the F.
According to standard logical practice, also emphasized by Frege and by
Russell5 , statements of this form are to be analysed as:
(23) a = ιxFx.
Thus,
(24) Quine is the author of Word and Object.
is to be analysed as:
(25) Quine = ιx(x is an author of W&O).
This is a statement of identity, and seems to describe the fact that Quine is
self-identical, although using two different identifications of Quine. This was
part of Russell’s motivation for denying that descriptions name or designate;
he felt that if the description in the right hand side of (25) named Quine,
then (25) would say exactly the same thing as
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(26) Quine = Quine
and would be a tautology6 .
Frege had faced this problem in “On Sense and Reference” and argued that the inference is fallacious. He held that in (25) Quine is presented
in different ways in the two sides of the identity statement and that there
is a fundamental difference in content between (25) and (26). Both state
Quine’s self-identity, but what they express is quite different due to the
difference in the identification of Quine in the right hand side. Frege puts
this in terms of the difference between sense and reference: (25) and (26)
have the same reference but different senses because the sense expressed by
‘Quine’ is different from the sense expressed by ‘the author of Word and
Object’. Thus Frege agrees with Russell that whereas (26) is (in Russell’s
terminology) a tautology, (25) is not. But Russell, who didn’t like Frege’s
notion of sense, felt that his analysis of descriptions gave a better and simpler account of the difference.
If we apply Russell’s theory of descriptions to (25) we get
(27) ∃x(x is an author of W &O & ∀y(y is an author of W &O ⇒ y = x) &
Quine = x),
which instead of referring to Quine’s self-identity describes an entirely different state of affairs involving the property of being an author of Word and
Object and of there being a unique such author. With this analysis Frege’s
distinction between sense and reference largely collapses7 , but Russell’s view
was that in (25) one is saying something about Quine’s being the author of
Word and Object rather than just using ‘the author of Word and Object’ to
identify Quine.
I entirely agree with this point of Russell’s if it is applied to (24),
but I agree with Frege on the analysis of (25) and (26) – although I would
say that (25) and (26) refer to a state of affairs, and to the same state of
affairs identifying it in different ways, while Frege would say that they refer
to the True. Where I disagree with both Frege and Russell is in the analysis
of (24) as having the form (25). In some cases one may actually mean (24)
as an identity statement, but I don’t think that this is the correct analysis
in general. In fact, (27) does give a better account of (24), but not because
it is a correct analysis of (25). If one means (24) as (25), then (27) is just
mixing things up.
Normally, when we say that a is the F we are predicating F of
a and we are also predicating uniqueness of a relative to that predication
of F. The proper way to read (22) is, as Russell himself says on several
occasions8 ,
(28) a is an F and nothing else is,
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which can be be expressed by
(29) F a & ∀y(F y ⇒ y = a)
or by
(29′ ) F a & ¬∃y(y 6= a & F y).
This can be seen as two predications about a, namely
(29-1) [F x](a)
and
(29-2) [∀y(F y ⇒ y = x)](a),
corresponding to the two parts of (28). Or, more accurately, as the single
predication
(30) [F x & ∀y(F y ⇒ y = x)](a)
which predicates is the F of a9 .
But (29) is logically equivalent to
(31) ∃x(F x & ∀y(F y ⇒ y = x) & a = x),
which is Russell’s analysis of (23). The difference is that on a natural reading
of (31) as predicating something of something this isn’t a predication about
a but, at least in part, a predication about F; we are predicating of F that it
applies to one and only one thing, and that that thing is a. Which suggests
that we can separate (31) into a part that predicates unicity of F and a part
that predicates F of a
(32) ∃!yFy & Fa,
which is logically equivalent to the alternative formulation of (31) as
(33) ∃!yFy & ∀y(F y ⇒ a = y).
Although there is a question as to what exactly one is talking
about and what one is predicating in these different formulations (31)-(33),
Russell’s formulation is better than Frege’s as an analysis of (22). In my
view, however, this is the result of two steps that are quite clearly incorrect;
namely, the analysis of (22) as (23) and the analysis of (23) as (31).
One can argue against Russell’s analysis (31) (and mine (30)) that
the uniqueness claim is not part of what is said in (22), but rather a presupposition of what is said. Frege did argue for something like this referring to
the example:
(34) Whoever discovered the elliptic form of the planetary orbits died in
misery.
His argument was that if the existence of a discoverer of the elliptic
form of the planetary orbits were part of what is asserted in (34), then the
negation of (34) should run:
(35) Either whoever discovered the elliptic form of the planetary orbits did
not die in misery or there was nobody who discovered the elliptic form of
the planetary orbits.
And a similar argument can be given for the statement
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(34′ ) The discoverer of the elliptic form of the planetary orbits died in
misery10 .
Although I agree with Frege on this particular example, his argument actually helps to confuse the issue even more by not distinguishing
statements of the form the F is G from statements of the form a is the F.
Take, for example,
(36) Russell is the author of Principia Mathematica.
I can certainly deny this on the grounds that Principia Mathematica was
written by Russell in collaboration with Whitehead and that therefore Russell is not the author of Principia Mathematica – i.e., he is not the only
thing that is an author of Principia Mathematica. Similarly, I can deny
that Quine is the author of Principia Mathematica on the grounds that
he isn’t an author of Principia Mathematica. So it seems that I can deny
something of the form a is the F either on the grounds that a is not an F
or on the grounds that a is not the only F; which corresponds precisely to
my analysis of (22) as (28)-(29) or (30).
This example also shows quite clearly, it seems to me, that Frege’s
analysis of statements of the form a is the F is untenable. For according to
Frege (36) is to be analysed as
(37) Russell = the author of Principia Mathematica,
which is neither true nor false because the description ‘the author of Principia Mathematica’ does not denote. Although I agree with this account of
(37), it just goes to show that it is an incorrect analysis of (36), because
(36) is clearly false.
Another source of misunderstanding in connection with this issue
derives from not drawing a clear distinction between predicate negation and
statement negation. If I take the predicate negation of (37)
(37 P¬) Russell 6= the author of Principia Mathematica,
this is also neither true nor false, for the same reason as before. But if I
take the statement negation
(37 S¬)¬(Russell = the author of Principia Mathematica),
in the sense of ‘it is not the case that’ or ‘it is not true that’, this is indeed
true, because (37) is neither true nor false, and hence not true, or not the
case. So in this sense I can deny (37) on the grounds that Russell is not the
only author of Principia Mathematica – which is what makes the description
non-denoting. But this does not justify the assertion of (37 P¬).
In my remarks about (36) I was using the predicate negation
(36 P¬) Russell is not the author of Principia Mathematica,
and the points I made about denial were in terms of the assertion of this
negation. For a statement of the form the F is G on the other hand, I cannot
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assert the predicate negation the F is not G on the grounds that there isn’t a
unique F. That’s precisely why Russell distinguishes the two negations and
denies statements of the form the F is G by means of statement negation.
The whole problem of the proper interpretation of descriptions is
confused because there is a fundamental difference between cases where the
description appears in subject position and cases where the description appears in predicate position. My view is that when the description appears in
subject position Frege’s referential analysis applies; but when the description appears in predicate position an analysis closer to Russell’s applies –
i.e., (29)-(30). Thus, there is an important asymmetry between statements
of the form (22) and statements of the form
(38) the F is G.
I agree with Frege that when we use something of the form (38)
the unicity of F is presupposed and we are predicating G of the referent of
‘the F’, if any. In these cases the F is used to identify something, namely the
thing that is F, if there is such a thing, and identifies nothing if there isn’t
a unique thing to which F applies. But here ‘the F’ is in subject position,
not in predicate position.
When ‘the F’ is in predicate position, as in (22), then the situation
is quite different, because we do in fact deny that a is the F either on the
grounds that a is not the only F or on the grounds that a is not an F.
But this doesn’t mean that F is part of the subject of discourse, as in
Russell’s analysis, nor that we are presupposing that there is a unique F,
as Frege suggests. We are predicating ‘is the F’ of a, in the sense that
we are predicating of a that it is an F and that it is the only thing that
is an F. Russell’s analysis gets closer to this than Frege’s, but by changing
the subject of discourse it also confuses use and mention. The difference
between (30)-(33) is that if we ask what is being said of what, in each case
we get a different answer11 .
Another interesting case is when we take statements of the form:
(39) the F is the G.
Standard practice is, as with (22), to interpret (39) as an identity statement. This is justified for those contexts where it is explicitly or implicitly
clear that what is asserted is that the F is the same as the G, because here
both the F and the G are subjects of the identity statement, but it is not
generally justified otherwise. In my view, therefore, we can distinguish the
different interpretations:
(40) the F = the G,
which can be analysed as
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(41) [x = y](ιxF x, ιxGx),
and
(42) the F is G and nothing else is,
which can be analysed as
(43) [Gx & ∀y(Gy ⇒ y = x)](ιxF x).
Frege’s theory gives a correct analysis of (41), but Russell’s theory analyses
(40) as
(44) ∃x((F x & ∀z(F z ⇒ z = x)) & ∃y((Gy & ∀z(Gz ⇒ z = y)) & x = y)),
which corresponds to a third reading of (39) as
(45) something is the F and something is the G and they are the same thing,
where is the F and is the G are both interpreted predicatively. Although I
think that statements of this form are perfectly legitimate, I do not think
that (44) is a correct analysis of (40)12 .
It is interesting that Russell’s basic insight about descriptions was
essentially an insight about use and mention. He saw quite clearly that
when one says something of the form a is the F, one is saying something
that involves F and not merely using F to identify a; which led him to
insist that the proper analysis of something of that form is given by (31)
– which was actually based on (28)-(29), as shown by the passages I quote
in notes 8 and 13. And in this case it worked reasonably well because the
description is in predicate position. That led him to the view that every
occurrence of the F in a sentence must be analysed as in (8). As a general
analysis of sentences with the F in subject position this is obviously wrong,
and everybody knows that it is wrong, but since it worked so well for (22)
it must be right. Thus he concluded that we are deceived by the surface
grammar and that we should trust the logical grammar13 .
Allegations that we are deceived about something are hard to answer, because they carry an imbedded mechanism that deflates our most
reasonable arguments – see, I told you, you are deceived. And if all our
friends and relatives tell us that we are deceived, we may end up agreeing
that we are deceived. What I am arguing is that in this particular case
we have been deceived as to the deceiving, and that Russell’s theory of descriptions does not give an adequate logical analysis of the use of definite
descriptions precisely because it does not distinguish descriptions in subject
position from descriptions in predicate position. And the same applies to
Frege’s theory. Russell had good insights about descriptions in predicate
position and generalized them to descriptions in subject position; Frege had
good insights about descriptions in subject position and generalized them
to descriptions in predicate position14 .
My view is that there are two distinct uses of ‘the F’ that are not
distinguished explicitly but that show themselves in these different theories
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of description and that are clear from the “surface” grammar. In terms of
the remarks about logical structure in Chapter 1 we can say that on the
one hand, the is a logical operator that applied to a first order predicate (or
property) F that applies to a unique thing gives that thing as result, and is
undefined otherwise. And on the other hand, the is a logical operator that
applied to any first order predicate (property) F yields another predicate
(property)
(46) [F x & ∀y(F y ⇒ y = x)](x),
which can apply to at most one thing and which extensionally coincides
with F if F is uniquely satisfied. Thus, the first operator goes from level 1
properties to level 0 objects, whereas the second operator goes from level 1
properties to level 1 properties. (The same analysis applies to higher orders.)
When the expression ‘the F’ occurs in subject position we generally have
the referring use – and it is this referring use that the description operator
‘ιxF x’ is normally meant to indicate. But when ‘the F’ occurs in predicate
position one is predicating the property (46) of the subject of discourse.
One can think of this use of ‘the F’ as an operator ‘!’ which applied to F
yields the property expressed by (46) as result – one could then write (46)
as
(46′ ) [!xFx](x).
Since the usual logical notation is extremely ambiguous with respect to subject and predicate positions, one ends up taking a uniform
interpretation which accounts for some cases but not for others. Thus Frege
takes a uniform referential interpretation and Russell takes essentially a
predicative interpretation. I think that a correct account must distinguish
the two aspects and that it is because of this that neither of these theories
works very well15 .
An important connection between the two uses of ‘the F’ I distinguished above can be stated by saying that the property (46) is the sense
expressed by the term ‘ιxF x’. Although for Frege the sense expressed by
a description ‘ιxF x’ (which for him is a name) is not a property but an
object, it is not at all clear which object this is. It is clear from his informal
explanations, however, that the sense expressed by ‘ιxF x’ is the manner
in which ‘ιxF x’ presents its reference. But how does ‘ιxF x’ present its
reference? It presents it as being an F which is unique, which is precisely
the content of (46). By taking the property (46) as the sense expressed by
‘ιxF x’ one can see the difference between Frege’s analysis and Russell’s in
that Russell substituted for Frege’s referential analysis of ‘ιxF x’ an analysis
in terms of the sense expressed by the description16 .
Another important connection between Frege’s analysis and Russell’s is that whereas they are not tv-logically equivalent, they are c-logically
106
equivalent – in the sense defined in (12) and (13) of Chapter 2. Differences
in truth value arise only when the property F does not apply to a unique
thing, in which case Frege’s analysis is truth-valueless and Russell’s analysis (with largest scope) is false. This means that in these cases the two
analyses are not materially equivalent. The analysis I suggested is also clogically equivalent to Frege’s and Russell’s, and agrees in truth value with
Frege’s when a non-unique description has at least one occurrence in subject
position, and with Russell’s when a non-unique description only occurs in
predicate position17 .
In my discussion so far I have not taken into account the philosophical content and aims of Russell’s theory. This is important because
Russell’s theory was neither primarily a theory about linguistic use nor simply a logical theory in a narrow formal sense, and its philosophical content
and aims may well override the objections that I have brought against it in
the last few pages. What were, then, Russell’s philosophical aims?
First and foremost was the problem of non-being: how to speak of
what is not? The theory was meant to solve this problem while preserving
the standard laws of classical logic, especially the principles of contradiction
and of excluded middle. Russell objected both to the Meinongian solution
that posited being of some sort for those cases were ordinary being was
missing – on account that, among other things, it violated the principle
of contradiction, – and to the Fregean solution that, for ordinary language,
denied truth values to sentences containing non-denoting terms – on account
that it violated the principle of excluded middle18 .
Russell saw his theory of descriptions as a brilliant application of
Ockham’s razor, the principle that calls for the elimination of superfluous
entities. He hardly ever mentions the theory without emphasizing this feature of it19 . And in this case the alleged entities were not just superfluous,
but were actually quite troublesome: the round square, the golden mountain, the present King of France, the winged horse captured by Bellerophon,
etc. Of course, it seemed that he was throwing the baby out with the bathwater, because out went also the author of Waverley, the first emperor of
the French, the teacher of Plato, the milkman, etc. What one has to show
is that one can throw out the bathwater and save the baby, which Russell
tried to do by his eliminative analysis of descriptive phrases.
This eliminative analysis was mainly justified by Russell’s epistemological distinction of knowledge by acquaintance and knowledge by
description. All statements must be reduced to statements that are about
entities with which we are acquainted. These entities are particulars (sense
data, conscious states, memories) and universals. Knowledge by description
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(of particulars and universals) is an essential extension of knowledge by acquaintance but must be founded on it20 . Thus, the philosophical content
and aims of Russell’s theory of descriptions were metaphysical, involving
logical, ontological, and epistemological considerations as well as linguistic
considerations.
Was Russell successful in achieving his primary aims? Well, in a
sense he was. His theory definitely maintains the principles of contradiction
and of excluded middle, at least in a certain form. As to the problem of
non-being, he solved it by means of two main moves. One was the theory of
descriptions itself – and the theory of incomplete symbols more generally –
which solves the problem of non-being for non-denoting descriptive phrases.
The other was the idea that proper names abbreviate definite descriptions.
This does yield a solution, because if ‘Pegasus’ abbreviates the description
‘the winged horse captured by Bellerophon’, then
(47) Pegasus flies,
which gets analysed as
(48) ∃x(x is a winged horse & x was captured by Bellerophon & ∀y((y is
a winged horse & y was captured by Bellerophon) ⇒ y = x) & x flies),
is meaningful and false, and
(49) Pegasus doesn’t fly,
which with negation as statement negation gets analysed as
(50) ¬(∃x(x is a winged horse & x was captured by Bellerophon & ∀y((y
is a winged horse & y was captured by Bellerophon) ⇒ y = x) & x flies),
is meaningful and true. And if we analyse (49) with negation as predicate
negation, then we get
(51) ∃x(x is a winged horse & x was captured by Bellerophon & ∀y((y is
a winged horse & y was captured by Bellerophon) ⇒ y = x) & x does not
fly),
which is meaningful and false. (50) saves the principle of excluded middle,
and the alternative analyses (50) and (51) are not really in conflict with the
principle of contradiction.
It is important to realize that this abbreviative analysis of names,
and the distinction between knowledge by acquaintance and knowledge by
description, is an essential component of Russell’s solution to the problem
of non-being. The crucial move is to deny that names and descriptions can
be used to denote things, or to name things, unless it is impossible that
there should be failures of denotation. If a name or description could be
used referentially, and one could be wrong about there being a reference,
then there might (and normally will) be cases in which one is wrong, and in
such cases our statements will be meaningless, or nonsensical. They will be
meaningless because the only function of an expression used referentially as
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a name is to name something, and therefore a name that doesn’t name has
no content whatever; it is a mere noise. But Russell held the very natural
view that statements containing ordinary names and descriptions which do
not denote according to the referential view are not nonsense; from which
he concluded that the referential view was wrong and that ordinary names
and descriptions are not really names in the strict sense.
Frege had actually reached a similar conclusion for names that do
not denote, and referred to them as mock proper names, from which he
concluded that in a logical language one must guarantee that all names and
descriptions denote. Since Russell saw Frege’s technical solution as being
artificial, and as avoiding the philosophical problems, he concluded that
one must eliminate the denoting function of all expressions except those
for which one can make a general (and somewhat “absolute”) logical and
philosophical case that they cannot fail to denote. Among these Russell
included what he called “logical proper names”, like ‘this’ and ‘that’, and
also (in a sense) the variables of quantification21 .
Russell’s intuitions about sense and nonsense were very closely
tied up to his intuitions about truth and falsity. He identified ‘not true’
with ‘false’ and held that a statement that is neither true nor false must be
nonsense22 . Thus his insistence in preserving the propositional form of the
principle of excluded middle and in analysing every (intuitively) meaningful
statement in such a way that it couldn’t fail to be either true or false. Given
these constraints, and the aim of solving the problem without appealing to
possible and impossible entities, Russell’s solution is philosophically quite
powerful.
But the elimination process for proper names is only a reductive
process, and for Russell’s solution to be successful it must be made plausible
that for every sentence the process terminates with a sentence containing no
terms for which problems of denotation can arise. This is not at all obvious,
however, and as we saw in the example I used above, the elimination of the
name ‘Pegasus’ led to the introduction of the name ‘Bellerophon’. What is
to guarantee that the continuing elimination process will not lead to circularities, for example? To someone who acquired the name ‘Bellerophon’ in the
context of certain philosophical discussions about the theory of descriptions,
as I suppose many of us did, ‘Bellerophon’ may simply abbreviate ‘whoever
captured Pegasus’ and the circle may close right there. Such problems give
reasonable grounds for doubt as to whether the elimination process can be
considered to be well-defined23 .
Some of the main evidence for Russell’s abbreviative theory of
names, and for the theory of descriptions itself, came from the analysis
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of singular existence and non-existence statements, where the problem of
non-being appears in the sharpest form. There is a use of the F in subject
position that doesn’t seem to be analysable as a denoting term, and that’s
when one says
(52) the F exists,
or, especially,
(53) the F does not exist.
As Frege before him, Russell maintained that existence cannot be
predicated of objects, and that if names stand for objects, then it wouldn’t
make sense to say such things as ‘Quine exists’, ‘Scott exists’, ‘France exists’,
etc. And if ‘the author of Word and Object’ is used as a name, then it
wouldn’t make sense to say that the author of Word and Object exists. But
this does make sense, and it is perfectly true according to Russell, because
what it means is
(54) ∃!x(x is an author of Word and Object).
Hence, in general, the proper analysis of (52) is
(55) ∃!xFx
and the proper analysis of (53) is
(56) ¬∃!xFx.
This has far reaching implications, because it also makes sense to
wonder whether Homer existed, whether God exists, etc., and this wouldn’t
be so if ‘Homer’, ‘God’, etc. were really names; so they must be abbreviated
descriptions to which the analysis in (55) and (56) can be applied. Thus
‘Homer’, for example, must abbreviate a description such as ‘the author of
the Iliad and the Odyssey’. Then to say
(57) Homer exists,
is to say
(58)∃!x(x is an author of the Iliad and the Odyssey),
and to say
(59) Homer does not exist,
is to say
(60)¬∃!x(x is an author of the Iliad and the Odyssey).24
One can thus truly deny that the present King of France exists, that the
round square exists, that Pegasus exists, that Sherlock Holmes exists, etc.
Although I agree that this is a very clever uniform solution to the
problem, it seems to me that its main aspects can be preserved without
appealing to the two rather unintuitive and problematic theses that definite descriptions are never used to name things, or to refer to things, and
that ordinary proper names always abbreviate definite descriptions – or are
meant as such an abbreviation.
110
Let me discuss first the question of treating existence as a predicate
(or property). What are the objections to this? One main objection is that
‘exists’ would be a sort of trivial predicate that would apply to everything,
and hence not a really significant predicate since it would have no content25 .
This objection applies equally well to self-identity, however, and logicians
have no qualms in using self-identity as a predicate. One can actually define
(61) x exists
as
(62) x = x,
in the sense that the first predicate applies to something if and only if
the second predicate applies to that thing26 . Moreover, if one follows-up
on Frege’s idea that the quantifiers are higher-level properties, such trivial
predicates (or properties) will arise everywhere in logic. The property of
self-subordination, for example, applies to every level 1 property – i.e., the
predicate
(63) [∀x(Zx ⇒ Zx)](Z)
applies to every level 1 property – and could thus be used to define the
predicate
(64) Z exists.
Unless one wants to do major surgery among logical predicates therefore,
one cannot eliminate ‘exists’ as a bona fide logical predicate. But even
aside from this I don’t think that the charge of triviality (and alleged nonsignificance) that would justify such elimination is correct.
I would argue for this on the grounds that Frege argued in “On
Sense and Reference” for the non triviality of true statements of identity of
the form ‘a = b’. A statement of the form
(65) a exists,
where ‘a’ stands for a name or description, is true if and only if the name
denotes, and whether this is true or not may involve hard empirical or
non-empirical research and be quite informative. If instead of a name we
have a description the F, then this research will naturally be based on F,
and it is generally the case that (52) is true if and only if (55) is true, as
Russell claims. For a name the research will involve the manner in which
the name is connected to its reference, if any, but this does not mean that
the name must abbreviate a definite description for such a connection to be
investigated27 .
Another main objection to allowing ‘exists’ as a predicate is precisely the problem of non-denoting names and descriptions. What happens
with existential statements involving names and descriptions that do not denote? My view is that when ‘exists’ is used as a predicate such statements
are neither true nor false. But then we have to deal with two further objec-
111
tions. One is Russell’s objection that this violates the principle of excluded
middle; the other is that it does not account for true negative existentials.
As I pointed out before, however, negative statements are generally ambiguous in that the negation can be interpreted either as predicate
negation or as statement negation. Thus, whereas
(66) [¬exists x](Pegasus)
is neither true nor false,
(67) ¬([exists x](Pegasus))
is true. And similarly for the two negations of (47), for example. So the
propositional form of the principle of excluded middle is maintained in virtue
of the distinction between predicate negation and statement negation.
Russell cannot really object to this solution because it is very similar to the kind of solution he offers for the principle of excluded middle. The
solutions depend on his distinctions of scope. The difference is that whereas
he sees the problem of scope as being tied up to the notion of description,
I see it as a very general problem about the logical structure of sentences.
What makes the case of true singular non-existence statements
rather peculiar is that whenever a name or description is used in subject
position there is a strong feeling that we are asserting something of something, which is what Frege maintained with his claim that there is a general
presupposition that the name or description denotes. Yet with a singular
non-existence statement we cannot be truly asserting non-existence of something. This is a central aspect of the paradox of non-being, and it is natural
to hold that if we are indeed truly asserting something of something, these
two somethings cannot be what they appear to be – i.e., that the statement must have an analysis in terms of a different subject and a different
predicate.
When the subject is a definite description the F Russell’s solution
for this alternative analysis is indeed very natural; we are asserting of F
that it is not uniquely instantiated. Although there seems to be a change
of subject, that’s precisely what we are looking for. And it is also quite
reasonable in the case of some uses of proper names where what is meant
by the denial can be naturally expressed by means of a description. The
problem is that in order to get a uniform solution for all cases Russell was
led to the two very radical generalizations I mentioned before that wipe out
all denoting uses of names and descriptions.
I will argue now that the sort of approach I am suggesting is better
than Russell’s in the sense that I can take advantage of Russell’s insights,
and Frege’s, and avoid some implausible consequences of their views. Since
I have already argued for this in connection with the distinction between de-
112
scriptions in subject position and in predicate position, I will concentrate on
names and descriptions in subject position and on the question of existence
statements.
To begin with, the distinction between predicate negation and
statement negation is already one of Russell’s insights that is essential for
his solution to the problems we are discussing. I see the difference between
these two kinds of negation as a change of subject that involves a change in
truth conditions. In predicate negation the subject remains the same, and
a predicate negation is true (false) if and only if the negated statement is
false (true). Thus predicate negations of truth-valueless statements are also
truth-valueless. In statement negation we are denying that something or
other is the case, and the subject now becomes the content of the statement
that is denied. A statement negation is true if and only if the statement
denied is not true – i.e., is either false or truth-valueless. Hence a statement
negation is always either true or false. I think that our intuitions about the
truth and falsity of statements involving non-denoting names and descriptions can often be justified as intuitions based on statement negation.
I conclude from this that the distinction between predicate negation and statement negation is enough by itself to give us a uniform solution for handling statements involving names and descriptions which do
not denote. Moreover, by allowing ‘exists’ as a predicate this solution also
handles statements of existence and non-existence in a uniform way. But
these statements do raise additional questions.
I quite agree with Frege that when we use a name or description
in subject position we normally presuppose that this name or description
denotes. In fact, it is this presupposition that leads us to disregard the distinction between predicate negation and statement negation, because when
the name or description denotes there is no difference in truth value. The
case of singular non-existence statements is especially puzzling (or paradoxical) precisely because the normal presupposition is in direct conflict with
what is asserted. If I assert that Pegasus doesn’t exist, I can’t be presupposing that ‘Pegasus’ denotes something of which I am denying existence.
Although some philosophers do follow this route, it seems rather paradoxical – as Parmenides and Plato as well as Russell and Quine emphasized.
One can indeed put the blame (or part of the blame) on the use of ‘exists’ as
a predicate, as Russell and Quine do, but I don’t think that this is the right
tack. What one must do, it seems to me, is recognize the logical character
of ‘exists’, which is to be universally applicable. From which it follows that
as self-identity, self-subordination, and other logical predicates that have
this universally applicable character, existence (in this sense) can never be
truly denied.
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In fact, it is curious that Quine rejects ‘exists’ as a legitimate
predicate28 , because he has himself forcefully made the case for the logical
universal character of exists with the famous quip with which he opens “On
What There Is”:
A curious thing about the ontological problem is its simplicity. It
can be put in three Anglo-Saxon monosyllables: ‘What is there?’ It
can be answered, moreover, in a word – ‘Everything’ – and everyone
will accept this answer as true. However, this is merely to say that
there is what there is. There remains room for disagreement over
cases; and so the issue has stayed alive down the centuries.
What Quine’s answer ‘Everything’ amounts to is
(68) ∀xx exists,
and I would say that everyone should indeed accept it as true precisely
because it is logically true. I see no good reason why the existential quantifier
should be considered a logical notion (or a logical property of properties)
and ‘exists’ as a predicate (or property) should not29 .
If we recognize the logical character of ‘exists’, it follows that in
many contexts our statements of existence or non-existence should not be
taken literally – just as we don’t take literally such pronouncements as ‘I am
not myself’ or ‘I am who I am’. But as long as we allow for truth-valueless
statements, it doesn’t follow that we cannot make such statements literally
or that there is anything especially problematic in making them.
Which brings me back to the analysis of statements of forms (52)
and (53). Russell’s analyses (55) and (56) are actually quite compatible with
my distinction between descriptions in subject position and descriptions
in predicate position, because my analysis allows an existential quantifier
interpretation of the ‘exists’ as well as a predicate interpretation of the
‘exists’. In the predicate interpretation the description must be interpreted
referentially, but in the quantifier interpretation the subject of the quantifier
must be a property, and the natural property to take is the property that I
expressed as (46). I think, in fact, that this gives an even better justification
for Russell’s analysis because the uniqueness claim derives from the nature
of the property that is existentially quantified. I.e., (55) can be expressed
as
(69) [∃xZx]([F x & ∀y(F y ⇒ y = x)](x)),
and (56) as
(70) [¬∃xZx]([F x & ∀y(F y ⇒ y = x)](x)).
That the existential quantifier reading is a very natural reading
can be seen from the fact that we often rephrase statements of forms (52)
and (53) as
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(71) there is such a thing as the F,
and
(72) there is no such thing as the F,
which formulated with a variable are
(73) there is an x such that x is the F,
and
(74) there is no x such that x is the F.
Analysing ‘x is the F’ as in (30) and symbolizing the quantifier and the
negation we get (69) and (70).
Even so, I think that there is no compulsion to interpret statements of the forms (52) and (53) as (69) and (70), because somebody may
literally mean to use ‘exist’ as a predicate – in which case my earlier analyses apply. Moreover, as Donellan has argued, there are circumstances in
which we can use a definite description to refer to a thing even though the
property used in the description does not individuate anything. The man
who stole your wallet may have been drinking ginger ale rather than champagne, but, nevertheless, you may succeed in referring to him by means of
the description ‘the man over there drinking champagne’30 . I think that
Donellan is quite right about this and that it has a bearing on the question
of existence statements as well.
Suppose that we are lost in the desert and have been “seeing”
oases all over the place. I say: It’s no use, it’s all an illusion. And you
reply: No, the oasis over there with the three camels exists. The property
you used may not individuate – they are horses rather than camels, say –
but you may have succeeded in referring and your statement may be quite
true. On the other hand you may be wrong, in which case we have a strong
intuition that what you said is false. But that’s partly because we generally
equate ‘not true’ (and ‘wrong’ in this sense) with ‘false’, which again largely
derives from the normal presupposition that our terms refer. I can account
for the intuition of falsity in this way, because it is true that what you said is
wrong (or not true); but I do agree that many intuitions get mixed up here.
Russell’s intuition is one of these, but his analysis does have the drawback
that it will make your statement wrong even if it was right.
And there are other intuitions that are relevant as well, because
even if one states an existential denial as (72), there is no compulsion to
interpret it as (74). By (72) somebody may mean (e.g., may paraphrase it
as)
(75) ‘the F’ does not denote anything.
This is also an interesting paraphrase, because if by “the F” one means to
be referring to the property of being the F, then (75) is again (70) – or close
– whereas if by “the F” one means to be referring to the singular term, then
115
(75) is rather something like
(76) ¬∃x(‘ιxFx’ denotes x).
This is quite relevant to the case of proper names because I can
deny that Pegasus exists meaning by this that the name ‘Pegasus’ does
not denote. Thus without appealing to an abbreviated description I can
truly deny that Pegasus exists either by means of statement negation, i.e.,
by saying that it is not the case that Pegasus exists (with ‘exists’ as a
predicate), or by the paraphrase in terms of denotation.
Let me now conclude with some general remarks about Russell’s
abbreviative theory of names. This could either be formulated as the strong
thesis that for every name there is a specific description which it abbreviates
in all contexts of use, or as the weaker thesis that for each context in which
a name is used there is a description which it is meant to abbreviate in
that context. Frege held related theses for senses, for he held that in a
well designed logical language every name should have a unique sense (and
denotation), whereas in a natural language the sense of a name would vary
with context (including in this the user of the name) and may have no
denotation. I think that a similar position should be attributed to Russell,
because in his general philosophical discussions he defends the contextual
thesis, whereas for a logical language such as the language of Principia
Mathematica names are only introduced into the language as abbreviations
for specific descriptions.
Russell’s ‘Homer’ example is interesting partly because it is not a
case of explicit fiction – unlike ‘Sherlock Holmes’ – and partly because the
association between ‘Homer’ and ‘the author of the Iliad and the Odyssey’
is very tight indeed. Thus, the biographical note of the Homer volume in
the Britannica Great Books of the Western World opens with the following
declaration31 :
Homer is not a man known to have existed, to whom the authorship
of the Iliad and the Odyssey is imputed. Homer is the author of
the homeric poems, a hypothesis constructed to account for their
existence and quality.
And in the “Homer” entry of The Encyclopedia Britannica (1961 edition)
we read:
The special difficulty about Homer is that, whereas David and Moses
have an independent existence, whether or not they wrote the works
ascribed to them, Homer has not: he is nothing but the author of
the Homeric Poems. The poems are facts and “Homer” a hypothesis
to account for them.
What do these statements mean?32 Although they are written in confusing
language some of the points seem reasonably clear. One point is that we
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don’t know whether the homeric poems are due to a single man. Nevertheless, the character (quality, etc.) of the poems suggests that they are the
work of a single man. This is the hypothesis. Now the question is how to
treat the name ‘Homer’.
Evidently, we didn’t just make up this name as shorthand for ‘the
author of the homeric poems’; the name ‘Homer’ comes historically associated with the poems. To say that Homer has no “independent existence”
is to say that the association with the poems is the only firm association
we have. This means that when we use the name ‘Homer’ we generally intend to refer to the author of the poems, which seems to support the claim
that ‘Homer’ abbreviates ‘the author of the Iliad and the Odyssey’ – as the
second passage clearly suggests. If we are discussing the poems themselves,
however, it doesn’t really matter, most of the time, whether there was one
author or many authors. We are basically interested in the texts, and the
name ‘Homer’ may be really shorthand for referring to the texts. Thus, a
context of the form ‘Homer says that . . .’ is not generally an abbreviation
for ‘the author of the Iliad and the Odyssey says that . . .’, but rather for
something like ‘it says in the Iliad that . . .’ or ‘it says in the Odyssey that
. . .’ or ‘it says in one of the homeric poems that . . .’.
We seem to gain nothing, therefore, by taking ‘Homer’ to be an
abbreviation for ‘the author of the Iliad and the Odyssey’ and eliminating
this by Russell’s analysis, because we don’t know whether there was a unique
author, and if there wasn’t, our true statements about what it says here or
there in the poems will no longer be true. If we are discussing the question
of the authorship of the poems and whether Homer existed or not, on the
other hand, as seems to be the case in the two passages I quoted, then the
claim that ‘Homer’ abbreviates ‘the author of the Iliad and the Odyssey’
does not seem plausible either. Just as I can hypothesize that there was a
unique author of the poems, I can hypothesize that there was indeed a man
called ‘Homer’ who was the author of the poems, and I can also hypothesize
that there was a man called ‘Homer’ who wasn’t the author of the poems
but who managed to pass himself off as the author of the poems. But the
latter hypothesis would be close to a contradiction in terms if ‘Homer’ were
just an abbreviation for ‘the author of the Iliad and the Odyssey’.
Although these considerations cast some doubt on the strong abbreviation thesis, they are not incompatible with Russell’s weaker thesis,
because he can easily agree that in the earlier contexts ‘Homer’ is used elliptically and should be replaced by a description “referring” to the texts –
though the names Iliad and Odyssey must be eliminated from these descriptions. And since he also suggests that in some contexts a name like ‘Homer’
may be meant as an abbreviation for the description ‘the man whose name
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was ‘Homer’ ’, he could agree with my discussion of the existence claim as
well33 . This may suggest that one can always appeal to this sort of description as a way out for those cases for which there is no other description
available – or description that one has in mind. But there are problems
with this suggestion.
Take the statement
(77) Homer was a poet,
and suppose that I have no description to substitute for ‘Homer’. To say
(78) the man whose name was ‘Homer’ was a poet
will not do, because there are many men whose name was ‘Homer’ and (78)
will be false according to Russell’s analysis – whereas (77) may be true. In
fact, even the description ‘the poet whose name was ‘Homer” (which is as
much as one can get out of that context) will not do, because
(79) the poet whose name was ‘Homer’ was a poet
may still be false – suppose that there were many poets whose name was
‘Homer’. Still, Russell would reply that if I state (77), then I must mean
something by ‘Homer’34 . Well, suppose that I remember from long gone
school days that Homer was a poet, but do not remember anything else.
Should one say that the appropriate description is ‘the man whose name
was ‘Homer’ such that I remember from long gone school days that he was
a poet’ ? The problem here is the ‘he’. To what does the ‘he’ refer? If we
formulate the description using variables we get: the x such that x was a
man called ‘Homer’ and I remember from long gone school days that x was
a poet. I don’t remember of any one man (or value of the variable) that he
was a poet, however; what I remember is that Homer was a poet. But if I
take the description to be ‘the man whose name was ‘Homer’ such that I
remember from long gone school days that Homer was a poet’, then I have
the problem of eliminating ‘Homer’ from (the end of) this description.
All this does not refute Russell’s view, however, because if one
does take the doctrine of knowledge by acquaintance and knowledge by description seriously, then the analysis of the content of statements should be
strongly relative to each individual speaker. The intersubjective character
of statements would be largely due to our acquaintance with universals,
including words, and our intuitions about the preceding examples – which
are based on the presumption that statements involving ordinary names are
true or false in terms of an objective (or intersubjective) notion of reference
– would be exposed as fallacious. The only really intersubjectively meaningful names would be those that abbreviate descriptions exclusively based
on universals with which we are acquainted35 .
Although I do not intend to pursue this line of discussion any
further at this point, I think that it brings out an important difficulty in
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discussing Russell’s view of names intuitively, for once we take seriously
into account the philosophical context in which it is inserted, we cannot
simply appeal to our ordinary intuitions about reference to refute it. The
discussion must necessarily shift to the question as to whether Russell’s
account of acquaintance in terms of sense data (memory, etc.), or some
alternative account, provides a sufficient basis for the analysis of knowledge
and language. Given the difficulties with the theory of sense data it is
unlikely that anyone would defend Russell’s theory on this basis today; but
that was a substantial part of his defence.
Russell also thought that there were no good alternatives to his
theory of names and descriptions, and that his objections to Frege’s theory
and Meinong’s theory showed that “it is imperative to effect . . . a reduction
of all propositions in which denoting phrases occur to forms in which no
such phrases occur”36 . What I have been suggesting is that by means of a
sharper analysis of subject-predicate form, and by allowing truth-valueless
sentences, one can approach some of the problems that Russell tackled in a
more natural way without dispensing with the denoting function of names
and descriptions.
I shall come back to Russell’s views in Chapter 5, and to the issues
raised in the present chapter in many later chapters, but first I want to
discuss the relation between descriptions and Frege’s thesis that truth is
denotation of truth values.
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Notes
1. See Davidson “Quotation”. I am not concerned here with an analysis of
quotation but with the more general understanding of the distinction between use
and mention suggested in the next few pages.
2. Gödel makes this observation in “Russell’s Mathematical Logic” (p. 215):
. . . a sentence involving the phrase “the author of Waverley” does
not (strictly speaking) assert anything about Scott (since it contains
no constituent denoting Scott), but is only a roundabout way of
asserting something about the concepts occurring in the descriptive
phrase.
3. For an account of theories of description and related issues see Linsky Referring
and Names and Descriptions. An important recent defence and development of
Russell’s theory of descriptions is Neale Descriptions. I shall restrict my discussion
to Russell and Frege, and the main point of the chapter is not to attack their views
but to sketch a new theory of descriptions that combines central aspects of their
views. For a shorter and more direct presentation see my paper “Descriptions:
Frege and Russell Combined”.
4. See The Basic Laws of Arithmetic, note 15 and section 11.
5. In “The Philosophy of Logical Atomism”, p. 245, Russell says:
In ‘Scott is the author of Waverley’ the ‘is’, of course, expresses
identity, i.e., the entity whose name is Scott is identical with the
author of Waverley. But, when I say ‘Scott is mortal’ this ‘is’, is
the ‘is’ of predication. It is a mistake to interpret ‘Scott is mortal’
as meaning ‘Scott is identical with one among mortals’, because
(among other reasons) you will not be able to say what ‘mortals’ are
except by means of the propositional function ‘x is mortal’, which
brings back the ‘is’ of predication. You cannot reduce the ‘is’ of
predication to the other ‘is’. But the ‘is’ in ‘Scott is the author of
Waverley’ is the ‘is’ of identity and not of predication.
6. Continuing the remarks in the previous quotation, Russell says (pp. 245-246):
If you were to try to substitute for ‘the author of Waverley’ in that
proposition any name whatever, say ‘c’, so that the proposition becomes ‘Scott is c’, then if ‘c’ is a name for anybody who is not Scott,
that proposition would become false, while if, on the other hand, ‘c’
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is a name for Scott, then the proposition will become simply a tautology. It is at once obvious that if ‘c’ were ‘Scott’ itself, ‘Scott is
Scott’ is just a tautology. But if you take any other name which is
just a name for Scott, then if the name is being used as a name and
not as a description, the proposition will still be a tautology. For
the name itself is merely a means of pointing to the thing, and does
not occur in what you are asserting, so that if one thing has two
names, you make exactly the same assertion whichever of the two
names you use, provided they are really names and not truncated
descriptions.
So there are only two alternatives. If ‘c’ is a name, the proposition
‘Scott is c’ is either false or tautologous. But the proposition ‘Scott
is the author of Waverley’ is neither, and therefore is not the same
as any proposition of the form ’Scott is c’, where ‘c’ is a name. This
is another way of illustrating the fact that a description is quite a
different thing from a name.
7. This was part of Russell’s point, however, and he argued explicitly against
Frege on this count in “On Denoting” pp. 48-50.
8. In “Knowledge by Acquaintance and Knowledge by Description” (p. 215) and
in The Problems of Philosophy (p. 53). In the latter he says:
The proposition ‘a is the so-and-so’ means that a has the property
so-and-so, and nothing else has. ‘Mr. A. is the Unionist candidate
for this constituency’ means ‘Mr. A. is a Unionist candidate for this
constituency, and no one else is’.
There is no suggestion here, or in the surrounding context, that ‘a is the so-andso’ must be analysed as an identity rather than as a predication. When Russell
introduces his account of definite descriptions in “On Denoting” (p. 44, see note
13 below) he gives a similar analysis. In fact, in “On Denoting” (p. 55) he makes
the following interesting remark:
The usefulness of identity is explained by the above theory. No one
outside of a logic-book ever wishes to say ‘x is x’, and yet assertions
of identity are often made in such forms as ‘Scott was the author of
Waverley’ or ‘thou art the man’. The meaning of such propositions
cannot be stated without the notion of identity, although they are
not simply statements that Scott is identical with another term, the
author of Waverley, or that thou art identical with another term,
the man. The shortest statement of ‘Scott is the author of Waverley’
seems to be ‘Scott wrote Waverley; and it is always true of y that
if y wrote Waverley, y is identical with Scott’. It is in this way that
identity enters into ‘Scott is the author of Waverley’; and it is owing
to such uses that identity is worth affirming.
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Since this analysis of ‘Scott is the author of Waverley’ is the predicative analysis,
what Russell is saying is that in the analysis of (22) identity enters in the second
predication (29-2).
9. Lyons considers this predicative interpretation in Semantics 1 when discussing
the example ‘Giscard d’Estaing is the President of France’ (p. 185):
. . . it might be understood to express a proposition that is comparable with such propositions as the following: that Giscard d’Estaing
comes from the Auvergne, that he likes playing tennis, and so on.
Under this interpretation . . . the phrase ‘the President of France’
is not being used to refer to an individual; it is being used with
predicative function to say something about the individual that is
referred to by means of the subject-expression, ‘Giscard d’Estaing’.
10. “On Sense and Reference”, p. 40. He argues as follows:
If anything is asserted there is always an obvious presupposition that
the simple or compound proper names used have reference. If one
therefore asserts ‘Kepler died in misery’, there is a presupposition
that the name ‘Kepler’ designates something; but it does not follow
that the sense of the sentence ‘Kepler died in misery’ contains the
thought that the name ‘Kepler’ designates something. If this were
the case the negation would have to run not
Kepler did not die in misery
but
Kepler did not die in misery, or the name ‘Kepler’ has no reference.
That the name ‘Kepler’ designates something is just as much a presupposition for the assertion
Kepler died in misery
as for the contrary assertion. Now languages have the fault of containing expressions which fail to designate an object (although their
grammatical form seems to qualify them for that purpose) because
the truth of some sentence is a prerequisite. Thus it depends on the
truth of the sentence:
There was someone who discovered the elliptic form of the
planetary orbits
whether the subordinate clause
Whoever discovered the elliptic form of the planetary orbits
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really designates an object or only seems to do so while having in
fact no reference. And thus it may appear as if our subordinate
clause contained as a part of its sense the thought that there was
somebody who discovered the elliptic form of the planetary orbits.
If this were right the negation would run: [(35)].
11. Although there are several readings of (31) in terms of subject-predicate
structure, an interesting way of seeing the difference between (30) and (31) is that
whereas in (30) the property [F x & ∀y(F y ⇒ y = x)](x) is predicated of a, in
(31) it becomes the subject of the existential quantification. We can thus think
of (31) as having the structure
(31a) [∃x(Zx & a = x)]([F x & ∀y(F y ⇒ y = x)](x)).
But we can also think of (31) as having any of the following structures:
(31b) [∃x(Zx & w = x)]([F x & ∀y(F y ⇒ y = x)](x), a),
(31c) [∃x(Zx & W x)]([F x & ∀y(F y ⇒ y = x)](x), [a = x](x)),
(31d) [∃x(Zx & ∀y(Zy ⇒ y = x) & a = x)](F ),
(31e) [∃x(Zx & ∀y(Zy ⇒ y = x) & w = x)](F, a),
(31f) [∃x(Zx & ∀y(Zy ⇒ y = x) & W x)](F, [a = x](x)).
The natural way of reading (32) is as a conjunction whose right conjunct
predicates F of a, but for the left conjunct we have two readings. One predicates
the logical property Unicity of F, i.e.,
(32-1) [∃y(Zy & ∀w(Zw ⇒ w = y))](F );
the other predicates of the property [F y & ∀w(F w ⇒ w = y)](y) that it is
instantiated, i.e.,
(32-1′ ) [∃yZy]([F y & ∀w(F w ⇒ w = y)](y)).
The same two readings apply to the left conjunct of (33), but its right conjunct is
naturally interpreted as a predication about F and a, or about F and the property
[a = y](y), or about the property [F y ⇒ a = y](y).
12. But this raises questions about identity and about the analysis of identity
statements. I have been assuming (as Frege and Russell do) that identity is a
legitimate relation among objects. If one denies this, however, then the proper
analysis of a statement of the form the F is (the same as) the G may be closer to
Russell’s than to Frege’s. For example, taking Unicity as a logical primitive (and
using ‘∃!x’ to express it) one can formulate (39) as
(44′ )∃!xFx & ∃!xGx & ∀x(F x ⇔ Gx),
which is logically equivalent to (44) and can be read as a predication about F and
G:
(44′′ ) F has unicity and G has unicity and F and G are mutually subordinate.
And, of course, (44) itself can be interpreted as being about F and G and as
predicating of them a complex logical property essentially equivalent to (44′′ ).
Another reasonable interpretation of (39) which has a close relation to
Russell’s analysis is
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(a)∀x(x is the F ⇔ x is the G).
If we analyse ‘x is the F’ and ‘x is the G’ predicatively, (a) can be formulated in
my notation as
(b)[∀x(Zx ⇔ W x)]([F x & ∀y(F y ⇒ y = x)](x), [Gx & ∀y(Gy ⇒ y = x)](x)),
and in ordinary notation as
(c)∀x((F x & ∀y(F y ⇒ y = x)) ⇔ (Gx & ∀y(Gy ⇒ y = x))),
which is essentially equivalent to (44) minus the existential claims. If I say that the
governor of California is the chairman of the Board of Regents of the University
of California, for instance, I may not be referring to anyone in particular, nor
stating an identity, nor implying that there is (at present, say) either a governor
of California or a chairman of the Board of Regents of the University of California.
A related case is Frege’s example (34), one reading of which (though
not the one he meant) is
(34′′ )∀x(x is the discoverer of the elliptic form of the planetary orbits ⇒ x died
in misery),
in which case the negation would run
(34′′ ¬)∃x(x is the discoverer of the elliptic form of the planetary orbits & x did
not die in misery).
To negate (34) simply by negating ‘x died in misery’ means that one is interpreting
(34) as (34′ ) rather than as (34′′ ), but even if one interprets (34) as (34′′ ) one
cannot negate it by denying the existence (or uniqueness) of a discoverer of the
elliptic form of the planetary orbits.
One can elaborate a lot further on examples like this, and I think that
an important advantage of my logical analysis (and notation) is that it allows
for a clear expression of this multiplicity of (natural) readings. A more precise
formulation of the notation is given in Chapter 6.
13. Even Hardy, who had a very good nose for logical issues, fell for these conclusions. He says (“Mathematical Proof”, p. 13):
It is one of Russell’s admitted achievements to have recognized in a
precise and explicit manner the immense importance of ‘incomplete
symbolism’ in logic and philosophy also, and so to have shown how
widely the correct analysis of a proposition may diverge from the
analysis of unreflecting common sense. . . . The ‘Waverley’ argument
applies to all propositions of the form ‘a is the b’, and shows that the
proposition cannot be analysed, as the words expressing it suggest,
into an assertion of identity between ‘a’ and ‘the b’. . . . It follows that
the analysis was wrong, and that there is no such object in reality as
‘the b’; ‘a is the b’ must be analysed in an entirely different manner.
The italics are mine, and everything he says about the ‘Waverley’ argument is
right except for that. Hardy is discussing contexts with descriptions ‘the b’ in
predicate position, and his considerations about them are correct, but it doesn’t
follow that there is no such object in reality as ‘the b’, and when he uses the
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description ‘the ‘Waverley’ argument’ (in subject position) he uses it to refer to
something in reality – namely, to a specific argument of Russell’s.
Russell makes the same “mistake” in his initial account of definite descriptions in “On Denoting”, p. 44. He begins by analysing the F in predicate
position in essentially the same way as in the passage I quoted in note 8:
Thus when we say ‘x was the father of Charles II’ we not only assert
that x had a certain relation to Charles II, but also that nothing else
had this relation. The relation in question, without the assumption
of uniqueness, and without any denoting phrases, is expressed by ‘x
begat Charles II’. To get an equivalent of ‘x was the father of Charles
II’, we must add, ‘If y is other than x, y did not beget Charles II’,
or, what is equivalent, ‘If y begat Charles II, y is identical with x’.
Hence ‘x is the father of Charles II’ becomes: ‘x begat Charles II;
and “if y begat Charles II, y is identical with x” is always true of y’.
But then he goes on to illustrate this analysis with an example in which ‘the father
of Charles II’ appears in subject position:
Thus ‘the father of Charles II was executed’ becomes: ‘It is not
always false of x that x begat Charles II and that x was executed
and that “if y begat Charles II, y is identical with x” is always true
of y’.
Russell comments that “[t]his may seem a somewhat incredible interpretation,”
but whatever may make it incredible is not the analysis of ‘x is the father of
Charles II’, which is entirely natural.
Perhaps the most natural way of looking at Russell’s analysis is to see
it as proceding through the intermediate step
∃x(x is the father of Charles II & x was executed),
in which ‘the father of Charles II’ appears in predicate position.
14. In Frege‘s case the main contributing factor to the faulty generalization was
the analysis of statements involving descriptions in predicate position as identity
statements. In Russell‘s case the situation is not so clear.
15. Of course, our modern logical notation was largely shaped by Frege and by
Russell, among others, and in both cases there was a tendency of rejecting subjectpredicate analysis as a traditional error incompatible with modern logic. I think
that this was a very useful view in that it eliminated many complications, but I
also think that it was quite wrong – although it was right about the limitations
of the traditional subject-predicate analysis. In my view what Frege did was
to sharpen subject-predicate analysis in essentially the way I am using it. And
Russell also helped sharpening it somewhat further by means of his distinctions
of scope (among other things).
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16. Senses are discussed in Chapter 11, which depends only on this chapter and
on the beginning of Chapter 9.
17. Thus, (23), (31) and (30), which are Frege’s analysis, Russell’s and mine of
statements of form (22), are c-logically equivalent. If F does not apply to a unique
thing, then (23) is truth-valueless and (31) and (30) are false. For statements of
form (38), Frege’s analysis and mine are the same, namely
(a) [Gx](ιxF x),
whereas Russell’s is
(b)∃x(F x & ∀y(F y ⇒ y = x) & Gx).
These are also c-logically equivalent, but if F does not apply to a unique thing,
then (a) is truth-valueless and (b) is false. For statements of form (39), Frege’s
analysis (41), Russell’s (44), and mine (43), are again c-logically equivalent. If
F does not apply to a unique thing, then (41) and (43) are truth-valueless and
(44) is false. If F does apply to a unique thing and G does not, then (41) is
truth-valueless and (43) and (44) are false. These points are discussed in terms of
interpretations for first order logic in Chapter 6.
18. Thus, about Meinong’s theory he says (“On Denoting”, p. 45):
This theory regards any grammatically correct denoting phrase as
standing for an object. Thus ‘the present King of France’, ‘the round
square’, etc., are supposed to be genuine objects. It is admitted that
such objects do not subsist, but nevertheless they are supposed to
be objects. This is in itself a difficult view; but the chief objection is
that such objects, admittedly, are apt to infringe the law of contradiction. It is contended, for example, that the existent present King
of France exists, and also does not exist; that the round square is
round, and also not round, etc. But this is intolerable; and if any
theory can be found to avoid this result, it is surely to be preferred.
And against Frege’s solution for ordinary language he argued (Ibid., p. 48):
By the law of excluded middle, either ‘A is B’ or ‘A is not B’ must be
true. Hence either ‘the present King of France is bald’ or ‘the present
King of France is not bald’ must be true. Yet if we enumerated the
things that are bald, and then the things that are not bald, we should
not find the present King of France in either list. Hegelians, who
love a synthesis, will probably conclude that he wears a wig.
As to Frege’s solution for logically perfect languages, that posited an arbitrary denotation for otherwise non denoting terms, Russell argued that “though
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it may not lead to actual logical error, is plainly artificial, and does not give an
exact analysis of the matter” (Ibid., p. 47).
19. Russell was a big fan of Ockham’s razor, and especially of its usefulness in
connection with logical constructions. At one point he makes the case for this
usefulness as follows (“Logical Atomism”, p. 326):
When some set of supposed entities has neat logical properties, it
turns out, in a great many instances, that the supposed entities can
be replaced by purely logical structures composed of entities that
have not such neat properties. In that case, in interpreting a body
of propositions hitherto believed to be about the supposed entities,
we can substitute the logical structures without altering any of the
detail of the body of propositions in question. This is an economy,
because entities with neat logical properties are always inferred, and
if the propositions in which they occur can be interpreted without
making this inference, the ground for the inference fails, and our
body of propositions is secured against the need of a doubtful step.
The principle may be stated in the form: ‘Wherever possible, substitute constructions out of known entities for inferences to unknown
entities.’
He goes on (pp. 326-330) to illustrate this with the principle of abstraction (or “the
principle which dispenses with abstraction”) and its application to the definition
of number in terms of classes; the elimination of classes in Principia Mathematica
by means of an extension of the theory of descriptions; the various eliminations
achieved by the theory of descriptions itself; applications to mathematics (points
and instants); to physics (particles of matter); to the mental (the subject). A
similar discussion is in pp. 270-281 of “The Philosophy of Logical Atomism”, and
(briefly) at the end of “On Denoting”.
In My Philosophical Development, p. 71, commenting on his definition
of the natural numbers in The Principles of Mathematics, Russell says:
But much more important than either of these two advantages is
the fact that we get rid of numbers as metaphysical entities. They
become, in fact, merely linguistic conveniences with no more substantiality than belongs to ‘etc.’ or ‘i.e.’. Kronecker, in philosophizing about mathematics, said that ‘God made the integers and the
mathematicians made the rest of the mathematical apparatus’. By
this he meant that each integer had to have an independent being,
but other kinds of numbers need not have. With the above definition of numbers this prerogative of the integers disappears and the
primitive apparatus of the mathematician is reduced to such purely
logical terms as or, not, all, and some. This was my first experience of the usefulness of Occam’s razor in diminishing the number of
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undefined terms and unproved propositions required in a given body
of knowledge.
20. That the distinction between knowledge by acquaintance and knowledge by
description is a central philosophical aspect of Russell’s discussion of denoting is
made clear at the beginning and at the end of “On Denoting” (pp. 41-42, 55-56).
This doctrine is elaborated in “Knowledge by Acquaintance and Knowledge by
Description”, The Problems of Philosophy (Chapter 5), and (with some shifts of
emphasis) in many later works up to and including Human Knowledge: Its Scope
and Limits (Part II, Chapter 4). In The Problems of Philosophy he says (p. 58):
Many universals, like many particulars, are only known to us by description. But here, as in the case of particulars, knowledge concerning what is known by description is ultimately reducible to knowledge concerning what is known by acquaintance.
The fundamental principle in the analysis of propositions containing
descriptions is this: Every proposition which we can understand must
be composed wholly of constituents with which we are acquainted.
21. In “The Philosophy of Logical Atomism”, p. 201, Russell says:
A name, in the narrow logical sense of a word whose meaning is
a particular, can only be applied to a particular with which the
speaker is acquainted, because you cannot name anything you are
not acquainted with. . . . This makes it very difficult to get any
instance of a name at all in the proper strict logical sense of the
word. The only words one does use as names in the logical sense are
words like ‘this’ or ‘that’. One can use ‘this’ as a name to stand for
a particular with which one is acquainted at the moment. We say
‘This is white’. If you agree that ‘This is white’, meaning the ‘this’
that you see, you are using ‘this’ as a proper name.
The question of quantification is rather complex and I won’t discuss it in
detail now. At the beginning of “On Denoting” Russell introduces quantification
as follows (pp. 42-43):
My theory, briefly, is as follows. I take the notion of the variable
as fundamental; I use ‘C(x)’ to mean a proposition in which x is
a constituent, where x, the variable, is essentially and wholly undetermined. Then we can consider the two notions ‘C(x) is always
true’ and ‘C(x) is sometimes true’. Then everything and nothing and
something (which are the most primitive of denoting phrases) are to
be interpreted as follows:
C(everything) means ‘C(x) is always true’;
C(nothing) means ‘ “C(x) is false” is always true’;
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C(something) means ‘It is false that “C(x) is false” is always true’.
Here the notion ‘C(x) is always true’ is taken as ultimate and indefinable, and the others are defined by means of it. Everything, nothing,
and something are not assumed to have any meaning in isolation,
but a meaning is assigned to every proposition in which they occur.
This is the principle of the theory of denoting I wish to advocate:
that denoting phrases never have any meaning in themselves, but
that every proposition in whose verbal expression they occur has a
meaning. The difficulties concerning denoting are, I believe, all the
result of a wrong analysis of propositions whose verbal expressions
contain denoting phrases.
Thus, the sentence
(a) everything is a man
is analysed by Russell as
(b) ‘x is a man’ is always true.
How should we interpret this? One interpretation is to think of (b) as being
about the propositional function ‘x is a man’ conceived as a property, or even as
a predicate, and saying that this property applies universally. This is essentially
Frege’s interpretation, which I have been using, but it does not seem to be Russell’s
interpretation – at least in the context I quoted. At various points (see the end
of the two quotations from Russell in note 13 , for example) he also says instead
of (b)
(c) ‘x is a man’ is always true of x.
This treats the variable ‘x’ as if it were a denoting term of whose “reference” we
can say that it is a man. The variable ranges over everything that exists, so to
speak, and we are saying of each such thing that it is a man. Another way to
put this (standard) interpretation of quantification is to say that the variable has
“values” and that we are saying of each such value that it is a man. This is what
I meant when I said that Russell treats variables as referring terms which cannot
fail to refer.
A related interpretation of (b) is to concentrate on the ‘true’ and to
think of ‘x is a man’ as if it were a variable proposition which we are saying to be
true for each determination of a “value” for the variable – we could also think of
‘x is a man’ as a “totality” of propositions all of which are said to be true.
In “Mathematical Logic as Based on the Theory of Types” Russell draws
a distinction between ‘any’ and ‘all’ which I think corresponds basically to the two
interpretations I distinguished above. Thus, he says (pp. 64-65):
If we say: ‘Let ABC be a triangle, then the sides AB, AC are together greater than the side BC’, we are saying something about
one triangle, not about all triangles; but the one triangle concerned
is absolutely ambiguous, and our statement consequently is also absolutely ambiguous. We do not affirm any one definite proposition,
but an undetermined one of all the propositions resulting from supposing ABC to be this or that triangle. This notion of ambiguous
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assertion is very important, and it is vital not to confound an ambiguous assertion with the definite assertion that the same thing
holds in all cases.
And a little later (pp. 65-66):
When we assert any value of a propositional function, we shall say
simply that we assert the propositional function. Thus if we enunciate the law of identity in the form ‘x = x’, we are asserting the
function ‘x = x’; i.e., we are asserting any value of this function.
Similarly we may be said to deny a propositional function when we
deny any instance of it. We can only truly assert a propositional
function if, whatever value we choose, that value is true; similarly
we can only truly deny it if, whatever value we choose, that value is
false. Hence in the general case, in which some values are true and
some false, we can neither assert or deny a propositional function.
If ϕx is a propositional function, we will denote by ‘(x).ϕx’ the
proposition ‘ϕx is always true’. . . . Then the distinction between
the assertion of all values and the assertion of any is the distinction
between (1) asserting (x).ϕx and (2) asserting ϕx where x is undetermined. The latter differs from the former in that it cannot be
treated as one determinate proposition.
But unless ‘ϕx is always true’ is an assertion about the propositional function ϕx,
it is not clear what the distinction is between truly asserting the propositional
function ϕx and asserting that ϕx is always true. Russell does not explain the
difference further and although he goes on to make a reference to Frege, he does
not refer to Frege’s view of quantifiers as higher level properties. He seems to
oscillate between the two readings, as shown by formulations like (c).
There are difficult issues involved in the interpretation of quantification
which are quite relevant (and closely related) to the questions we are discussing,
but to take them up now would lead us too far afield.
22. The interconnection between Russell’s intuitions about truth, falsity, denotation, sense, and nonsense is quite clear in many passages in “On Denoting” – for
example when he objects to Frege’s view in p. 46:
One of the first difficulties that confront us, when we adopt the view
that denoting phrases express a meaning and denote a denotation,
concerns the cases in which the denotation appears to be absent. If
we say ‘the King of England is bald’, that is, it would seem, not a
statement about the complex meaning ‘the King of England’, but
about the actual man denoted by the meaning. But now consider
‘the King of France is bald’. By parity of form this ought to be
about the denotation of the phrase ‘the King of France’. But this
phrase, though it has a meaning provided ‘the King of England’
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has a meaning, has certainly no denotation, at least in any obvious
sense. Hence one would suppose that ‘the King of France is bald’
ought to be nonsense; but it is not nonsense, since it is plainly false.
Frege would say that ‘the King of France is bald’ has a meaning (or sense) and that
it is for this reason that it is not “nonsense”. He would agree that the sentence is
not true, though he would claim that it is not false either precisely because ‘the
King of France’ does not denote. But Russell’s intuition is that what is not true
is false, or that ‘not true’ should be identified with ‘false’. In “Mr Strawson on
Referring” he comments on this point (p. 125):
Mr Strawson objects to my saying that ‘the King of France is wise’
is false if there is no King of France. He admits that the sentence
is significant and not true, but not that it is false. This is a mere
question of verbal convenience. He considers that the word ‘false’
has an unalterable meaning which it would be sinful to regard as
adjustable, though he prudently avoids telling us what this meaning
is. For my part, I find it more convenient to define the word ‘false’
so that every significant sentence is either true or false. This is a
purely verbal question; and although I have no wish to claim the
support of common usage, I do not think that he can claim it either.
Suppose, for example, that in some country there was a law that
no person could hold public office if he considered it false that the
Ruler of the Universe is wise. I think an avowed atheist who took
advantage of Mr Strawson’s doctrine to say that he did not hold this
proposition false, would be regarded as a somewhat shifty character.
I think that this is both unfair and misleading, because even if one identifies
‘false’ with ‘not true’ one can (and should) still distinguish two senses of ‘false’
(or cases of falsity). Namely, those cases (as in the two examples) such that the
predicate negation is also false from cases in which the predicate negation is true.
As I discuss later in the text (but can be seen from (51)), this is essentially what
Russell does; so it is partly a verbal matter, but not in the sense he meant it. And
this is also relevant to his remarks about Frege’s view.
23. One specific important case of the application of these techniques is the
elimination of incomplete symbols in Principia Mathematica. But even here, a
very technical and precise domain, it is not clear that the procedure is well-defined.
In “Russell’s Mathematical Logic” (p. 212) Gödel comments:
What is missing, above all, is a precise statement of the syntax of
the formalism. Syntactical considerations are omitted even in cases
where they are necessary for the cogency of the proofs, in particular
in connection with the “incomplete symbols”. These are introduced
not by explicit definitions, but by rules describing how sentences
containing them are to be translated into sentences not containing
them. In order to be sure, however, that (or for what expressions)
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this translation is possible and uniquely determined and that (or to
what extent) the rules of inference apply also to the new kind of
expressions, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations.
The matter is especially doubtful for the rule of substitution and of
replacing defined symbols by their definiens. If this rule is applied
to expressions containing other defined symbols it requires that the
order of elimination of these be indifferent. This however is by no
means always the case (ϕ!û = û[ϕ!u], e.g., is a counter-example). In
Principia such eliminations are always carried out by substitutions
in the theorems corresponding to the definitions, so that it is chiefly
the rule of substitution which would have to be proved.
24. Russell says (Introduction to Mathematical Philosophy, pp. 178-179):
We may inquire significantly whether Homer existed, which we could
not do if “Homer” were a name. The proposition “the so-and-so exists” is significant, whether true or false; but if a is the so-and-so
(where “a” is a name), the words “a exists” are meaningless. It
is only of descriptions – definite or indefinite – that existence can
be significantly asserted; for if “a” is a name, it must name something: what does not name anything is not a name, and therefore,
if intended to be a name, is a symbol devoid of meaning, whereas
a description, like “the present King of France,” does not become
incapable of occurring significantly merely on the ground that it
describes nothing, the reason being that it is a complex symbol of
which the meaning is derived from that of its constituent symbols.
And so, when we ask whether Homer existed, we are using the word
“Homer” as an abbreviated description: we may replace it by (say)
“the author of the Iliad and the Odyssey”. The same considerations
apply to almost all uses of what look like proper names.
25. Frege argues for something like this in The Foundations of Arithmetic, pp.
39-40:
. . .in calling the things units we are supposed to be adding to our
description of them; under the influence of the grammatical form, we
are regarding “one” as a word for a property and taking “one city” in
the same way as “wise man”. In that case a unit would be an object
characterized by the property “one” and would stand to “one” in the
same relation as “a sage” to the adjective “wise”. Now reasons have
already been given as conclusive against the view that number is a
property of things; but there are several further arguments against
the present suggestion in particular. It must strike us immediately
as remarkable that every single thing should possess this property. It
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would be incomprehensible why we should still ascribe it expressly to
a thing at all. It is only in virtue of the possibility of something not
being wise that it makes sense to say “Solon is wise”. The content
of a concept diminishes as its extension increases; if its extension
becomes all-embracing, its content must vanish altogether. It is not
easy to imagine how language could have come to invent a word for
a property which could not be of the slightest use for adding to the
description of any object whatsoever.
26. See Frege “Dialogue with Pünjer on Existence”, p. 62. One of Frege’s main
points in this dialogue is the distinction between ‘exists’ as a predicate and the
existential quantifier; about which I agree, of course.
27. That a statement of the form ‘a exists’ is true if and only if the name denotes is
related to Frege’s discussion of identity statements as follows. Take the sentence
‘Odysseus exists’, for example. According to Frege the name ‘Odysseus’ has a
sense (at least in a given context) which may or may not present something, but
whether the sense does present something is not a trivial question. Thus the
justification of such statements may depend on empirical information and the
statements cannot be considered to be analytic.
It is something of an incongruity in Frege’s discussion in “On Sense and
Reference” that he seems to lose sight of this point, because it is a consequence
of his thesis that sentences containing non-denoting expressions are neither true
nor false. In fact, given Frege’s conception of analytic, a priori, and a posteriori
in The Foundations of Arithmetic (pp. 3-4), and given the principle (S) that the
sense of a complex expression depends only on the senses of the parts and not on
their denotation (Chapter 2, p. 76), we cannot generally say that (the thoughts
expressed by) sentences of the form ‘a = a’ are analytic, or that they can be
established a priori, even when the name denotes, because this may depend on
factual considerations – which would make such sentences a posteriori according
to Frege’s characterization. Thus, neither the thought that Odysseus exists nor
the thought that Odysseus is self-identical can be said to be analytic even if
‘Odysseus’ denotes. The difference between statements of the form ‘a exists’ or
of the form ‘a = a’ and statements of the form ‘a = b’ is that once we know that
the name substituted for ‘a’ denotes (or that its sense presents something), we
know that the former are true, whereas for the latter we must know that the name
substituted for ‘a’ denotes, that the name substituted for ‘b’ denotes, and that
they denote the same thing. We can also put the point by saying that statements
of the forms ‘a exists’ and ‘a = a’ cannot be false, whereas statements of the form
‘a = b’ can be false.
28. In “Meaning and Existential Inference”, p. 167, Quine concludes:
The theoretical advantages of [reconstructing proper names trivially
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as descriptions] are overwhelming. The whole category of singular
terms is thereby swept away, so far as theory is concerned; for we
know how to eliminate descriptions. In dispensing with the category
of singular terms we dispense with a major source of theoretical confusion . . . In particular, we dispense altogether, in theory, with the
perplexing form of notation ‘a exists’; for we know how to translate
singular existence statements into more basic logical terms when the
singular term involved is a description.
29. Of course, it doesn’t follow from the fact that (68) is a logical truth that
any particular existence statement is true, although it does follow that no such
statement is false. As I interpret it, (68) is a statement about the logical property
Existence, not about particular things. But even if one interprets the variable
in (68) as in some sense “ranging over” things, it doesn’t follow that particular
existence statements must be true – because names need not denote, whereas the
variable is interpreted as always “denoting” something.
30. See Donellan “Reference and Definite Descriptions”.
31. Op. Cit., vol. 4, p. v.
32. These are intensional contexts (‘know’, ‘imputed’, etc.) and another of Russell’s main points in favor of his theory of descriptions was that
(a) George IV wished to know whether Scott was the author of Waverley,
does not mean the same thing as
(b) George IV wished to know whether Scott was Scott,
but that it should be analysed as
(c) George IV wished to know whether ∃x(x wrote Waverley &
∀y(y wrote Waverley ⇒ y = x) & Scott = x).
(See “On Denoting” pp. 47-48, 52).
One of the problems with (b) is the earlier one of interpreting something
of the form a is the F as a = the F and then substituting in the right hand side.
I.e., in my view the proper analysis of (a) is
(d) George IV wished to know whether [x!(x is an author of Waverley)](Scott).
Although this is not enough to handle (a) and must be coupled with further
analyses of the occurrence of ‘Scott’ in (d), Russell has an analogous problem for
the occurrence of ‘Scott’ in (c).
Even though I do not agree with Russell’s formulations, I think that
he had good insights about intensional contexts as well and that he was right to
emphasize this as an important aspect of his theory.
33. In The Problems of Philosophy, pp. 58-59, Russell says:
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We must attach some meaning to the words we use, if we are to speak
significantly and not utter mere noise; and the meaning we attach to
our words must be something with which we are acquainted. Thus
when, for example, we make a statement about Julius Caesar, it is
plain that Julius Caesar himself is not before our minds, since we
are not acquainted with him. We have in mind some description of
Julius Caesar: ‘the man who was assassinated on the Ides of March’,
‘the founder of the Roman Empire’, or, perhaps, merely ‘the man
whose name was Julius Caesar’. (In this last description, Julius
Caesar is a noise or shape with which we are acquainted.) Thus our
statement does not mean quite what it seems to mean, but means
something involving, instead of Julius Caesar, some description of
him which is composed wholly of particulars and universals with
which we are acquainted.
34. In spite of the following remarks I agree with this point of Russell’s in Chapter
11.
35. Russell expresses essentially this view in Human Knowledge: Its Scope and
Limits, p. 108:
It thus appears that two people are more likely to have the same
“this” if it is somewhat abstract than if it is fully concrete. In
fact, broadly speaking, every increase of abstractness diminishes the
difference between one person’s world and another’s. When we come
to logic and pure mathematics, there need be no difference whatever:
two people can attach exactly the same meaning to the word “or”
or to the word “371,294”. This is one reason why physics, in its
endeavor to eliminate the privacy of sense, has grown progressively
more abstract. This is also the reason for the view, which has been
widely held by philosophers, that all true knowledge is intellectual
rather than sensible, and that the intellect liberates while the senses
keep us in a personal prison. In such views there is an element of
truth, but no more, except where logic and pure mathematics are
concerned; for in all empirical knowledge liberation from sense can
only be partial. It can, however, be carried to the point where two
men’s interpretations are nearly certain to be both true or both false.
36. “On Denoting”, p. 45.
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Chapter 4
Arguments for Frege’s Thesis
It is obvious that if we accept that sentences have denotation and
accept Frege’s principle (R) of substitutivity of reference, then we must say
that the following sentences have the same denotation:
(1) Socrates was snubnosed,
(2) The teacher of Plato was snubnosed,
(3) The Greek philosopher who died in 399 B.C. by drinking hemlock was
snubnosed.
This is no more surprising than the fact that all the singular terms in question refer to the same object. If Frege’s thesis that sentences denote truth
values is accepted, however, these sentences also have the same denotation
as any other true sentence; for example,
(4) Copper oxide is green.
But whereas in the case of (1)-(3) we can say that the denotation has something to do with Socrates’ snubnosedness – it is the circumstance of Socrates
being snubnosed, or the state of affairs of Socrates being snubnosed, or the
fact that Socrates was snubnosed, – this isn’t true with (4), which seems to
have nothing whatever to do with either Socrates or snubnosedness. That’s
why Frege’s conclusion seems so odd.
Several philosophers and logicians have tried to strengthen Frege’s
arguments, either to make it more plausible that Frege’s conclusion indeed
follows from his premisses, or to get an actual proof of that conclusion from
allegedly acceptable premisses, or at least to establish the hypothetical that
if sentences denote, then all sentences with the same truth value denote the
same thing – which would cast serious doubt on the possibility of a theory
of facts, states of affairs, etc. as the denotation of sentences1 .
Thus Church tries to motivate Frege’s conclusion by showing how
we could argue on the basis of substitutions that two sentences with the
same truth value have the same denotation2 . He does not argue in general
but through a specific example; namely, the sentences
(5) Sir Walter Scott is the author of Waverley,
and
(8) The number of counties in Utah is twenty-nine.
These sentences seem sufficiently unrelated for us to feel that if we can get
from one to the other by substitutions, then very likely the same can be done
for any other true sentences. And even if we can’t do it for all other pairs
of sentences with the same truth value, what could possibly be the state of
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affairs, or circumstance, or fact, that is denoted by these two sentences?3 As
it happens, however, the sentences are not as unrelated as it seems, due to
the fact that Waverley consists of twenty-nine novels. And this fact is used
essentially in Church’s argument, which is thereby considerably weakened.
But still, even with this common feature, the sentences (5) and (8) are not
at all like (1)-(3) and the problem of specifying something other than the
True as their shared denotation would remain.
The two steps of Church’s argument are:
(6) Sir Walter Scott is the man who wrote twenty-nine Waverley novels
altogether,
and
(7) The number, such that Sir Walter Scott is the man who wrote that
many Waverley novels altogether, is twenty-nine.
There is no question that if (5)-(8) are interpreted as identities, then by
Frege’s principle (R) together with
(9) ‘the author of Waverley’ has the same denotation as ‘the man who wrote
twenty-nine Waverley novels altogether’,
and
(10) ‘the number such that Sir Walter Scott wrote that many Waverley
novels altogether’ has the same denotation as ‘the number of counties in
Utah’,
(5) has the same denotation as (6), and (7) has the same denotation as (8).
The problem is the transition from (6) to (7). What is the relation between
these two? Church says that if they are not actually synonymous, they are
so nearly so as to ensure sameness of denotation.
What Church is appealing to here is something like Frege’s principle that the denotation of an expression is determined by its sense; i.e.,
that the denotation is a function of the sense. So if (6) and (7) express the
same sense, or very nearly so, then they have the same denotation. Whether
they actually express the same sense, or nearly so, in Frege’s sense of ‘sense’
seems to me rather problematic; but even if they are nearly synonymous in
some sense of ‘synonymy’, there should be some argument that this sense
of ‘synonymy’ does ensure sameness of denotation.
The structure of the transformations in Church’s argument seems
roughly the following:
(5′ ) Sir Walter Scott = ιx(x is an author of Waverley)
(6′ ) Sir Walter Scott = ιx(x is a man & x wrote 29 Waverley novels altogether)
(7′ ) ιy(y is a number & Sir Walter Scott = ιx(x is a man & x wrote y
Waverley novels altogether)) = 29
(8′ ) ιy(y is a number & Utah has y counties) = 29
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The transitions (5′ ) − (6′ ) and (7′ ) − (8′ ) are applications of (R) and appropriate true identities, but (6′ ) and (7′ ) cannot be equated by (R) or by any
other principles stated by Frege with the possible exception of the principle
that sense determines denotation. Church’s problem is to establish that (6′ )
and (7′ ) have the same denotation. Do they really mean the same thing, or
nearly the same thing, as Church claims? It seems to me that from the point
of view of meaning one can raise several questions about the argument.
As I argued in the last chapter, to say that Sir Walter Scott is the
author of Waverley is not to say (5′ ) but, rather, to say that Sir Walter
Scott is an author of Waverley and nothing else is an author of Waverley.
Or, more precisely,
(5P) [x is an author of Waverley & ∀y(y is an author of Waverley ⇒ y =
x)](Sir Walter Scott),
which, using the notation I introduced in Chapter 3 (46′ ) can be abbreviated as
(5P) [x!(x is an author of Waverley)](Sir Walter Scott).
Similarly, I would say that what we normally mean by (6) is expressed more
accurately by
(6P) [x!(x is a man & ∃!29z(z is a Waverley novel & x wrote z))](Sir Walter
Scott).
Now in an ordinary sense (7) can be taken to mean nearly the same thing
as
(7a) The number of Waverley novels that Sir Walter Scott wrote is twentynine,
and this can be interpreted as
(7P) [y!(y numbers [z is a Waverly novel written by Sir Walter Scott](z))](29).
I think that ordinarily one would take (7) to be a rather contrived way of
expressing something like (7a). Similarly, one can interpret (8) as
(8P) [y!(y numbers [z is a county in Utah](z))](29),
or, perhaps, as
(8E) ∃!29z(z is a county in Utah)4 .
On this interpretation of Church’s argument not only it seems
rather unlikely that (6P) and (7P) mean the same thing, or nearly the same
thing, but one can’t even use Frege’s principle (R) to go from (5P) to (6P)
and from (7P) to (8P)5 .
I think that if one understands by ‘S ′ has the same meaning as S’
roughly that
(CM) one can normally communicate by S ′ what one wants to communicate
by S, and conversely,
then it is reasonable to say that (5P)-(8P) have the same meaning as the
corresponding identities:
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(5I) Sir Walter Scott = ιx(x is an author of Waverley),
(6I) Sir Walter Scott = ιx(x is a man & ∃!29z(z is a Waverley novel & x
wrote z),
(7I) ιy(y numbers [z is a Waverly novel written by Sir Walter Scott](z)) =
29,
(8I) ιy(y numbers [z is a county in Utah](z)) = 29.
I wouldn’t agree, however, that this is a sense of sameness of meaning that
preserves denotation; either between (5P)-(8P) and (5I)-(8I), or between
(6I) and (7I).
Church can reply, no doubt, that he didn’t mean either of these
analyses but that he meant the ordinary identities
(5*) Sir Walter Scott is identical to the author of Waverley,
(6*) Sir Walter Scott is identical to the man who wrote twenty-nine Waverley novels altogether,
(7*) The number, such that Sir Walter Scott is identical to the man who
wrote that many Waverley novels altogether, is identical to twenty-nine,
and
(8*) The number of counties in Utah is identical to twenty-nine;
which, by parity with Church’s formulation of (7), could perhaps be rephrased as
(8**) The number such that there are that many counties in Utah is identical to twenty-nine.
So we can consider the question whether (6*) and (7*) say, or mean, the
same thing.
Now it looks as if in (6*) one is talking about the identity of two
men, or of one man who has been identified in two different ways. That’s
the point of (6*): it says that it is the same man. And the same goes for
(5*). The difference between (5*) and (6*) is that (6*) contains an extra
bit of information in the right hand side. With (7*), on the other hand,
it looks as if one is talking about the identity of two numbers, or of one
number that has been identified in two different ways; it says that it is the
same number. And the same goes for (8*)-(8**), although the difference in
the identifications used in (7*) and (8*)-(8**) is now considerable. Does it
really seem plausible that (6*) and (7*) are nearly synonymous in a sense
from which one can infer that they have the same denotation? The sentences
certainly seem to be referring to different features (or aspects) of reality.
What is the appropriate sense of ‘synonymy’ to which Church is appealing?
Let’s take ‘synonymy’ as based on Frege’s notion of sense, which
is the context in which Church seems to be working. Do (6*) and (7*)
express the same sense, or nearly the same sense? I think that following
up on the preceding considerations one can make a plausible case that on
140
Frege’s view of senses as manners of presentation (6*) and (7*) express
rather different senses. The senses expressed by ‘Sir Walter Scott’ and
by ‘the man who wrote twenty-nine Waverley novels altogether’ are (or
contain) manners of presentation of a man, and it is part of these senses
that they are presentations of a man. The sense expressed by an identity
statement is that the two senses expressed by the terms in the statement
are presentations of the same object. Thus, I think that the main content
of the Fregean sense expressed by (6*) is that two presentations of a man
are presentations of the same man. And by the analogous argument for
(7*), I think that the main content of the sense expressed by (7*) is that
two presentations of a number are presentations of the same number. The
connection between the senses expressed by (6*) and by (7*) derives from
the way in which the objects are presented. But since (7*) nearly contains
(6*), it may seem reasonable to claim that the sense expressed by (6*) is
part of the sense expressed by (7*). And it may also seem reasonable to
claim that the sense expressed by (7*), or nearly the sense expressed by
(7*), is part of the sense expressed by (6*). In fact, the sense expressed by
(6*) seems to be part of the sense which identifies twenty-nine in the left
hand side of (7*), and the sense expressed by (7*) seems to be part of the
sense that identifies Sir Walter Scott in the right hand side of (6*). Now
the question is: Does it follow from this that the sense expressed by (6*) is
nearly the same as the sense expressed by (7*)?
Without a more accurate account of the structure of the sentences
and of their senses it is hard to argue conclusively, but it is reasonably
clear that the two senses are directed at rather different aspects of reality.
According to Frege, to judge (6*) is to recognize the truth of the sense
expressed by (6*), and to assert (6*) is to manifest this recognition. And
the same for (7*). But what am I judging when I judge (6*)? I am judging
that two senses identify the same man. But if the sense expressed by (7*)
were the same, or nearly the same, as the sense expressed by (6*), then I
would be making the same judgement, or nearly the same judgement, when
I judge (7*); and I would be making the same assertion, or nearly the same
assertion, when I assert (6*) and when I assert (7*). These conclusions do
not seem at all plausible to me6 .
To this one could reply that in order to judge (7*) I must also judge
(6*), and that in order to judge (6*) I must also judge (7*). Thus either
the senses are the same and the complex judgement involved is actually the
same in both cases, though it may appear to be different, or else the senses
are different but there is a strong mutual dependence so that I can’t judge
either one without judging the other.
141
I think, however, that a more accurate account is that judging
(6*) presupposes, or perhaps even involves, the subordinate judgement that
Waverley consists of twenty-nine novels altogether; whereas judging (7*)
presupposes (or involves) the subordinate judgement that Sir Walter Scott
is the man who wrote a certain number of Waverley novels. Thus the main
judgements are quite different and do not presuppose (or involve) each other.
I conclude, therefore, that there are reasonable grounds for denying that (6*)
and (7*) express the same sense7 .
More generally now, it seems to me that in order to make good
a claim of synonymy, in a sense of ‘synonymy’ from which one could infer
sameness of denotation, it must at least be clear what one is talking about
in the various sentences. (6) is already somewhat ambiguous as between
a predicative interpretation and an identity interpretation, but (7) is much
worse. What is Church talking about in (7)? Is he talking about the number
of Waverley novels that Sir Walter Scott wrote, or is he talking about the
fact that Sir Walter Scott wrote those novels? Or both, maybe? That’s why
the commas, with (6) essentially within them. This clause is doing double
duty; on the one hand it is helping to qualify the initial ‘the number’, and
on the other hand it is appealing to that phrase and to the ‘twenty-nine’ in
order to make a statement of its own. That’s why (6) and (7) seem to be
saying nearly the same thing.
So even for his example Church’s argument does not seem to make
it plausible that the two sentences have the same denotation. And how is
one to go from (1) to (4) by a similar method? We don’t have a clue.
But Church had given earlier a very general argument for Frege’s
thesis; in fact, he says that it was this argument that led him to revise
his view that the denotation of sentences are propositions to Frege’s view
that the denotation of sentences are truth values8 . Although the argument
is given in the context of Carnap’s formal semantics and depends on the
specific definitions of various semantic notions used by Carnap, it is basically
the following.
Any true sentence R is logically equivalent to
(11) {x : x = x & ¬R} = Ø,
which is synonymous with
(12) Ø = Ø.
Moreover, logically equivalent sentences are synonymous and synonymous
sentences have the same denotation. Therefore, any two true sentences are
synonymous and have the same denotation. And the same can be shown
for false sentences9 .
142
If ‘synonymy’ is interpreted as sameness of Fregean sense, however,
then this result would only establish Frege’s thesis about truth values at
the cost of completely destroying his theory of senses and his distinction
between sense and denotation. From Frege’s point of view the result is
much too strong, and it is clear from his treatment of identity statements
in “On Sense and Reference” that (11) and (12) do not have the same sense
– it seems also clear that one should not assume either that an arbitrary
true sentence R has the same sense as (11), but I will come back to this
point later. In fact, since there is no reasonable sense of ‘synonymy’ as
sameness of meaning, or sameness of Fregean sense, or sameness of content,
for which the result is at all plausible, why not simply take it as a reductio
ad absurdum of the principles and definitions that lead to it? This may
be Church’s later position, for in Introduction to Mathematical Logic 1 he
wants to preserve mainly the denotational part of the result and join it to
a Fregean semantics of senses.
Davidson reformulates Church’s argument directly for denotation.
He argues that if we accept the principle that logically equivalent expressions
have the same denotation, accept Frege’s principle of substitutivity (R), and
assume that sentences have denotation, then sentences that have the same
truth value have the same denotation. His argument is as follows10 .
Suppose that R and S are sentences that have the same truth value.
Then, the class {x : x = x & R} is the universal class if R is true and is
empty if R is false. Therefore, R is logically equivalent to
(13) {x : x = x & R} = {x : x = x}.
Since S has the same truth value as R, we have that {x : x = x & S} is the
same class, universal or empty, as {x : x = x & R}. Hence, by (R), we can
substitute the term on the left hand side of (13) to get
(14) {x : x = x & S} = {x : x = x},
which, in turn, is logically equivalent to S11 .
One thing that makes it possible for these arguments to be completely general, applying to any sentences and both to true and to false
sentences, is the switch from definite descriptions of individuals to set descriptions. This is somewhat analogous to Frege’s idea of always having a
denotation for a definite description which does not otherwise denote.
The main philosophical question about these arguments, in this
more streamlined form, concerns the justification of the principle that logically equivalent sentences have the same denotation, about which very little
is said. Maybe the implicit justification is that logical equivalence is such
a strong relation that it must guarantee sameness of denotation. But this
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begs the question, because one of the issues is precisely the relationship
between logical structure and denotation.
Let’s take a specific instance of Davidson’s argument for the sentences (1) and (4) as R and S, respectively. The two steps are:
(15) {x : x = x & Socrates was snubnosed} = {x : x = x}
and
(16) {x : x = xa & Copper oxide is green} = {x : x = x}.
It is quite true that (1) is true if and only if (15) is true and that (4) is
true if and only if (16) is true, and that this is guaranteed by logical form.
But in (1) we are saying something about Socrates, that he was snubnosed,
and in (4) we are saying something about copper oxide, that it is green,
whereas in (15) and (16) we are using these truths in a basically irrelevant
way (aside from being truths) in our identifications of the universal class
in the left hand side. In fact, in normal circumstances one would wonder
why the universal class should be identified in this way. And the point may
be that with these irrelevant (true) additions we still identify the universal
class. Does it seem plausible that a notion that equates (1) with (15) and
(4) with (16) preserves denotation?
But couldn’t one argue that logically equivalent sentences have the
same denotation because they express the same Fregean sense? Although
Frege does suggest in a letter to Husserl that with certain provisos logically
equivalent sentences express the same sense, it would be very problematic to
use his criterion in the present case12 . One can clearly see this by appealing
to intensional contexts, for one cannot generally substitute (1) by (15) in
contexts of the form ‘X said that S’, ‘X asserted that S’, ‘X believes that
S’, ‘X knows that S’, etc. preserving truth value. It may be true that
(17) Aristotle said that Socrates was snubnosed,
and yet
(18) Aristotle said that {x : x = x & Socrates was snubnosed} = {x : x = x}
is certainly false.
In fact, even much more natural logically equivalent sentences do not seem
to be intersubstitutable in such contexts preserving truth value. It is at
least arguable that
(19) John asserted that there is at least one prime number different from 6,
need not have the same truth value as
(20) John asserted that there is at least one prime number and that if 6 is
a prime number, then there are at least two prime numbers13 .
I would actually take Davidson’s reformulation of Church’s argument as prima facie evidence that the hypothesis that logical equivalence
preserves either meaning, or sense, or denotation, is wrong. And this is
partly what Davidson is trying to do. He thinks that it is reasonable to as-
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sume that if sentences denote, then logically equivalent sentences have the
same denotation. From which, by his argument, Frege’s thesis follows – at
least in the sense that there are only two entities denoted by sentences and
that all true sentences denote one of these entities and all false sentences
denote the other. But he also thinks that it is not reasonable to suppose
that the meaning of a sentence is given by its denotation. Therefore he concludes that “if the meaning of a sentence is what it refers to, all sentences
alike in truth value must be synonymous – an intolerable result”14 .
It is quite clear, actually, that logical equivalence only preserves
truth value under the assumption that every sentence is either true or false.
If we allow names and descriptions that do not denote, and sentences that
are neither true nor false, then logical equivalence in the sense of mutual
logical consequence does not preserve truth value. I have argued for this
in Chapter 2 in connection with Frege’s claim that a sentence S and a
sentence of the form ‘S is true’ have the same content. For such pairs are
c-logically equivalent but need not be materially equivalent. And I have
argued in Chapter 3 that for sentences of the form ‘a is the F’, Frege’s
analysis, Russell’s analysis and my analysis are c-logically equivalent but
need not be materially equivalent. Of course, if we take logical equivalence
in the sense of having the same truth value in all interpretations of the
non-logical terms – i.e., as tv-logical equivalence – then logical equivalence
preserves truth value. But, even so, it is not at all clear that this guarantee
of sameness of truth value is a guarantee of sameness of denotation. If the
true sentences are those that denote a state of affairs, false sentences are
those whose predicate negation denotes a state of affairs, and the others are
neither true nor false, then a guarantee that two sentences S and S ′ have
the same truth value in every interpretation of their non-logical terms, is
a guarantee that either both S and S ′ denote, or their predicate negations
denote, or none denote in every interpretation of their non-logical terms.
Even if this is the case, however, it does not follow that the denotations
must be the same. Let me illustrate with a different example.
Consider the descriptions
(21) ιx(Rxa & ∃!ySya)
and
(22) ιx(Sxa & ∃!yRya).
These descriptions denote or lack denotation under exactly the same interpretations of their non-logical terms; namely, they denote if, and only if,
∃!xSxa and ∃!xRxa. Nevertheless, their denotation need not be the same.
A simple case in which their denotation differs is to take as R the relation
‘father of’ and as S the relation ‘mother of’ for human beings.
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It seems to me, therefore, that the claim that logical equivalence
preserves denotation, if it is to be admitted as at all plausible, requires a
much more careful examination than either Church or Davidson have given
it15 .
In my view the subtlest general argument for Frege’s thesis is suggested by Gödel in his paper on Russell’s logic. The argument is not explicitly formulated by Gödel so there may be some question as to what exactly
he had in mind. I shall first present a formulation of the argument, and
then comment on the issues that are raised by it16 .
Gödel assumes Frege’s principle (R) and Frege’s principle about
definite descriptions
(D) If there is a unique object satisfying the property (or condition) P, then
the definite description ιxP x denotes that object.
Gödel then says that “the only further assumptions one would need to obtain
a rigorous proof” of Frege’s thesis are the following:
(G1) ‘P a’ and the proposition ‘a is the object that has the property P and
is identical to a’ mean the same thing,
and
(G2) Every proposition “speaks about something”, i.e., can be brought to
the form Pa.
In addition, says Gödel, one only needs the following fact:
(G3) For any two objects a, b, there exists a true proposition of the form
Pab as, e.g., a 6= b or a = a & b = b.
The crux of the matter here is the principle (G1). One could state it in the
form:
(G1′ ) A sentence of the form Pa means the same thing as a sentence of the
form a = ιx(P x & x = a).
Although what one needs is the following:
(G1*) If sentences denote, then a sentence of the form Pa has the same
denotation as a sentence of the form a = ιx(P x & x = a).
Also (G2) should be formulated in the form:
(G2*) If sentences denote, then any sentence has the same denotation as
some sentence of the form Pa.
Presumably, for the justification of (G1*), and maybe also of (G2*),
Gödel is appealing to something like Frege’s principle that sense determines
denotation: if those sentences mean the same thing, then they have the same
denotation. This raises some of the same issues as in Church’s argument
(5)-(8). But there is also the following problem.
If a sentence of the form Pa is true, then the description ιx(P x & x =
a) denotes the same thing as a, and the sentence a = ιx(P x & x = a) is
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also true. But if the sentence Pa is false, then the definite description
ιx(P x & x = a) denotes nothing according to Frege, and the sentence
a = ιx(P x & x = a) will be neither true nor false.
One can argue however, that if Pa is false, then ¬P a is true; and,
therefore, that a = ιx(¬P x & x = a) is also true. And it is natural to
suppose that if both true and false sentences have denotation and the negations of two false sentences have the same denotation, then the sentences
themselves have the same denotation. This raises some questions that I will
discuss later, but given Gödel’s assumption (G2), and treating negation as
predicate negation, it seems that we can restrict ourselves to true sentences
without loss of generality. So, in the formulation of Gödel’s argument I
consider only true sentences.
Let’s suppose that R and S are any true sentences, and that they
have denotation. One can then argue as follows.
By (G2*) we have that
(23) R
has the same denotation as
(24) F a,
for some a and some F. Also by (G2*),
(31) S
has the same denotation as
(30) Gb,
for some b and some G. We now fill the gap from (24) to (30).
By (G1*), (24) has the same denotation as
(25) a = ιx(F x & x = a).
and (30) has the same denotation as
(29) b = ιx(Gx & x = b).
By (R) now, we can substitute the right hand side of (25) by ιx(x = a & b =
b & x = a), which has the same denotation, getting
(26) a = ιx(x = a & b = b & x = a).
Similarly, (29) has the same denotation as
(28) b = ιx(a = a & x = b & x = b).
Finally, we can use (G1*) in the following way. Let Px be the
property [x = a & b = b](x). Then (26) is the same thing as
(26′ ) a = ιx(P x & x = a),
and by (G1*) this has the same denotation as
(26′′ ) P a,
which is the same thing as
(27) a = a & b = b.
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Similarly, using as Qx the property [a = a & x = b](x), we get
from (28) to
(28′ ) b = ιx(Qx & x = b),
and then by (G1*) to
(28′′ ) Qb,
which is again the same thing as (27).
Schematically, and in order, the sequence of transformations is the
following:
(23) R
(24) F a
(25) a = ιx(F x & x = a)
(26) a = ιx(x = a & b = b & x = a)
(27) a = a & b = b
(28) b = ιx(a = a & x = b & x = b)
(29) b = ιx(Gx & x = b)
(30) Gb
(31) S.
As I said, I don’t know whether this is exactly the formulation
of the argument that Gödel intended, but it is a rather elegant formal
argument17 . If R and S were false, one could start with their negations
and go on exactly as before. But if one tried to do it with false sentences
directly, then it is not clear how one would justify the various steps. The
step from (25) to (26), for example, depends on the two descriptions denoting the same object – and, at any rate, (26) is true. If descriptions always
denote however, then one can formulate Gödel’s argument directly for false
sentences R and S. Suppose that a normally non-denoting description ιxF x
denotes the set of F’s and that neither a nor b is the empty set. Then we
can replace (26)-(28) by:
(26F) a = ιx(x 6= a & b 6= b & x = a)
(27F) a 6= a & b 6= b
(28F) b = ιx(a 6= a & x 6= b & x = b)
The two descriptions denote the empty set, as do the descriptions in (25)
and (29). So everything is false and the principles that are used to justify
each step are exactly the same as before18 .
I will now make two preliminary remarks about the argument, and
then go on to the central issue.
First, instead of (27) one could use a 6= b, as Gödel suggests:
(266=) a = ιx(x 6= b & x = a)
(276=) a 6= b
(286=) b = ιx(a 6= x & x = b).
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This revised argument is a little more elegant than the earlier one but it has
the disadvantage that one has to assume that a 6= b is true, which wasn’t
necessary before19 .
Second, although the principle (G2*) is essential for a rigorous
proof, for the purposes of refuting a theory of denotation of sentences in
terms of states of affairs one can really drop it. If Frege’s thesis holds for all
true sentences of the form Pa, then there is no way of having a reasonable
theory of denotation for sentences in terms of states of affairs, or facts, or
anything of the sort.
But, anyway, how can one justify (G2*)? Take a sentence of the
form
(32) ∀xFx.
One thing one could do is use
(33) ∀x(x 6= a ⇒ F x) & F a,
or simply
(34) ∀xFx & a = a.
But this is very problematic unless one assumes that logically equivalent
sentences have the same denotation; in which case one might as well use
Davidson’s argument.
A better alternative is to say that (32) speaks about something,
namely the property F. This means interpreting (32) as
(35) [∀xZx](F).
This is a much more natural suggestion, and one can generalize it to obtain
a uniform formulation of the argument for all sentences. The principles
(G1)-(G1*) would have to be extended to apply to properties as well as to
objects, but this does not bring up any additional difficulties20 . Still, there
is a question as to what exactly Gödel means by “every proposition speaks
about something” and by “can be brought to the form ϕ(a)”.
Let’s consider now the principles (G1)-(G1*). I must acknowledge
that for the longest time I was completely unable to get a good grasp on
the issues raised by these principles. It was, in fact, the attempt to analyse
these principles that led me to take seriously the idea of states of affairs as
denotation for sentences.
Let’s take a specific example; say
(36) Quine is a philosopher.
According to (G1) this means the same thing as
(37) Quine is the object that has the property of being a philosopher and
is identical to Quine,
and, according to (G1′ ), this means the same thing as
(38) Quine = ιx(x is a philosopher & x = Quine).
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Finally, (G1*) concludes that if these sentences have denotation, then (38)
has the same denotation as (36).
Intuitively it is hard to deny that (36) and (37) mean the same
thing – at least in the sense suggested earlier in (CM) that they can be used
to communicate the same thing. (37) may be an unnecessarily roundabout
way of saying that Quine is a philosopher, but it seems to say just that
nonetheless.
Although I have already anticipated my basic objection to the
move from (G1) to (G1′ ) in Gödel’s argument – namely, that the descriptive
predicate cannot be treated as a singular term – I must emphasize again
how persuasive this move is. We have all been brought up on the idea that
it would be a confusion to interpret the first ‘is’ in (37) as a predicative ‘is’;
it must be the ‘is’ of identity. This is almost second nature to us, and it is
supposed to be the only reasonable interpretation. Let’s now look a little
more carefully at this example.
We start with (36); Quine is a philosopher. That Quine is identical
to Quine is entirely obvious, especially since we know that Quine exists. On
top of that it’s also an alleged logical truth. We feel, therefore, that by
saying
(39) Quine is a philosopher that is Quine
we don’t really add anything to (36) because we are merely emphasizing
the obvious. We also know that to be a philosopher that is Quine can only
be to be one thing; namely, Quine. Moreover, even if we didn’t know that
Quine is a philosopher, or even if he weren’t, (39) is telling us just that;
so (39) says the same thing as (36). But if Quine, and only Quine, can be
a philosopher that is Quine, there can be no objection to saying that (39)
says the same thing as
(40) Quine is the philosopher that is Quine.
And since to be the philosopher that is Quine is to be the object that has the
property of being a philosopher and is identical to Quine, we can rephrase
(40) as (38). But what is the relation between Quine and the object that
has the property of being a philosopher and is identical to Quine? They are
identical, of course. The object that has the property of being a philosopher
and is identical to Quine is Quine. So we get to (38).
I puzzled for many years over these points. I considered denying
every one of the principles involved in Gödel’s argument, but I couldn’t
get over their intrinsic plausibility. And I couldn’t get over the intrinsic
plausibility of the argument itself – except for the conclusion. At the end,
it was one of Gödel’s remarks that gave me the main clue to an answer.
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After suggesting his argument for Frege’s thesis, Gödel examines
Russell’s theory of descriptions and argues that it avoids Frege’s conclusion
by abandoning principle (D). This is obvious, and it was part of Russell’s
point in making up the theory. If we apply Russell’s elimination technique
to (25) and (26), we cannot get from one to the other anymore. But then
Gödel says (Op. Cit., p. 213):
As to the question in the logical sense, I cannot help but feel that the
problem raised by Frege’s puzzling conclusion has only been evaded
by Russell’s theory of descriptions and that there is something behind it that is not completely understood.
Why is it that Gödel feels that Russell’s theory evades the problem but does
not solve it? This has nothing to do with formal considerations, especially
since Gödel goes on to say that there is “one purely formal respect in which
one may give preference to Russell’s theory of descriptions”21 . So I worried
about Russell’s theory of descriptions as well, and, eventually, got to the
ideas presented in the last chapter.
Let’s look again at the transition from (36) to (38). To begin with,
the predicates ‘is a philosopher’ and ‘is a philosopher that is Quine’ are not
extensionally equivalent, so one cannot claim that they say (or mean) the
same thing. It is not clear therefore that (39) interpreted as the predication
(39P1) [x is a philosopher that is Quine](Quine)
says the same thing as (36). But one can also interpret (39) either as the
predication
(39P2) [x is a philosopher & Quine = Quine](Quine),
or as the conjunction
(39C) Quine is a philosopher & Quine = Quine.
That Quine is identical to Quine is obviously true, but only because
we know that Quine exists; otherwise the natural thing to say is that it isn’t
true (nor false). That Quine exists is a fact, but it is not a logical fact; it
is a contingent fact. Thus, if we don’t assume that Quine exists, or that
‘Quine’ denotes, it is not reasonable to say that Quine = Quine is a logical
truth.
Therefore, the predicate ‘is a philosopher and Quine = Quine’
says something – namely, that Quine = Quine – that the predicate ‘is a
philosopher’ doesn’t say, and we can only claim that they are extensionally
equivalent on the basis of our factual knowledge that Quine exists. So I don’t
think that either these predicates or that (39P2) and (36) say (or mean) the
same thing. Similarly, since (39C) says something that (36) doesn’t say, I
wouldn’t agree that (39C) says (or means) the same thing as (36).
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The truth of (36) also involves the fact that Quine exists, however;
if Quine didn’t exist, then (36) wouldn’t be true. So the truth of (36) and
the truth of (39) stand or fall together – for any of the versions of (39).
But it doesn’t follow from this that they say the same thing. What clearly
follows is that they must have the same truth value; i.e., that they are either
both true or both false or both neither true nor false in virtue of logical form.
We go on now to the next step, (40). Given the condition ‘is Quine’
in the predicate ‘is a philosopher that is Quine’ one can agree that this predicate says the same thing as the predicate ‘is the philosopher that is Quine’.
But even here one can question the transition; ‘is Quine’ could be taken
as a predicate rather than as an identity, in which case what would justify
the definite article are certain characteristics of such predicates. Take the
predicate ‘is a number that is an even prime’, for instance. This predicate
does not say the same thing as the predicate ‘is the number that is an even
prime’, but the characteristics of the predicate ‘is an even prime’ justify the
definite article. I would agree that there is a difference between the two
cases, and I am not pressing the point, but I think that the claim that this
sort of transition preserves meaning is not as obvious as it seems22 .
We now take the predicate ‘is the philosopher that is Quine’ and
separate the definite description ‘the philosopher that is Quine’. On the
natural view of descriptions given in principle (D) this description denotes
Quine. The plausibility of this move is reinforced by formulating ‘the
philosopher that is Quine’ as ‘the object that has the property of being
a philosopher and is identical to Quine’. As I argued in the last chapter,
however, this denotational view of descriptions is only natural when the
description is taken by itself, or as subject of a sentence, not when the
descriptive phrase is part of a predicate.
We see, therefore, that the conclusions about sameness of meaning
and sameness of denotation are quite problematic for a number of different
reasons. My conclusion is that the plausibility of Gödel’s argument derives
partly from an interplay between predications, descriptions, names and sentences which is not clearly analysed, and partly from an intuitive notion of
sameness of meaning which is not clearly analysed either.
Let’s now take a particular exemplification of the argument going
from ‘Quine is a philosopher’ to ‘Nixon is a lawyer’ and consider it from a
more formal point of view.
(24P) [x is a philosopher](Quine)
(25I) Quine = ιx(x is a philosopher & x = Quine)
(26I) Quine = ιx([x = Quine & Nixon = Nixon] & x = Quine)
(27P-1) [x = Quine & Nixon = Nixon](Quine)
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(27P-2) [Quine = Quine & x = Nixon](Nixon)
(28I) Nixon = ιx([Quine = Quine & x = Nixon] & x = Nixon)
(29I) Nixon = ιx(x is a lawyer & x = Nixon)
(30P) [x is a lawyer](Nixon)
The only steps that seem to me unobjectionable are the transitions from
(25I) to (26I) and from (28I) to (29I), which are clear applications of Frege’s
principle (R). All the other steps except for the transition from (27P-1) to
(27P-2) depend on Gödel’s principle (G1*). And even if we were to grant
(G1*), the transition from (27P-1) to (27P-2) involves two different readings
of (27)23 . Although one can claim that in some sense these two readings
have the same meaning, I don’t think that this is a sense of meaning from
which it follows that the denotations are the same. I would argue for this
along the same lines I argued in connection with Church’s transition from
(6*) to (7*). In fact, Church’s transition is akin to going directly from (26I)
to (28I). I also don’t agree, of course, that (24P) and (25I) have the same
meaning (or denotation) or that (29I) and (30P) have the same meaning
(or denotation).
If we formulate the whole argument in terms of predication, then
instead of the I-steps we have:
(25P) [x!(x is a philosopher & x = Quine)](Quine)
(26P) [x!([x = Quine & Nixon = Nixon] & x = Quine)](Quine)
(28P) [x!([Quine = Quine & x = Nixon] & x = Nixon)](Nixon)
(29P) [x!(x is a lawyer & x = Nixon)](Nixon)
If predicates denote sets, then the two transitions above as well as (26P)(27P-1) and (27P-2)-(28P) would be justified by (R), but the others wouldn’t.
If predicates denote properties, then even these transitions wouldn’t be justified either by (R) or by sameness of meaning because my earlier arguments
about predicates can be used to cast reasonable doubt on the claim that
these pairs of predicates denote the same property24 .
Looking back at the various arguments, I conclude that there are
substantial grounds for rejecting them as establishing Frege’s thesis. What
is interesting about the arguments is that they show the need for a more
careful examination and elaboration of the main notions involved in them:
denotation, meaning, sense, logical equivalence, predication, descriptions,
etc25 .
From Frege’s point of view the arguments are a two-edged sword,
for although Frege did not distinguish the denotation of sentences aside
from truth value, he did want to make finer distinctions concerning the
sense of sentences; but these distinctions are endangered by the assumptions
of the various arguments. This is particularly obvious if one concludes,
as in Church’s general argument, that all true sentences are synonymous
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and all false sentences are synonymous, but it is also fairly clear in the
other arguments. My impression is that any argument for truth values that
appeals to Frege’s principle that sense determines denotation is likely to
raise difficulties for Frege’s theory of sense.
From Russell’s point of view the arguments may be taken as further
confirmation that the assumption that names and descriptions denote, and
that sentences denote, lead to absurd conclusions. That’s why I think that
he may fairly claim that his general theory of names and descriptions does
solve the problem. But the solution is too radical and unintuitive, so as an
answer to Frege’s specific conclusion about truth values it seems more an
evasion than a solution. What I have tried to show is that one can largely
keep Frege’s distinction between sense and denotation and use some aspects
of Russell’s solution in conjunction with that to avoid the conclusion that
truth values are the denotation of sentences. Whether this is a satisfactory
approach will depend on the development of the notions I mentioned before.
154
Notes
1. These arguments (by Church, Gödel, Davidson, and Quine) have been dubbed
by Davidson “the slingshot” (see Barwise and Perry “Semantic Innocence and
Uncompromising Situations”, p. 400). An essential component of my attack on
these arguments is the theory of descriptions sketched in the last chapter. My aim
is to reject the arguments, and the conclusion that sentences denote truth values,
while preserving the basic outlook of Frege’s theory of sense and reference.
2. Introduction to Mathematical Logic 1, pp. 24-25.
3. Church says (Op. Cit., p. 25):
Now the two sentences, [(5)] and [(8)], though they have the same
denotation according to the preceding line of reasoning, seem actually to have very little in common. The most striking thing that
they do have in common is that they both are true. Elaboration of
examples of this kind leads us quickly to the conclusion, as at least
plausible, that all true sentences have the same denotation.
4. ‘∃!29x’ is read ‘there are exactly 29 x’ and the whole sentence can be taken
to be an abbreviation for a sentence involving 29 existential quantifiers and one
universal quantifier. For example,
∃x∃y(x 6= y & F x & F y & ∀z(F z ⇔ (z = x ∨ z = y))
says that there are exactly two F’s.
Both the analysis of (6P) and (8E) using quantification, and the analysis
of (7P) and (8P) using the bracketed predicates are based on Frege’s idea that
statements of number are statements involving concepts (or properties). I don’t
want to make an issue of what the proper analysis of these sentences should be,
but this can certainly be relevant to the question of what (if anything) is the
denotation of the sentences.
5. If one takes the denotation of the predicates in these sentences to be sets, then
one can use (R) to show that (5P) has the same denotation as (6P) and (7P) has
the same denotation as (8P). Thus, according to the set-theoretic conception of
states of affairs in Chapter 1, each of these pairs denotes the same state of affairs;
but (6P) and (7P) do not. If the predicates refer to properties (or concepts) on the
other hand, then one can’t use (R) and the states of affairs denoted by (5P)-(8P)
are all different.
6. ‘X judges that S’ and ‘X asserts that S’ are intensional contexts and Frege’s view
is that sentences expressing the same sense are intersubstitutable in such contexts
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preserving truth value – which follows from (R) and from the assumption that in
an intensional context a sentence denotes its (ordinary) sense. I can reinforce my
argument by taking other intensional contexts like ‘X said that S’, ‘X knows that
S’, etc. Consider, for instance, the following variation on Russell’s example about
George IV (using the original sentences (6) and (7)):
(a) George IV wished to know whether Sir Walter Scott is the man who wrote
twenty-nine Waverley novels altogether,
(b) George IV wished to know whether the number, such that Sir Walter Scott is
the man who wrote that many Waverley novels altogether, is twenty-nine.
It seems quite clear that these two sentences do not necessarily have the same
truth value – from which one can conclude that (6) and (7) do not express the
same sense.
7. On a natural understanding of (a) in the example in the previous note one would
assume that George IV already knows (or presumes) that there is a unique man
who wrote Waverley and that Waverley consists of twenty-nine novels altogether;
what he wishes to know is whether Sir Walter Scott is the man who wrote twentynine Waverley novels altogether. On a natural understanding of (b) one would
assume that George IV already knows (or presumes) that Sir Walter Scott is
the man who wrote Waverley and that Waverley consists of a certain number of
novels; what he wishes to know is whether the number such that Sir Walter Scott
is the man who wrote that many novels altogether is twenty-nine.
This example shows that (6) (or (6*)) is not really part of the sense that
identifies a number in the left hand side of (7) (or (7*)), and that (7) (or (7*)) is
not really part of the sense that identifies Sir Walter Scott in the right hand side
of (6) (or (6*)). That’s why one should analyse the sentences more precisely in
order to get a better grasp of their sense. Of course, also (a) and (b) have several
interpretations and should be analysed more precisely; I informally emphasized
one aspect of these sentences that seems enough to indicate why they need not
have the same truth value. (My student Rodrigo Bacellar has pointed out that the
analysis of these examples is actually a little more complicated, because Waverley
could consist of a certain number of novels larger than twenty-nine – say, thirty –
of which Scott wrote twenty-nine.)
8. The argument is in his review “Carnap’s Introduction to Semantics”, pp. 299300. In p. 299 he says:
Carnap takes it as an assumption that the designata of sentences
are propositions, and makes this his primary usage (although he
does also mention the possibility of truth-values as designata of sentences). However, if a language, in addition to certain other common properties, contains an abstraction operator ‘(λx)’ such that
‘(λx)(...)’ means ‘the class of all x such that . . .’, then – independently of the question whether the language is intensional or exten-
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sional – it is possible to prove that the designata of sentences must
be truth-values rather than propositions. For the sake of uniformity,
it therefore seems desirable to take the designatum of a sentence always to be a truth-value. On this point the reviewer confesses to
have changed his own former opinion, but not without compelling
reason.
9. Church concludes the argument as follows (Op. Cit., p. 300):
Thus we have a means by which any true sentence (as illustrated
in the case of [R]) can be shown to be synonymous with [(12)].
Hence we have a means of showing any two true sentences to be
synonymous. By a similar method any two false sentences can be
shown to be synonymous. Therefore finally no possibility remains
for the designata of sentences except that they be truth-values.
10. See “Truth and Meaning”, p. 19. Davidson points out (note 3) that “the
argument does not depend on any particular identification of the entities to which
sentences are supposed to refer.”
11. If one wants to avoid the use of the universal class in the argument, one can
use a version formulated by Quine in “Three Grades of Modal Involvement”, pp.
163-164. With R and S as before, we can substitute the two steps by
(13′ ){x : x = Ø & R} = {Ø}
and
(14′ ){x : x = Ø & S} = {Ø}.
12. See Chapter 2 note 20 for Frege’s statement of this criterion. In his review of
Carnap, Church accepts a stronger criterion for synonymy (Op. Cit., p. 300):
. . . Carnap assumes (page 92) that L-equivalent sentences are synonymous – and, of various ways that suggest themselves of settling the synonymy of sentences, this seems indeed one very natural
choice.
13. This example is based on the logical equivalence of
(a) ∃x(F x & x 6= a)
and
(b) ∃xFx & (F a ⇔ ∃x∃y(x 6= y & (F x & F y))).
Another example with a similar structure is:
(c) John asserted that there was at least one person who was a great philosopher
and that this person was not Napoleon,
and
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(d) John asserted that there was at least one person who was a great philosopher,
and that if Napoleon was a great philosopher, then there were at least two great
philosophers.
It is also clear that logical truths are not intersubstitutable in such
contexts preserving truth value. Thus,
(e) John said that for any object x, x is identical to itself
does not necessarily have the same truth value as
(f) John said that for any objects x, y, z, if x is identical to y and y is identical to
z, then x is identical to z;
which raises another problem for the Carnap-Church criterion.
14. Op. Cit., p. 19.
15. See Barwise and Perry Situations and Attitudes, pp. 24-26. This question is
discussed in Chapter 6 in terms of the distinction between c-logical equivalence,
tv-logical equivalence, denotational equivalence (sameness of denotation in all
interpretations) and d-logical equivalence (which involves also the denotation of
the predicate negation of sentences).
16. “Russell’s Mathematical Logic”, p. 214 note 5. The full passage is the
following (pp. 213-214):
An interesting example of Russell’s analysis of the fundamental logical concepts is his treatment of the definite article “the”. The problem is: what do the so-called descriptive phrases . . . denote or signify
and what is the meaning of the sentences in which they occur? The
apparently obvious answer that, e.g., “the author of Waverley” signifies Walter Scott, leads to unexpected difficulties. For, if we admit
the further apparently obvious axiom, that the signification of a
composite expression, containing constituents that have themselves
a signification, depends only on the signification of these constituents
(not on the manner in which this signification is expressed), then it
follows that the sentence “Scott is the author of Waverley” signifies the same thing as “Scott is Scott”; and this again leads almost
inevitably to the conclusion that all true sentences have the same
signification (as well as all false ones).5 Frege actually drew this conclusion; and he meant it in an almost metaphysical sense, reminding
one somewhat of the Eleatic doctrine of the “One”. “The True” –
according to Frege’s view – is analysed by us in different ways in
different propositions; “the True” being the name he uses for the
common signification of all true propositions.
[Note 5] The only further assumptions one would need in order to obtain a rigorous proof would be: (1) that “ϕ(a)” and the proposition
“a is the object which has the property ϕ and is identical with a”
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mean the same thing and (2) that every proposition “speaks about
something,” i.e., can be brought to the form ϕ(a). Furthermore one
would have to use the fact that for any two objects a, b, there exists
a true proposition of the form ϕ(a, b) as, e.g., a 6= b or a = a · b = b.
I don’t know whether there was any relation between the argument in
Church’s review and this argument of Gödel’s. The two papers were published
nearly at the same time, and Gödel thanks Church for stylistic help with his paper
(note 50), but neither one refers to the other in connection with these arguments.
In any case, it is clear that, unlike Church, Gödel did not mean to endorse Frege’s
thesis by means of this argument, but merely suggests it as an argument for it.
17. A closely related reconstruction of Gödel’s argument is formulated by Wedberg
in A History of Philosophy vol. 3, pp. 119-121 (note 15). Also Neale in “The
Philosophical Significance of Gödel’s Slingshot” has a related reconstruction as
well as further references to the literature.
18. To avoid specific assumptions about the denotation of the names a and b
one could adopt Dana Scott’s idea in “Existence and Definite Descriptions” of
treating non-denoting descriptions as denoting an extraordinary object that is
outside the domain of interpretation. This has the effect that two (normally)
non-denoting descriptions will denote the same thing, and that no (normally)
non-denoting description will denote the same thing as any (normally) denoting
name or description. This is fine from a formal point of view, because one can
think of the extraordinary object as a sort of formal addition (like points at infinity
in geometry) for purposes of simplification, but from a more general point of view
it is not clear what such an object would be – the one (and only) non-existent
object, perhaps?
19. This is the version used by Neale (Op. Cit., pp. 777-779).
20. But for Frege there would be difficulties because he holds that one cannot
denote a property (or concept) by means of a definite description. Aside from
this, however, we could say that the sentence
(a) Everything is a man
means the same thing as
(b) The property of being a man is the property that has the property of applying
to everything and is identical to the property of being a man.
Another question one can raise is whether a relational sentence such as
(c) Frege taught Carnap
means the same thing if interpreted as stating a relation between Frege and Carnap
and as stating a property of Frege (or Carnap). It may be natural to assume so
– in which case the argument generalizes for relational sentences via this further
assumption – though it is not clear that this is a sense of sameness of meaning
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that preserves denotation. But even without this assumption Gödel’s principle
can be applied to (c) in the form:
(d) Frege is the object that has the relation ‘taught’ to Carnap and is identical to
Frege.
21. Ibid., my italics. This is what Gödel says:
There seems to be one purely formal respect in which one may
give preference to Russell’s theory of descriptions. By defining the
meaning of sentences involving descriptions in the above manner, he
avoids in his logical system any axioms about the particle “the”, i.e.,
the analyticity of the theorems about “the” is made explicit; they
can be shown to follow from the explicit definition of the meaning
of sentences involving “the”. Frege, on the contrary, has to assume
an axiom about “the”, which of course is also analytic, but only in
the implicit sense that it follows from the meaning of the undefined
terms. Closer examination, however, shows that this advantage of
Russell’s theory over Frege’s subsists only as long as one interprets
definitions as mere typographical abbreviations, not as introducing
names for objects described by the definitions, a feature which is
common to Frege and Russell.
22. Consider the sort of situation described by Dennett in “Where Am I?”. If it
is not clear that there is just one thing that is Dennett, then it is not clear that
we can go from ‘is a philosopher that is Dennett’ to ‘is the philosopher that is
Dennett’.
23. And it doesn’t do any good to appeal to the form of the argument that uses
(27 6=), i.e., a 6= b, because one would still have to read this in two different ways;
namely:
(276= P-1) [x 6= Nixon](Quine)
and
(276= P-2) [Quine 6= x](Nixon).
24. In terms of the set-theoretic states of affairs of Chapter 1 the states of affairs
denoted by the P-sentences are:
(24P) <set of philosophers, Quine>
(25P)-(27P-1) < {Quine}, Quine >
(27P-2)-(29P) < {Nixon}, Nixon>
(30P) <set of lawyers, Nixon>
And the states of affairs denoted by the I-sentences are:
(25I)-(26I) <extension of Identity, Quine, Quine>
(28I)-(29I) <extension of Identity, Nixon, Nixon>
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25. Thus I’m basically in agreement with Neale’s conclusions (Op. Cit., pp. 815817) concerning the importance of Gödel’s argument for a theory of facts and a
theory of descriptions – and the interconnections between the two. As I mentioned
above, it was precisely Gödel’s argument that led me to the theory of descriptions
in Chapter 3. This theory blocks Gödel’s argument in a somewhat similar way as
Russell’s theory, because the descriptions are in predicate position, yet it treats
descriptions referentially when in subject position. Hence, the problem of substitutivity that Neale raises for the Russellian – argument (76), p. 816 – does
not apply to my theory because in this argument the descriptions are in subject
position and Frege’s principle (R) holds.
Another conclusion of Neale’s is that “logical equivalence is not the
most important issue when it comes to the force of slingshot arguments” (p. 817).
Although I agree that it is not the only main issue, as one may be tempted to
conclude from Church’s general argument and from Davidson’s argument, in my
view it is an equally important issue. The distinction between several notions of
logical equivalence mentioned in note 15 above, including the distinction between
c-logical equivalence and tv-logical equivalence, which does not depend on the
question of the denotation of sentences, plays a basic role in my treatment of many
central problems – as I have already illustrated in this chapter and in chapters 2
and 3.
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Chapter 5
Objections to Facts
Although in recent times the notion of fact, or of state of affairs,
has made something of a come-back, for many years it was considered to be
a rather disreputable notion1 . What is it about facts, or states of affairs,
or similar entities, that is so problematic? As I see it, there are three main
reasons for questioning such entities.
To begin with, there is the claim that various arguments show
that if one ascribes denotation to sentences, then all distinctions collapse
to the two truth values. So the most natural interpretation of facts, or
states of affairs, as the denotation of sentences seems to be blocked by
these arguments. By discussing in some detail the arguments given by
Frege, Church, Davidson, and Gödel, I have tried to show that they do not
establish that conclusion.
It is also not true that the only interpretation of states of affairs,
or facts, is as the denotation of sentences. The logical atomists made facts
one of their central ontological categories, but developed an interpretation
in which it was claimed that sentences did not denote facts. The relation
between a sentence and a fact was a relation of picturing, or of pointing
to or away from a fact, that was supposed to be quite different from the
relation of denotation. This leads to what I consider to be the second main
reason for objecting to facts, states of affairs, and similar entities; namely,
the failure of logical atomism.
It is quite clear that one of the difficulties with the philosophy of
logical atomism had to do with problems in the account of facts. Are there
negative facts? Disjunctive facts? General facts? How is one to handle
negation? And falsity? In his lectures on logical atomism, Russell was time
and again faced with these questions, often by the audience, but wasn’t
really able to give a satisfactory reply to them. So the demise of logical
atomism also contributed to the demise of the notion of fact2 .
The third reason for questioning the notion of fact derives from
some more or less direct objections – emphasized by Quine among others –
concerning both the nature of facts and their usefulness in explaining truth3 .
Quine’s objections to facts are essentially the following. First, facts are not
well-defined; they lack a criterion of identity. Second, they are unnecessary;
they serve merely as useless intermediaries between sentences and the world.
Third, they arise from confusions of use and mention.
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On the question of identity, Quine argues initially against a conception of facts as concrete entities4 :
There is a tendency . . . to think of facts as concrete. This is fostered
by the commonplace ring of the word and the hint of bruteness,
and is of a piece, for that matter, with the basic conception that
it is facts that make sentences true. Yet what can they be, and be
concrete?
Consider the sentences
(1) Fifth Avenue is six miles long,
and
(2) Fifth Avenue is a hundred feet wide.
These sentences “presumably state different facts; yet the only concrete or
at any rate physical object involved is Fifth Avenue.” Fair enough; and
what follows is that facts are not physical objects. One wouldn’t have much
use for them if they were, and Russell certainly did not confuse facts with
physical objects. But if facts are not physical, and concrete, then they are
abstract, and in that case they “are in the same difficulty over a standard
of identity as propositions were seen to be.”
As to the usefulness of facts Quine claims that “surely they cannot
be seriously supposed to help us explain truth.” Why are sentences (1) and
(2) true? Because they refer (or point) to a fact? Not so;
they are true because of Fifth Avenue, because it is a hundred feet
wide and six miles long, because it was planned and made that way,
and because of the way we use our words; only indirection results
from positing facts, in the image of sentences, as intermediaries.
Quine’s further charge that facts arise from a confusion of meaning
with reference is directed at Russell. He says5 :
Russell was receptive to facts as entities because of his tendency to
conflate meaning with reference. Sentences, being meaningful, had
to stand to some sort of appropriate entities in something fairly like
the relation of naming. Propositions in a nonsentential sense were
unavailable, having been repudiated; so facts seemed all the more
needed.
I think that Quine is right about Russell, although I believe that Russell’s
confusions of meaning with reference were largely due to his theory of descriptions, and to his theory of naming more generally, and that it was partly
because of this that he wasn’t able to handle the notion of fact adequately.
164
But, aside from Russell, Quine is obviously not sensitive to any
kind of intuition that a sentence is somehow, as a whole, directed at some
aspect of the world in a specific way. Why is it, to put it in a non Russellian
way, that a denoting definite description describes (or refers to) an object
and a true sentence does not describe (or refer to) anything? And if a true
sentence does describe (or refer to) something, what is it that it describes
(or refers to)? (There is the problem of false sentences, of course, but this
is an exact counterpart to the problem of non-denoting descriptions.) This
is a natural question which cannot be simply dismissed as irrelevant to the
question of truth.
Quine’s objections are hard to assess because they are rather generic and, except for the points against Russell, are not directed at any
specific account of the notion of fact. But it is quite true that facts are often
characterized derivatively from sentences as what accounts for the truth of
a sentence – and propositions are also often characterized derivatively from
sentences as what is expressed by a sentence. Moreover, the word ‘fact’ is
a dangerous word because it suggests both an ontological interpretation of
facts as referents of propositions and a propositional interpretation of facts
as true propositions. This is one of the reasons that one may start talking
about facts in the ontological sense and end up identifying (or confusing)
facts with true propositions – another reason being the problem with false
propositions. I agree that this sort of approach to facts and propositions
is quite unclear as to the nature of these entities and is open to Quine’s
reservations6 . The more general question one can raise, however, is whether
Quine’s considerations are in principle difficulties for an ontological account
of facts (or states of affairs) as the referents of propositions (and sentences).
I will make some preliminary remarks about this leaving a more systematic
discussion of the issues for later chapters.
The difficulty over a standard of identity is a very general one that
applies to all sorts of entities. As Kripke has pointed out, only (perhaps)
in mathematics do we have rigorous criteria of identity7 . And that’s in fact
what Quine often has in mind; extensional entities such as sets. But there
are famous difficulties over criteria of identity also for physical objects, for
example, or persons, or events, or sensations. One may argue, and Quine
does argue, that such entities as properties, propositions and facts are not
even as well off as physical objects, or persons, but this isn’t really clear.
From a theoretical point of view the problems seem to me equally complex,
and from a practical point of view we distinguish properties and propositions
just as easily as we distinguish physical objects.
Thus, to take one of Quine’s examples, even if we assume that the
property of being a kidneyed creature and the property of being a hearted
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creature apply to the same things, we have no more difficulty in distinguishing these two properties as in distinguishing the property of being a
kidney from the property of being a heart or in distinguishing kidneys from
hearts. If we identify facts (or states of affairs) as combinations of a property with some objects and/or properties, then there is no more difficulty
in distinguishing facts as there is in distinguishing properties and objects.
What accounts for the difference between the facts denoted by (1) and (2)
is the difference between the property of being a hundred feet wide and the
property of being six miles long. Of course, if properties and objects are
constitutive of facts (in some sense), then facts will inherit any infirmities
over identity criteria that properties and objects may have; so the question
may indeed reduce to the question of identity for properties and for objects.
A somewhat separate question is whether the identification of facts
with combinations of properties with objects and/or properties is itself problematic. Quine may ask as to the individuation of the fact that is denoted by
a given true sentence. My view is that this depends on the analysis of logical
form, which may involve considerations about ambiguity, vagueness, etc.,
but which is a central aspect of any account of truth, including Quine’s. In
Philosophy of Logic, for instance, he postulates a logical grammar of logical
forms, for which one may suppose that there is a criterion of identity, yet
he has no criterion for determining in general which is the logical form of
an ordinary sentence, or for when two sentences have the same logical form.
This distinction is quite relevant to many of the issues Quine discusses.
In his objections to propositions Quine’s basic point is that we lack
a criterion for determining when two sentences express the same proposition. But there are at least three separate problems. One is whether, given
some notion of proposition, there is a criterion of identity for propositions.
Another is whether there is a criterion of individuation for the proposition
that is expressed by a sentence. A third is whether there is a criterion of
identity for determining when two sentences express the same proposition.
These problems are interconnected, but one may have a good criterion of
identity for propositions without having a good criterion of individuation
for the proposition expressed by a sentence or for when two sentences express the same proposition8 . Similarly, one may have some sort of criterion
of identity for physical objects without having a criterion of individuation
for the physical object (if any) that is denoted by a singular term, or for
when two singular terms denote the same physical object.
Thus, consider Quine’s identification of physical objects with sets
of quadruples of real numbers9 . Although there is a criterion of identity
for such sets, there is no criterion of individuation that determines which
set of quadruples corresponds to a specific physical object as ordinarily
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individuated. Which set corresponds to Fifth Avenue, for instance? In
fact, it seems quite unlikely that one could ever specify such a set for any
physical object. How can one disregard the problem of identity if for most
(or all) physical objects as ordinarily individuated there are infinitely many
different sets of quadruples of real numbers for each of which the question
as to whether it is that object cannot be settled at all?
One can’t even settle the question of which set of sets of quadruples
of real numbers may correspond to a given ordinary physical object. I agree
that for certain purposes one can disregard such questions, but to argue, as
Quine does, that one can thereby “drop” (or “abandon”) physical objects
“in favor of the corresponding classes of quadruples of numbers” (Op. Cit.,
p. 17) seems to me very questionable indeed both from an ontological and
from an epistemological point of view.
In “On the Individuation of Attributes” (pp. 100-101), where
Quine is identifying physical objects with aggregates (or swarms) of molecules, he distinguishes between having a criterion of “individuation” for
physical objects – which is what I called having a criterion of identity for
when two physical objects are the same – from having a criterion of “specification” for which aggregate of molecules corresponds to a given physical
object (Fifth Avenue, say) as ordinarily specified. He claims that the latter
is a problem of “vagueness of boundaries”:
Faulty individuation has nothing to do with vagueness of boundaries. We are accustomed to tolerating vagueness of boundaries.
We have little choice in the matter. The boundaries of a desk are
vague when we get down to the fine structure, because the clustering of the molecules grades off; the allegiance of any particular
peripheral molecule is indeterminate, as between the desk and the
atmosphere. However, this vagueness of boundaries detracts none
from the sharpness of our individuation of desks and other physical objects. What the vagueness of boundaries amounts to is this:
there are many almost identical physical objects, almost coextensive
with one another, and differing only in the inclusion or exclusion of
various peripheral molecules. Any one of these almost coextensive
objects could serve as the desk, and no one the wiser; such is the
vagueness of the desk. Nevertheless they all have their impeccable
principle of individuation; physical objects are identical if and only if
coextensive. Where coextensiveness is not quite fully verifiable, neither is identity, but the identity is still well defined, though the desk
is not. Specification is one thing, individuation another. Physical
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objects are well individuated, whatever else they are not. We know
what it takes to distinguish them, even where we cannot detect it.
Even if one were to agree that coextensiveness is a good criterion of identity
for physical objects in this sense, which I think is questionable, there are
problems with this sort of identification.
Let’s take the desk I am using right now, which happens to be
the desk I bought four months ago. According to Quine’s identifications,
these descriptions (‘the desk I’m using right now’, ‘the desk I bought four
months ago’) denote sets of quadruples of numbers, or certain aggregates
of molecules. Why should these sets or aggregates be the same? If somehow sets or aggregates were assigned directly as denotation to names and
descriptions within the limits of vagueness (with “no one the wiser”), then
there is no reason why names and descriptions that denote the same “ordinary” physical object (this desk, as normally understood) should denote
the same set or aggregate. All the precision of Quine’s criteria of identity
would help not one whit in determining when I am talking about the same
thing. In order to avoid this conclusion one must require that the set or
aggregate that is assigned to names and descriptions that denote the same
“ordinary” physical object must also be the same. Which means that the
criterion of identity for when two names or descriptions denote the same
set or aggregate depends (among other things) on the criterion of identity
for when two names or descriptions denote the same “ordinary” physical
object. Since there is no sharp criterion of identity for when names and
descriptions denote the same “ordinary” physical object, nor is there any
criterion of specification at all for sets or aggregates as denotation of names
and descriptions, all the alleged precision of these ways of treating physical
objects seems to me quite illusory.
It is true that in the preceding discussion I have switched to linguistic descriptions whereas Quine often talks in terms of (proxy) functions
that to “ordinary” physical objects assign sets of quadruples of numbers or
aggregates of molecules. This doesn’t help much, however, because such
talk of functions presupposes that identity is well defined for “ordinary”
physical objects – the same “ordinary” physical object cannot be assigned
different sets or aggregates.
The question is not whether this desk consists of molecules or not,
for we can all agree to that, but the relevance of this to our ordinary identifications of physical objects. To suppose that a criterion of identity in
terms of molecular structure is relevant to our ordinary identifications of
physical objects seems to me obviously false (in general). The distinction
between “ordinary” physical objects and “molecular” physical objects seems
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to me spurious, and the idea of a function from “ordinary” physical objects
to “molecular” physical objects at best a misleading way of speaking of
a function from ordinary descriptions of physical objects to molecular descriptions of physical objects. Even if such molecular descriptions could be
associated to ordinary descriptions, which I think is generally out of the
question, they would necessarily be vague and would neither identify a specific aggregate of molecules nor would allow for a specific description of such
an aggregate to be extracted from them. As I see it the vagueness resides
here, not in my desk (and its boundaries) nor in my descriptions of it as the
desk I’m using right now or as the desk I bought four months ago. Thus
Quine’s considerations seem to me completely irrelevant to the question of
identity for desks. And, as I mentioned before, the identification of physical
objects with sets of quadruples of numbers seems to me a lot worse; in all
respects10 .
I conclude, therefore, that the identification of facts with combinations of properties with objects and/or properties is likely to be no more
problematic in terms of identity conditions than some of the identifications
to which Quine appeals.
About the usefulness of facts in explaining truth I have already
made some remarks in earlier chapters. Although the best reply to this
objection is to develop an account of truth that shows the usefulness of
facts in approaching various problems, including the problem of truth, I
also think that Quine’s remarks are somewhat high-handed. He is not just
appealing to Fifth Avenue, to its planners and makers, and to the way we
use our words, but to Fifth Avenue’s being a hundred feet wide and to Fifth
Avenue’s being six miles long. Part of the way in which we use our words
is that when we state something like (2) we mean to be referring to (or
describing) some feature of the world that consists in Fifth Avenue’s being
a hundred feet wide. Why should Fifth Avenue be a feature of reality and
Fifth Avenue’s being a hundred feet wide not be a feature of reality? I
agree that we can disagree as to how to carve up reality in our metaphysical
theories, but the way we use our words does not settle this issue Quine’s
way. Nor does the appeal to the planners and makers of Fifth Avenue,
because Fifth Avenue may have been planned and made to be a hundred
feet wide and yet not be a hundred feet wide for any of a variety of reasons.
Quine’s view is that (2) is true because the predicate ‘is a hundred
feet wide’ is true of the physical object Fifth Avenue, or because this predicate has an extension to which Fifth Avenue belongs. That’s what saying
that Fifth Avenue is a hundred feet wide amounts to for him. He may agree
that being a hundred feet wide is a feature of Fifth Avenue, and he may
even agree that Fifth Avenue’s being a hundred feet wide is a feature of
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the world; his disagreement is about conferring ontological status to such
“features”. I agree that there is a relevant metaphysical issue here, but
the claim that to divide reality into such features, as well as into physical objects (and other objects), is to posit intermediaries that result only
in indirection does not seem to me justified. One can just as well make
the same claim about Quine’s explanations in terms of predicates and sets.
Doesn’t indirection result from saying that (2) is true because the predicate
‘is a hundred feet wide’ is true of Fifth Avenue – or because Fifth Avenue
belongs to the set of objects that are a hundred feet wide? To the extent
that these help us explain truth, I think that even the account in terms of
set-theoretic states of affairs is a better option. Although this account is
artificial, for several reasons, it is no more artificial than the conception of
sets as the denotation of predicates, or than the conception of predicates as
some sort of primitives that are true of or false of things.
The account of states of affairs in terms of sets, which should
be acceptable to Quine, helps illustrate how one can have a conception
of something like facts as the denotation of sentences which is helpful in
understanding truth as a form of reference and gives reasonable identity
conditions for the denotation of sentences. I actually think – and argue in
Chapter 10 – that the issue of identity is quite complex even for sets and that
the best way to conceive of sets as the extension of predicates is in terms
of states of affairs (and properties); so I am only offering the set-theoretic
account of states of affairs as a partial answer to Quine’s objections.
Although I basically agree with Quine’s dictum that “there is no
entity without identity”, my tentative conclusion from the preceding considerations is that there are certain questions that must be examined more
carefully before one can decide on the worth of his arguments for or against
postulating various sorts of entities. Nevertheless, I do agree that Quine’s
allegations concerning Russell’s notion of fact are largely justified. So let
me now turn to the question of logical atomism11 .
As developed by Russell, logical atomism is a natural outgrow of
the ideas first presented in “On Denoting”. His most sustained account of
it is in the lectures published as “The Philosophy of Logical Atomism” on
which I will center my discussion12 . I shall first outline briefly the main
points I wish to examine in his account of facts and of truth.
Facts, according to Russell, belong to the objective world (183).
They are complex entities that have constituents and components (190-192).
The constituents are particulars (199), and the components are qualities
and relations (196). In combination these give raise to a hierarchy of atomic
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facts constituted by a particular and a quality, two particulars and a binary
relation, three particulars and a ternary relation, etc. (198-199).
Besides atomic facts there are also general facts and existential
facts corresponding to such propositions as ‘all men are mortal’ and ‘there
are men’, respectively (183-184, 236). Are there also molecular facts corresponding to molecular propositions? Russell holds, albeit somewhat reluctantly, that there are negative facts; for example, the fact that Socrates is
not alive (211). On the other hand, he denies that there are conjunctive
facts, or disjunctive facts, or conditional facts (211, 216).
It is very important for Russell that propositions are not names for
facts. He was convinced of this by Wittgenstein, who pointed out that to
each fact there correspond two propositions (187). Consider, for example,
the fact that Socrates is dead. To this fact correspond the propositions
‘Socrates is dead’ and ‘Socrates is not dead’. The very same fact makes
one true and the other false. But they are equally meaningful, and if one
of them named the fact, the other would be a mere noise and hence not
meaningful. That a proposition may correspond to a fact in these two ways
Russell illustrates with the following picture (208):
−−−→
True: Prop. Fact
−−−→
False: Fact Prop.
Not only are not propositions names of facts, but facts cannot be named
at all. A name can only name a particular, and if it doesn’t it’s just a
noise, not a name (187-188). Particulars are named by such words as ‘this’,
which mean nothing aside from being a proper name for a present object of
attention (201, 222).
What particulars there are is for Russell an empirical matter that
does not concern the logician as such (199), but he defines (199, 200):
(3) Particulars = terms of relations in atomic facts. Df.,
and
(4) Proper names = words for particulars. Df.
He also states, though not formally as a definition,
(5) Relations = components of atomic facts,
where ‘component’ is characterized by the “provisional definitions” (196):
(6) That the components of a proposition are the symbols we must understand in order to understand the proposition,
and
(7) That the components of the fact which makes a proposition true or
false, as the case may be, are the meanings of the symbols which we must
understand in order to understand the proposition.
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Although Russell uses the term ‘proposition’ extensively in his
lectures, he holds that interpreted abstractly (or logically) propositions are
“curious shadowy entities” which are not part of the inventory of the world.
The role of true propositions is supposed to be taken over by facts in a more
or less direct way, while false propositions “must . . . be taken to pieces,
pulled to bits, and shown to be simply separate pieces of one fact in which
the false proposition has been analysed away” (223-224)13 .
With this bare outline as a guide to some salient points in Russell’s
account, let’s consider some of the issues.
Russell argues that there must be general facts essentially as follows. Suppose that there are three particulars a, b, and c, that have a
certain quality Q. We have then three atomic facts, say <Q, a>, <Q, b>,
and <Q, c> – I am not attributing any specific interpretation to this notation. Suppose, moreover, that a, b, and c are all the particulars in the
world that have the quality Q. That this is so, says Russell, is a further fact
“about” the world; and it is not an atomic fact, but a general fact.
The proposition that a, b, and c are all the particulars that have
the quality Q, can be expressed as
(8) ∀x(Qx ⇔ (x = a ∨ x = b ∨ x = c)),
while the proposition (or conjunction of propositions) that a, b and c have
the quality Q, can be expressed as
(9) Qa & Qb & Qc.
It is quite obvious that these propositions are not logically equivalent and
that (8) cannot be inferred from (9), for (9) could be true and (8) false. It
is also obvious that if I know that a has Q, b has Q, and c has Q, I don’t
thereby know that a, b, and c are all the particulars that have the quality
Q. So, conversely, if I know that a, b, and c are all the particulars that have
the quality Q, then I know something beyond knowing that a has Q, that b
has Q, and that c has Q. The question is whether from these differences in
the propositions (or sentences, or statements), and in their truth conditions,
one can infer that there is a difference in the facts that account for their
truth.
This is basically what Russell wants to argue, but his arguments
are not quite convincing. His first argument for the existence of general
facts is the following (183-184):
. . . it would be a very great mistake to suppose that you could describe the world completely by means of particular facts alone. Suppose that you had succeeded in chronicling every single particular
fact throughout the universe, and that there did not exist a single
particular fact of any sort anywhere that you had not chronicled,
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you still would not have got a complete description of the universe
unless you also added: ‘These that I have chronicled are all the particular facts there are’. So you cannot hope to describe the world
completely without having general facts as well as particular facts.
Since facts are not supposed to be what one uses to describe the world but
what the world is made of, or part of that, the bearing of these considerations on the existence of general facts (as opposed to general propositions,
or sentences, or statements) is somewhat unclear. Besides, this argument
raises some questions of its own.
Suppose I succeed in chronicling all the facts (particular, general,
etc.); do I still have to add ‘These that I have chronicled are all the facts’ to
obtain a complete description of the world? If I do, then I haven’t chronicled
all the facts and my statement is false. Hence, if I have chronicled all the
facts, the statement that these are all the facts cannot state a new fact. But
why should the statement that such and such chronicling of particular facts
chronicles all the particular facts state a new fact, then? If a chronicling of
all facts (assuming that this makes sense) is a complete description of the
world, it will be so simply in virtue of being a chronicling of all facts. But if
particular facts are all the facts there are, then a chronicling of all particular
facts (assuming that this makes sense) will be a complete description of the
world simply in virtue of being a chronicling of all facts.
One can argue that the statement ‘These are all the facts’ is not
really meaningful in that it involves a circularity (or impredicativity) – which
would follow from Russell’s views about types – whereas the statement
‘These are all the particular facts’ is meaningful and should be true. This
issue is rather complex and I shall not pursue it now, but in any case it
is clear that if Russell’s argument from complete description establishes
anything, and makes sense, it must involve additional considerations.
In a later lecture Russell argues that “[y]ou cannot ever arrive at
a general fact by inference from particular facts, however numerous” (235),
and goes on to use this consideration to reinforce the previous conclusion
(236):
It is perfectly clear, I think, that when you have enumerated all the
atomic facts in the world, it is a further fact about the world that
those are all the atomic facts there are about the world, and that
it is just as much an objective fact about the world as any of them
are. It is clear, I think, that you must admit general facts as distinct
from and over and above particular facts. The same thing applies to
‘All men are mortal’. When you have taken all the particular men
that there are, and found each one of them severally to be mortal,
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it is definitely a new fact that all men are mortal; how new a fact,
appears from what I said a moment ago, that it could not be inferred
from the mortality of the several men that there are in the world.
We see here again a certain ambiguity between facts and propositions. That
the proposition that all men are mortal is different from the totality of
propositions that this man is mortal for each particular man, and cannot
be inferred from them simpliciter, is indeed obvious; as shown by Russell’s
arguments. What is not obvious is what the truth of ‘all men are mortal’
involves in addition to the mortality of each particular man. Well, replies
Russell, it involves that they are all the men; that none was left out. But
we are assuming that the particular facts leave none out, so the general fact
seems to add nothing.
If a universal quantification is interpreted as a (possibly infinite)
generalized conjunction, which is essentially the standard interpretation,
then the same reasons one can give for denying that there is a conjunctive
fact that accounts for the truth of (9) can be given to deny that there is a
general fact that accounts for the truth of (8). The difference between (8)
and (9) would lie in that whereas the truth of (9) derives simply from the
atomic facts <Q, a>, <Q, b> and <Q, c>, the truth of (8) derives from
these facts plus all the negative (particular) facts ¬ <Q, e> (say), where
e is a particular other than a, b and c.
As I pointed out in Chapter 3 (note 21), Russell does make a
distinction between ‘any’ and ‘all’, with ‘any’ interpreted essentially as a
generalized conjunction and ‘all’ interpreted as involving a propositional
function. Although the analysis of ‘all’ is not clearly distinguished from the
analysis of ‘any’ in ontological terms, the connection between general facts
and propositional functions is tentatively brought up by Russell in a passage
immediately following the second passage I quoted above (236-237):
Of course, it is not so difficult to admit what I might call existencefacts – such facts as ‘There are men’, ‘There are sheep’, and so
on. Those, I think, you will readily admit as separate and distinct
facts over and above the atomic facts I spoke of before. Those facts
have got to come into the inventory of the world, and in that way
propositional functions come in as involved in the study of general
facts. I do not profess to know what the right analysis of general facts
is. It is an exceedingly difficult question, and one that I should very
much like to see studied. I am sure that, although the convenient
technical treatment is by means of propositional functions, that is
not the whole of the right analysis. Beyond that I cannot go.
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The main question, therefore, is how to interpret the notion of ‘all’ in relation to the structure of general facts.
In his discussion of Russell’s vicious circle principle Gödel says that
“one may, on good grounds, deny that reference to a totality necessarily implies reference to all single elements of it or, in other words, that “all” means
the same as an infinite logical conjunction”, and he refers to suggestions of
Langford and Carnap “to interpret “all” as meaning analyticity or necessity
or demonstrability”14 . This opens up a whole range of interpretations, some
ontological – if one appeals to some necessary intrinsic feature of men upon
which their mortality depends, for example – and some epistemological – if
one appeals to demonstrability. But such interpretations would not really
reflect Russell’s view, because although the analysis of ‘all’ must be different from the analysis of ‘any’ as a generalized conjunction, there must be a
direct connection between the two.
The most natural ontological interpretation of general facts that
establishes a direct connection with the generalized conjunction account is
to take the fact that all men are mortal as involving an extensional relation
between the properties of being a man and of being mortal. The relation
is the logical relation of Subordination which relates properties in terms of
their extension15 . If one takes propositional functions to be properties, this
Fregean view of quantification – quantification over objects is a second level
property, or concept, that applies to, or relates, first level properties in terms
of their extension – seems to me perfectly compatible with Russell’s views
about general facts; and it also shows that general facts can be conceived
as not being much different in kind and structure from atomic facts.
What is actually somewhat misleading, at least in this conception,
is the terminology ‘general fact’, for general facts are just as specific as particular facts. The difference between the facts corresponding to ‘all men are
mortal’ and to ‘Frege taught Carnap’ is that the former involves a relation
between two properties whereas the latter involves a relation between two
objects – that is, there is a difference of type, or level. It is quite true
that what accounts for the property ‘man’ being extensionally subordinate
to the property ‘mortal’ is that each particular man is individually mortal,
but if one leaves the account of the general fact simply at that, then the
distinction between the general fact and the totality of particular facts becomes unclear. Saying that one is talking about all the men does not really
help much, whereas switching to properties – or even to sets – makes the
distinction clear. The general fact exists because the particular facts do,
but it is a different fact because it involves different components16 .
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Similarly for existential facts. The fact corresponding to the proposition that some men are dentists involves a logical relation that extensionally relates the properties ‘man’ and ‘dentist’ and is different from each
particular fact (or totality of particular facts) consisting of a man who is a
dentist; yet the existence of the existential fact depends on there being at
least one such particular fact. Russell’s reason for claiming that the fact
corresponding to ‘There are men’ is a new fact is essentially that ‘There are
men’ states something about the propositional function ‘x is a man’ rather
than about individual men, but now we have a specific interpretation of
how the propositional function ‘x is a man’ is involved in the fact.
The distinction between the facts that account for the truth of
(8) and (9), or the propositions conjoined in (9), can be conceived as the
distinction between a general fact involving the property [∀x(Zx ⇔ (x =
a ∨ x = b ∨ x = c))](Z) and the quality Q, with (8) interpreted as the
predication
(10) [∀x(Zx ⇔ (x = a ∨ x = b ∨ x = c))](Q),
and the atomic facts <Q, a>, <Q, b> and <Q, c>17 .
Although I am not suggesting that this is Russell’s view, I think
that it is a natural generalization of his account of atomic facts that overcomes the main difficulty in his discussion of general facts, which derives
from attempting to infer their existence from considerations about general
propositions without having a clear conception of their structure and of how
they relate to universal propositions.
The same problem affects Russell’s discussion of negative facts.
To the question “How do you define a negative fact?”, he replied: “You
could not give a general definition if it’s right that negativeness is an ultimate” (216). This was felt to be rather odd, and as Russell tells us “nearly
produced a riot” at Harvard (211).
Russell’s main argument for negative facts is an argument against
explaining negation in terms of incompatibility with “positive” facts. Let’s
take an example. Quine is not now in this room. He is, say, at home in
Boston. The feeling one has is that there is a fact involving Quine’s location right now, and that is the fact that he is at home in Boston. Given
this fact, there are indefinitely many locations at which Quine is not right
now; he is not in this room, he is not on top of the Sugar Loaf, he is not
in Beijing, etc. That he isn’t in any of these places seems to be due not to
some mysterious “negativeness” but, rather, to the fact that he is at home
in Boston right now, and that his being at home in Boston right now precludes his being on top of the Sugar Loaf right now. So, one avoids these
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indefinitely many “negative” facts in favor of one “positive” fact and some
notion of incompatibility.
Russell makes two main points about this. One is that if one takes
a negative existential, for example that there isn’t a hippopotamus in this
room, then the situation is more complex because the corresponding fact
“cannot be merely that every part of this room is filled up with something
that is not a hippopotamus” (213-214). What we would be forced to do,
he claims, is to substitute molecular facts for negative facts, which would
bring no advantage if the idea is to eliminate both molecular and negative
facts.
This example also brings out a certain counterpart to the earlier
Quine case. In the Quine example one was bothered by all the indefinitely
many negative facts and was happy to rule them out through the relation of
incompatibility of the propositions to one positive fact. In the hippopotamus example on the other hand, one has to appeal to indefinitely many
positive facts to rule out the postulation of a single negative fact.
The second point concerns the relation of incompatibility, about
which Russell says (214):
It is clear that no two facts are incompatible. The incompatibility
holds between the propositions, . . . and therefore if you are going to
take incompatibility as a fundamental fact, you have got, in explaining negatives, to take as your fundamental fact something involving
propositions as opposed to facts. It is quite clear that propositions
are not what you may call “real”. If you were making an inventory
of the world, propositions would not come in.
With respect to the first point one could argue that if one takes all
the atomic facts involving all the particulars in the room, then the existence
of a hippopotamus in the room would be incompatible with them (or with
the corresponding propositions). One could argue that this would not work
for the statement that, say, there isn’t a gold atom in the room; in order to
exclude that one would have to say that all those facts about the particulars
in the room are all the facts. What this involves is a general fact though, not
a molecular fact. In fact, since a negative existential is a universal negative,
it is not surprising that negative existentials will involve general facts.
Actually, one would have to do the same thing about the lack of a
hippopotamus in the room, because one is up against Russell’s second point.
What the facts exclude is not the fact that there is a hippopotamus in the
room, because there is no such fact to be excluded or not excluded, but
the proposition that there is a hippopotamus in the room. And, as Russell
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argued before, an existential proposition is not excluded by particular facts;
one needs a general fact.
Of course, Russell’s main point against the relation of incompatibility is that it is a relation between propositions, and that propositions are
not part of the inventory of the world. But this seems to me quite irrelevant
because instead of taking propositions in the abstract sense one can take
sentences, or statements, or whatever Russell is using to talk truly or falsely
about the world. One can then agree with Russell about the existence of
particular and general facts and take the relation of incompatibility to be a
relation between propositions in the non-abstract sense.
In any case, I think that none of this is convincing as an argument
for the existence of negative facts, because, as in the case of general facts,
one needs to have an account of what is involved in negative facts. Without
an account of the structure of negative facts one can argue forever and not
settle the issue. And it seems to me that the most natural account is in
terms of negative (or complementary) properties and relations, generalizing
again Russell’s account of atomic facts.
It is important to notice that if one thinks of the components of
facts as sets and relations in the extensional sense, then there is no special problem about negativeness. Suppose that the universe of particulars
U = {a, b, c, d, e}, and that Q = {a, b, c}. Then the fact that a is Q is
< Q, a >, i.e., < {a, b, c}, a >, and the fact that d is not Q is < U − Q, d >,
i.e., < {d, e}, d >. There is no negativeness here. Evidently, if Q is a quality, then one cannot treat it extensionally, and one cannot obtain another
quality from it either by complementation or by negation. But one can get
a negative property such as not-white; though this is something that Russell
doesn’t do.
I think that this interpretation through negative properties is the
best interpretation for negative facts, but it does require a subject-predicate
analysis of propositions (or sentences) in general and a clear distinction
between predicate negation and sentential negation. I would imagine that
this is one of the reasons why Russell did not consider treating negative
facts in this way.
Let us turn now to the question of truth. Russell’s argument for
saying that propositions don’t name facts was that to a single fact there
correspond two propositions; the affirmation and the negation of the fact,
so to speak. The true proposition is the one that points to the fact, and
the false proposition is the one that points away from the fact. This is
actually quite close to Frege’s postulation of the truth values the True and
the False, except that instead of postulating truth values as two different
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objects denoted by propositions, Russell postulates them as two different
connections between propositions and reality.
That Russell’s is a postulation rather than an explanation is quite
clear because he has no analysis of how a proposition points to a fact or
how it points away from a fact. He says (208-209):
Supposing you have the proposition ‘Socrates is mortal’, either there
would be the fact that Socrates is mortal or there would be the fact
that Socrates is not mortal. In the one case it corresponds in a way
that makes the proposition true, in the other case in a way that
makes the proposition false. This is one way in which a proposition
differs from a name.
Thus, as in Frege’s introduction of the True as the circumstance that a
thought is true and of the False as the circumstance that a thought is
false, Russell postulates a way of correspondence that makes a proposition
true and a way of correspondence that makes a proposition false. What
is missing in Frege’s case is an account of the truth values as objects, and
what is missing in Russell’s case is an account of the two ways in which the
proposition may correspond to the fact. How do propositions point to or
away from facts18 ?
From the provisional definitions (6) and (7) one would expect that
a proposition points to or away from a fact due to a correlation between
the structure of the proposition and the structure of the fact. For a simple
sentence such as ‘Socrates is mortal’ the correlation seems quite obvious, for
the meaning of the predicate ‘is mortal’ is the property of being mortal and
the meaning of the name ‘Socrates’ is the individual Socrates, which are
the component and constituent of the corresponding fact. Although this is
actually incorrect, because the sentence ‘Socrates is mortal’ should have an
extremely complex analysis that would not correspond to an atomic fact, it
will apply to a sentence such as ‘this is red’, where the meaning of ‘this’ is
a present momentary red patch.
Russell’s notion of meaning is really a notion of reference – and the
word ‘meaning’ is so used in the recent translations of Frege’s ‘Bedeutung’.
We can say, therefore, that the sentence (or proposition) ‘this is red’ points
to the fact consisting of the particular red patch and the quality red in the
sense that it means that fact; and it means that fact because its component words ‘this’ and ‘is red’ mean the constituent and component of that
fact. Thus, at least as far as such examples go, Russell’s notion of meaning
behaves rather like Frege’s notion of reference in that the meaning of the
complex expression ‘this is red’ would seem to depend on (or be functionally
related to) the meanings of the component expressions ‘this’ and ‘is red’.
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And (7) suggests that this may hold in general. Hence, even though Russell
claims that true propositions do not name facts, it seems to me quite consistent with his view that they point to facts in the sense that they mean
facts, or refer to facts.
Suppose now that the particular patch in question is blue rather
than red. How is the correlation between the false proposition ‘this is red’
and the corresponding fact to be made? The corresponding fact is not
the fact consisting of the particular patch and the quality blue, but is the
negative fact that the patch is not red. Since Russell gives no analysis
of negative facts we can’t really tell how the meaning of ‘red’ enters into
the fact and how ‘this is red’ points away from the fact. But even if the
negative fact had the property not-red as component and the particular
patch as constituent, it is not clear in what sense ‘this is red’ points away
from this fact. It is not because the property not-red is not the meaning of
‘red’, because the quality blue is also not the meaning of ‘red’ and yet ‘this
is red’ does not point away from the fact consisting of the quality blue and
the particular patch. The notion of negation must play an essential role in
the explanation of pointing away, but Russell’s account does not make this
role explicit.
If one analyses negative facts as I suggested above, however, then a
natural account for the notion of pointing away would be the following. The
proposition ‘this is red’ points away from the negative fact consisting of the
particular patch and the property not-red in that its predicate negation ‘this
is not red’ points to (means, refers to) that fact. In general, a proposition
points away from a fact if and only if its predicate negation points to that
fact. Of course, what this amounts to is an explanation of truth and falsity
in terms of reference – or meaning in Russell’s sense. For its generality the
explanation requires that all sentences (propositions) should be analysed
as having subject-predicate form and that all facts should have the sort of
structure that Russell attributes to atomic facts.
On this account a false proposition does not mean (or refer to) a
fact, which is as it should be, but has an indirect connection to a fact through
its predicate negation. Russell’s “pointing away” – and the picture with the
arrow – suggests a more direct connection, but even as a figure of speech it is
not altogether clear. My explanation is somewhat like saying that to point
away from an object is to point in a direction which if reversed points to that
object, which seems to be essentially what Russell has in mind. This reduces
the notion of pointing away to the notions of pointing and of reversal of
direction, where for propositions reversal of direction is figuratively achieved
by predicate negation. That Russell’s notion of pointing away does require
a specific appeal to the notion of negation is shown by the fact that when
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I say ‘this is red’ where the ‘this’ means a particular blue patch I am not
pointing away from the fact consisting of that patch and the quality blue.
It seems clear, in any case, that whatever analysis one may want to give
of Russell’s ways of correspondence cannot depend merely on the imagery
of pointing (to something or away from something) but must depend on
structural considerations concerning both propositions and facts and on
specific connections between parts of propositions and parts of facts – as
Russell’s provisional definitions (6) and (7) suggest.
I conclude from the preceding remarks that Russell’s account of
truth and falsity can be reasonably formulated as a referential account of
truth and falsity, with true propositions characterized as those that refer to
a fact and false propositions characterized as those whose predicate negation
refers to a fact. What seems to me to confuse the issue is Russell’s insistence
that facts cannot be named either by true propositions or “in any other
way”19 .
On the face of it, it seems curious that one can name a particular
by ‘this’ but cannot name a fact by ‘this’. If I say ‘this is intolerable’
referring to the fact that a student is standing on my desk, I seem to be
saying that this, the fact present to my attention consisting of the student
standing on the desk, is intolerable, and therefore naming that fact by ‘this’
just as much as I name a flower by ‘this’ when I say ‘this is white’. Hence,
in an ordinary sense of ‘naming’ it is not at all clear that one cannot name
facts. The problem is that Russell’s view has nothing to do with ordinary
considerations about naming but is based on the idea that names can only
name simples20 .
Russell would deny that when I say ‘this is white’, referring to
the flower, I am naming the flower. Complex expressions do not name,
precisely because they are complex and can be analysed in simpler terms;
but simple expressions do not name anything that can be described by a
complex expression, because the analysis shows that the “thing” that is
allegedly described is not really real – it is, rather, a logical construct. So
complex names name nothing, and simple names name simples. Since a fact
is complex it cannot be named, and since a flower is complex it cannot be
named either. So what Russell is actually doing is postulating a very narrow
notion of naming (and name) that restricts names to words like ‘this’ and
‘that’ which by assumption can only name particulars – as his definition (4)
suggests.
What Russell ends up saying is that facts cannot be named because
“they are not there to be named”, although he also appeals to “the fact that
you cannot name them” as a reason for facts not being “properly entities
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at all”21 . This is quite problematic, however, even aside from the obvious
circularity involved. That qualities and relations cannot be named by ‘this’
does not show that they are not properly entities, and apparently one can
refer to them, or mean them. Moreover, both particulars and relations are
characterized in terms of atomic facts in definitions (3) and (5), which is
rather odd if particulars and relations are properly entities and facts are
not.
It seems to me, therefore, that the question concerning the connection between propositions and facts in the account of truth and falsity is not
so much that to each fact there correspond two propositions, but that one
doesn’t know in what terms the question should be approached; especially
since neither propositions nor facts are properly entities.
My brief examination of Russell’s formulation of logical atomism
was not intended to refute it as a philosophical theory but to give some
plausibility to the conclusion that it wasn’t the notion of fact as such that led
to trouble. I am actually sympathetic to some central aspects of the project,
though not in the way it was carried out. There should have been a logical
development of the structure of facts (general facts, negative facts, etc.), of
quantification and other logical notions in relation to facts, of propositions
(or whatever it is that points to or away from facts) and of these two ways
of correspondence, and of naming, meaning, meaningfulness, etc. But there
wasn’t; there was no theory of facts at all. What there was, and Quine
is quite right about this, was a tremendous confusion between facts and
propositions, knowledge and reality, use and mention22 .
The more general conclusion that I draw is that although there
are reasons to be suspicious of an analysis of truth in terms of facts (or
states of affairs), neither Frege’s arguments, nor the additional arguments
by Church, Davidson, and Gödel, nor the demise of logical atomism, show
that there is anything especially problematic in the way of developing such
an analysis along the lines suggested in Chapter 1.
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Notes
1. Dummett, for instance, commenting on Frege’s thesis that sentences denote
the True and the False, says (Frege: Philosophy of Language, p. 182):
It is generally agreed that, if Frege had to ascribe reference to sentences at all, then truth-values were by far the best thing he could
have selected as their referents: at least, he did not go down the
dreary path which leads to presenting facts, propositions, states of
affairs or similar entities as the referents of sentences.
Although facts, states of affairs, propositions, etc. play an important role in the
work of many philosophers, a significant factor in the recent revival of interest in
such entities was the development of situation semantics by Barwise and Perry.
2. The rise and fall of logical atomism is analysed in some detail in Urmson
Philosophical Analysis – my interpretation of some of the issues is different from
his, however. The failure of logical atomism is important as an argument against
facts both because of the central role that facts played in logical atomism and
because logical atomism is supposed to have been a particularly well developed
logical and metaphysical theory. Thus Urmson says (p. 4):
Logical atomism was presented as a superior metaphysics which was
to replace inferior ones, not as an attack on metaphysics as such. Indeed, as presented in the documents of its hey-day, logical atomism
is one of the most thorough-going metaphysical systems yet elaborated. . . . For breadth of sweep, clarity, detailed working-out, and
consistency it can have few rivals.
I don’t agree with the last claim and I think that the notion of fact in particular
was very poorly worked-out by the logical atomists, both from a logical and from
a metaphysical point of view. This is what I argue in the later part of this chapter
in connection with Russell.
3. One of the thrusts of Quine’s philosophy since the late 40’s was the development
of various arguments to sweep away with all sorts of allegedly dubious philosophical entities: facts, propositions, meanings, possible objects, etc. Quine’s main
tools have been Ockham’s razor and criteria of identity, and the enormous influence of his philosophy during the 50’s and 60’s all but buried such entities as
propositions and facts. Since the 70’s however, Quine’s influence on some of these
issues has lessened, and propositions and facts began to surface again.
4. The following quotations are from Word and Object, p. 247.
5. “Russell’s Ontological Development”, p. 82 – see also p. 83.
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6. But this depends on the characterization of propositions. In Chapter 12 I
introduce a notion of proposition on the basis of which it is quite natural to
characterize states of affairs as true propositions.
7. Naming and Necessity, p. 43.
8. If propositions are characterized as what is expressed by a sentence, then the
problems of identity do indeed collapse into the problem of determining when two
sentences express the same proposition.
9. See, for example, “Things and Their Place in Theories”, pp. 17-18.
10. This is an attempted way out by Quine of the problem of identity for physical objects – see “Whither Physical Objects?”. I come back to this question in
Chapter 10.
11. I discuss the question of criteria of identity at some length in chapters 9
and 10 – and in some later chapters as well. Quine’s dictum about identity –
which he formulates in various works and I quoted from “On the Individuation of
Attributes”, p. 102 – derives essentially from Frege’s views in The Foundations
of Arithmetic, and there is an initial discussion of it in connection with Frege in
Chapter 8.
12. All numbers in parentheses in the rest of the chapter are references to this
work.
13. The whole passage is the following:
Time was when I thought there were propositions, but it does not
seem to me very plausible to say that in addition to facts there are
also these curious shadowy things going about such as ‘That to-day
is Wednesday’ when in fact it is Tuesday. I cannot believe they go
about the real world. It is more than one can manage to believe,
and I do think no person with a vivid sense of reality can imagine it.
. . . To suppose that in the actual world of nature there is a whole set
of false propositions going about is to my mind monstrous. I cannot
bring myself to suppose it. I cannot believe that they are there in
the sense that facts are there. There seems to be something about
the fact that ‘To-day is Tuesday’ on a different level of reality from
the supposition ‘That to-day is Wednesday’. When I speak of the
proposition ‘That to-day is Wednesday’ I do not mean the occurrence
in future of a state of mind in which you think it is Wednesday, but
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I am talking about the theory that there is something quite logical,
something not involving mind in any way; and such a thing as that
I do not think you can take a false proposition to be. I think a
false proposition must, wherever it occurs, be subject to analyses,
be taken to pieces, pulled to bits, and shown to be simply separate
pieces of one fact in which the false proposition has been analysed
away. I say that simply on the ground of what I should call an
instinct of reality.
14. “Russell’s Mathematical Logic”, p. 219.
15. I.e., the fact that all men are mortal is the fact that I expressed as (23i )
in Chapter 1. A somewhat similar line is taken by Neale in “The Philosophical
Significance of Gödel’s Slingshot”, p. 768.
16. In the actual case of the subordination of the property ‘man’ to the property
‘mortal’ I would hold that there is an intrinsic relation as well, but in my view
this is a different fact involving a different relation of subordination.
17. One can interpret (8) in a different way so that the fact corresponding to (8)
involves a relation between Q, a, b and c corresponding to the predication
(10′ ) [∀x(Zx ⇔ (x = u ∨ x = v ∨ x = w))](Q, a, b, c).
And one can also interpret (9) in different ways; e.g.
(9′ ) [Qx & Qy & Qz](a, b, c).
The fact corresponding to (9′ ) would have as component a ternary relation that
holds between three particulars just in case these particulars have the quality Q.
I see no reason to disqualify such relations and such facts.
18. In an earlier passage Russell says (187-188):
There are two different relations, as you see, that a proposition may
have to a fact: the one the relation that you may call being true to
the fact, and the other being false to the fact. Both are equally essentially logical relations which may subsist between the two, whereas
in the case of a name, there is only one relation that it can have to
what it names. A name can just name a particular, or, if it does not,
it is not a name at all, it is a noise. It cannot be a name without
having just that one particular relation of naming a certain thing,
whereas a proposition does not cease to be a proposition if it is false.
It has these two ways, of being true and being false, that together
correspond to the property of being a name. Just as a word may be
a name or be not a name but just a meaningless noise, so a phrase
which is apparently a proposition may be either true or false, or may
185
be meaningless, but the true and false belong together as against the
meaningless. That shows, of course, that the formal logical characteristics of propositions are quite different from those of names, and
that the relations they have to facts are quite different, and therefore
propositions are not names for facts.
19. Russell continues (188):
You must not run away with the idea that you can name facts in
any other way; you cannot. You cannot name them at all. You
cannot properly name a fact. The only thing you can do is to assert
it, or deny it, or desire it, or will it, or wish it, or question it, but
all those are things involving the whole proposition. You can never
put the sort of thing that makes a proposition to be true or false
in the position of a logical subject. You can only have it there as
something to be asserted or denied or something of that sort, but
not something to be named.
20. This is the idea of logical atoms (179):
The reason that I call my doctrine logical atomism is because the
atoms that I wish to arrive at as the sort of last residue in analysis
are logical atoms and not physical atoms. Some of them will be
what I call ‘particulars’ – such things as little patches of colour or
sounds, momentary things – and some of them will be predicates or
relations and so on.
There are actually two versions of the logical atoms in Russell. One, the absolute
version, according to which the logical atoms are “known only inferentially as the
limits of analysis”; the other, a relativistic version, according to which one can
take as atoms any “objects of some one type, even if these objects are not simple”
(“Logical Atomism”, p. 337):
When I speak of ‘simples’ I ought to explain that I am speaking of
something not experienced as such, but known only inferentially as
the limit of analysis. It is quite possible that, by greater logical skill,
the need for assuming them could be avoided. A logical language
will not lead to error if its simple symbols (i.e. those not having
any parts that are symbols, or any significant structure) all stand
for objects of some one type, even if these objects are not simple.
The only drawback to such a language is that it is incapable of
dealing with anything simpler than the objects which it represents
by simple symbols. But I confess it seems obvious to me (as it did to
Leibniz) that what is complex must be composed of simples, though
the number of constituents may be infinite.
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21. In the last lecture Russell sums up (270):
One purpose that has run through all that I have said, has been the
justification of analysis, i.e., the justification of logical atomism, of
the view that you can get down in theory, if not in practice, to ultimate simples, out of which the world is built, and that those simples
have a kind of reality not belonging to anything else. Simples, as I
tried to explain, are of an infinite number of sorts. There are particulars and qualities and relations of various orders, a whole hierarchy
of different sorts of simples, but all of them, if we were right, have
in their various ways some kind of reality that does not belong to
anything else. The only other sort of object you come across in the
world is what we call facts, and facts are the sort of things that are
asserted or denied by propositions, and are not properly entities at
all in the same sense in which their constituents are. That is shown
in the fact that you cannot name them. You can only deny, or assert, or consider them, but you cannot name them because they are
not there to be named, although in another sense it is true that you
cannot know the world unless you know the facts that make up the
truths in the world; but the knowing of facts is a different sort of
thing from the knowing of simples.
22. A careful recent reformulation of some aspects of the metaphysics of logical
atomism is developed by Armstrong in A Combinatorial Theory of Possibility.
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Chapter 6
Truth, Denotation, and Interpretations
I will now formulate some of the logical ideas involved in the previous chapters in terms of a formal syntax and a set-theoretic semantics for
first order logic1 .
The vocabulary of the language consists of the connectives
¬, &, ∨, ⇒, ⇔;
the quantifiers
∀, ∃;
identity
=;
infinitely many individual constants
a, b, c, . . . ;
infinitely many individual variables
u, v, w, . . . ;
infinitely many n-ary predicate constants (n > 0)
An , B n , C n , . . . ;
infinitely many n-ary predicate variables (n > 0)
U n, V n, W n, . . . ;
parentheses, brackets, and comma.
Although this language has predicate variables, these variables
cannot be quantified, so it is essentially a first order language – or an unquantified second order language. The syntactic notions of formula, scope of
a quantifier, free and bound variable, etc., are the usual ones2 . A closed formula is one that has no free variables, and since I will use the term ‘sentence’
in a special sense, it is important not to call closed formulas ‘sentences’.
We have two kinds of predicates:
(1) For n > 0, an n-ary first order predicate is an expression of the form
[ϕ](α1 , . . . , αn ), where ϕ is a formula that contains free only the distinct
individual variables α1 , . . . , αn .
(2) For m > 0 and n ≥ 0, an m+n-ary second order predicate is an expression
of the form [ϕ](ρ1 , . . . , ρm , α1 , . . . , αn ), where ϕ is a formula that contains
only the distinct predicate variables ρ1 , . . . , ρm , and that contains free only
the distinct individual variables α1 , . . . , αn .
And we have sentences:
(3) A first order sentence is an expression of the form [ϑ](β1 , . . . , βn ), where
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ϑ is an n-ary first order predicate and β1 , . . . , βn are individual constants
(not necessarily distinct).
(4) A second order sentence is an expression of the form [ϑ](π1 , . . . , πm , β1 ,
. . . , βn ), where ϑ is an m + n-ary second order predicate, π1 , . . . , πm are
first order predicates (not necessarily distinct) of the same arity as the corresponding predicate variables in ϑ, and β1 , . . . , βn are individual constants
(also not necessarily distinct).
To each closed formula of ordinary first order logic there corresponds a certain set of first order and/or second order sentences, some of
which give different readings of its logical structure. For example, as in
Chapter 1, corresponding to the closed formula
(5) Rab
we have the following nine sentences
(6) [[Rxy](x, y)](a, b)
(7) [[Rxb](x)](a)
(8) [[Rax](x)](b)
(9) [[Zab](Z)]([Rxy](x,y))
(10) [[Zxb](Z,x)]([Rxy](x, y), a)
(11) [[Zax](Z,x)]([Rxy](x, y), b)
(12) [[Zxy](Z, x, y)]([Rxy](x, y), a, b)
(13) [[W x](W, x)]([Rxb](x ), a)
(14) [[W x](W, x)]([Rax](x ), b).
The standard reading of (5) is (6), but each of the others “interprets” the logical structure of (5) in some way. The difference between
them is a difference as to what they are saying about what. Of course, since
one can use different variables, there are infinitely many sentences that are
readings of (5), but they split into equivalence classes of (6)-(14). We can
say that these are the canonical readings of (5) in the subject-predicate language, (6)-(8) being first order readings and (9)-(14) second order readings.
As another example, consider the closed formula
(15) ∀x(F x ⇒ Gx).
Among the different readings of (15) we have
(16) [[∀x(V x ⇒ W x)](V, W )]([F x](x), [Gx](x)),
which is the most natural reading of (15), and
(17) [[∀xWx](W )]([F x ⇒ Gx](x)),
which corresponds to the usual open sentence reading of the universal quantifier. We also have
(18) [[∀x(F x ⇒ W x)](W )]([Gx](x)),
and
(19) [[∀x(V x ⇒ Gx)](V )]([F x](x)).
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Among the predicates we can distinguish the purely logical predicates which do not contain non-logical constants. The predicates of (12)(14) and (16)-(17) are logical predicates in this sense. Similarly, we can
have purely logical sentences in which no non-logical constants occur; for
instance,
(20) [[∀xZxx](Z)]([x = y](x, y))3 .
In order to do a denotational semantics for this language we must
first define the states of affairs that can be the denotations of sentences:
(21) Let D be a non-empty set and n > 0. A level 1 state of affairs over
D is any sequence < R, d1 , . . . , dn >, where d1 , . . . , dn ∈ D, R is an n-ary
relation on D (i.e., R ⊆ Dn ), and < d1 , . . . , dn >∈ R.
(22) Let D be a non-empty set, n ≥ 0, and m > 0. A level 2 state of affairs
over D is a sequence < K, A1 , . . . , Am , d1 , . . . , dn >, where d1 , . . . , dn ∈ D,
each Ai is a ki -ary relation on D, K is an m+n-ary relation ⊆ P(Dk1 )×. . .×
P(Dkm ) × Dn – if n = 0, this last term drops out – and < A1 , . . . , Am , d1 ,
. . . , dn >∈ K.
We may say that the level 1 and level 2 states of affairs over D
are jointly the elementary states of affairs over D. It is important to notice
that the totality of elementary states of affairs over a non-empty set D is
completely determined by D (and by the notion of power set).
The semantics of the language is carried out in a straightforward
way using the ordinary semantic constructions for first order logic. The
notion of interpretation, or structure, is the standard one, and we can define
satisfaction for formulas in the usual way4 . We then extend the notion of
denotation so that each n-ary first order predicate denotes an n-ary relation
on the domain D of the interpretation, and each second order m + n-ary
predicate denotes a set of m + n-ary tuples consisting of m ki -ary relations
on D and n elements of D. (I am assuming for the moment that every nonlogical constant denotes in every interpretation.) If dℑ is the denotation
function of an interpretation ℑ, each sentence either denotes an elementary
state of affairs in an obvious way or does not denote anything, and we define
truth in ℑ as follows:
(23) A first order sentence
σ = [[ϕ](α1 , . . . , αn )](β1 , . . . , βn )
is true in ℑ if and only if dℑ (σ) is defined in ℑ; i.e.,
< dℑ ([ϕ](α1 , . . . , αn )), dℑ (β1 ), . . . , dℑ (βn ) >
is a level 1 state of affairs of ℑ. σ is false in ℑ if and only if its predicate
negation
[[¬ϕ](α1 , . . . , αn )](β1 , . . . , βn )
is true in ℑ.
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(24) A second order sentence
σ = [[ϕ](ρ1 , . . . , ρm , α1 , . . . , αn )](π1 , . . . , πm , β1 , . . . , βn )
is true in ℑ if and only if dℑ (σ) is defined in ℑ ; i.e.,
< dℑ ([ϕ](ρ1 , . . . , ρm , α1 , . . . , αn )), dℑ (π1 ), . . . , dℑ (πm ), dℑ (β1 ), . . . , dℑ (βn ) >
is a level 2 state of affairs of ℑ. σ is false in ℑ if and only if its predicate
negation
[[¬ϕ](ρ1 , . . . , ρm , α1 , . . . , αn )](π1 , . . . , πm , β1 , . . . , βn )
is true in ℑ.
In this formulation truth is denotation and the elementary states
of affairs are a generalization, via Frege, of Russell’s facts. And in a sense
we have atomic, molecular (negative, conjunctive, disjunctive, etc.), and
general states of affairs conceived set-theoretically on a par with the usual
set-theoretic semantics of first order logic. We can also generalize this formulation to higher order logic by introducing as many levels of predicate
variables and constants as the order of the logic. Thus, for second order
logic we can introduce n-ary second order predicate constants
An , B n , C n , . . . ,
n-ary second order predicate variables
U n, V n, W n, . . . ,
and define third order predicates, third order sentences, and level 3 states
of affairs. The second order closed formula
(25) ∀Z∀x(Zx ⇒ Zx)
can then be expressed as the third order sentence
(26) [[∀ZYZ](Y)]([∀x(Zx ⇒ Zx)](Z)),
which denotes a level 3 state of affairs in every interpretation5 .
If all one is interested in is in the truth or falsity of the original
closed formulas, then all the different readings of these formulas in the
subject-predicate language may be of no interest. But then, under the
assumption that all terms denote, the different readings of a given closed
formula will either be all true or all false, and nothing is lost. If one is
only interested in certain questions about truth and falsity, therefore, one
may gain nothing from the complications introduced above, and since the
increase in notational complexity is considerable it is not surprising that
the distinctions are not normally made6 .
The usual notions of logical truth, logical consequence and logical
equivalence are defined exclusively in terms of truth value. Thus:
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(27) A sentence σ is a logical truth if, and only if, σ is true (i.e., dℑ is
defined) in every interpretation ℑ;
(28) A sentence σ is a logical consequence of a set of sentences Γ if, and only
if, for any interpretation ℑ, if all the sentences in Γ are true in ℑ, then σ is
also true in ℑ.
For logical equivalence I will distinguish two notions:
(29) A sentence σ is c-logically equivalent to a sentence σ ′ if, and only if, σ
and σ ′ are logical consequences of each other,
where a sentence is a logical consequence of another sentence if it is a logical
consequence of its unitary set,
and
(30) A sentence σ is tv-logically equivalent to a sentence σ ′ if, and only if, σ
and σ ′ have the same truth value in every interpretation ℑ.
It follows that the various readings ϕ∗ of a closed formula ϕ that is a logical
truth are all logical truths, that the various readings ϕ∗ of a closed formula
ϕ that is a logical consequence of a set of closed formulas Γ are logical
consequences of any set Γ∗ of readings of the formulas in Γ, and that the
various readings ϕ∗ and ψ∗ of two logically equivalent closed formulas ϕ
and ψ are logically equivalent sentences according to both definitions. The
distinctions that we built into the logical notation have no effect because
we are only concerned with the truth value of sentences. If we take our
denotational semantics seriously, however, and especially if we allow for the
possibility of non-denoting terms and predicates, then the situation becomes
more interesting. Let’s consider the case of logical equivalence that came
up in Chapter 4.
It is quite obvious that logical equivalence as defined in (29) and
(30) does not preserve denotation in our semantics of states of affairs. We
can define a notion of denotational equivalence as follows:
(31) A sentence σ is denotationally equivalent to a sentence σ ′ if and only
if for every interpretation ℑ, either dℑ (σ) and dℑ (σ ′ ) are both defined in ℑ
and dℑ (σ) = dℑ (σ ′ ), or dℑ (σ) and dℑ (σ ′ ) are both undefined in ℑ.
The various readings (6)-(14) and (16)-(19) are not denotationally equivalent because in each case the state of affairs denoted is different.
Since denotational equivalence only makes distinctions that are
reflected in the denotation that the sentences have in various interpretations,
and since in each interpretation all false sentences denote the “same thing”
(i.e., nothing), denotational equivalence behaves as if we had something like
Frege’s the False as the common denotation of all false sentences. Thus,
whereas the sentences
(32) [[x = x](x)](a)
and
193
(33) [[x = x](x)](b)
are not denotationally equivalent, because they will denote different states
of affairs if the denotation of ‘a’ is different from the denotation of ‘b’, their
predicate negations
(34) [[x 6= x](x)](a)
and
(35) [[x 6= x](x)](b)
are denotationally equivalent.
Given our definition of falsity, however, we can introduce a stronger
notion of denotational logical equivalence as follows:
(36) A sentence σ is d-logically equivalent to a sentence σ ′ if and only if for
every interpretation ℑ, dℑ (σ) and dℑ (σ ′ ) are defined and dℑ (σ) = dℑ (σ ′ ),
or dℑ (¬σ) and dℑ (¬σ ′ ) are defined and dℑ (¬σ) = dℑ (¬σ ′ ), or none of
dℑ (σ), dℑ (σ ′ ), dℑ (¬σ) and dℑ (¬σ ′ ) are defined,
where the negations indicate predicate negation. We can justify the additional condition on negations by parity with the definition of tv-logical
equivalence.
This definition characterizes logical equivalence as a generalized
material equivalence. If the totality of interpretations consists of only one
interpretation, then tv-logical equivalence is material equivalence in that
interpretation. As we take more interpretations into account we get finer
distinctions, but the basis of these distinctions is still the notion of material
equivalence. Since two sentences are materially equivalent if and only if they
have the same truth value, a natural counterpart of material equivalence in
terms of states of affairs is to require that the states of affairs that account
for the truth or falsity of the sentences be the same. Generalizing on this
stricter notion of material equivalence in an interpretation we get the notion
of d-logical equivalence.
This notion makes finer distinctions than the notions of logical
equivalence defined in terms of truth values, and we may consider to what
extent differences between these notions reflect assumptions about the denotation of non-logical terms and predicates in an interpretation. The standard assumption, which I have been adopting so far in this chapter, is that
all non-logical constants denote in every interpretation. If we drop this
assumption we enlarge the totality of interpretations in a way that has a
considerable effect on the notions defined in terms of truth value.
A simple way to modify the semantics uniformly is the following.
Individual constants need not denote and predicate constants need not denote. Any predicate (and term) that involves a non-denoting non-logical
constant does not denote. Any sentence that involves either a non-denoting
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predicate or a non-denoting subject does not denote – but such sentences
aren’t false; they are neither true nor false.
We can see that with this semantics sentences such as (32) are not
logical truths anymore – the only logical truths are sentences like (20) which
do not involve non-logical constants. Moreover, neither (32) and (33) nor
(34) and (35) are tv-logically equivalent, because in some interpretations
one of them may be true (or false) and the other truth-valueless. Naturally,
with this semantics sameness of truth value means that in each interpretation the sentences are both true, or both false, or both neither true nor
false. All the definitions were formulated so as to allow for truth-valueless
sentences, but whereas tv-logical equivalence becomes more discriminating,
d-logical equivalence is not affected by this change of semantics. Nevertheless the two notions do not coincide, for sentences (6)-(8) and (16)-(17) are
still tv-logically equivalent. And neither of them coincides with c-logical
equivalence, because whereas (32) and (33) are not c-logically equivalent,
(34) and (35) are still c-logically equivalent7 .
The notion of d-logical equivalence is a natural notion in a semantics where sentences denote, and in a semantics where sentences denote
the truth values the True and the False, tv-logical equivalence is essentially
denotational equivalence. In fact, given the subject-predicate analysis, the
definition of falsity, and the definition of d-logical equivalence, we could do
without the False altogether, for tv-logical equivalence is d-logical equivalence when true sentences denote the True. We could then think of a sentence denoting the True as the sentence denoting some (unspecified) aspect
of the world.
It is also worth noting that denotational equivalence can be defined
not only for sentences but for any denoting expressions such as terms and
predicates – and d-logical equivalence can also be directly generalized to
predicates. Since the only denoting terms we have considered so far are individual constants, denotational equivalence for terms is not an interesting
notion – because no two individual constants are denotationally equivalent.
But if we add descriptions and functional constants and variables, then we
do get an interesting notion that has many of the features of a notion of
logical equivalence. And it is also natural to distinguish terms (and predicates) that denote in the same interpretations from terms (and predicates)
that have the same denotation or no denotation in every interpretation.
Although the first notion is trivial if we assume that terms and predicates
always denote, it is not trivial otherwise.
In fact, since our semantics is entirely denotational we can extend
the notion of logical consequence to a more general relation. Let ∆ be a
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set of sentences, predicates, and/or closed terms, and η be a sentence, a
predicate or a closed term. Then:
(37) η is a logical consequence of ∆ if and only if for every interpretation ℑ,
if all elements of ∆ denote in ℑ, then η denotes in ℑ.
I will use the usual symbol ‘|=’ for logical consequence to indicate this relation, and will use ‘|= =|’ to indicate a broader notion of c-logical equivalence,
which for sentences coincides with c-logical equivalence as introduced earlier. Then, even though (32) is not a logical truth, we can assert that it is
true (i.e., denotes) in every interpretation in which ‘a’ denotes:
(38) a |= [[x = x](x)](a).
Similarly, whereas
(39) 6|= [[(F x ∨ ¬F x)](x)](a),
and
(40) [[∀xZx](Z)]([F x](x)) 6|= [[F x](x)](a),
we can assert
(41) [F x](x), a |= [[(F x ∨ ¬F x)](x)](a),
and
(42) [[∀xZx](Z)]([F x](x)), a |= [[F x](x)](a).
But let me now turn to descriptions and sentential logic.
We can introduce a description operator in our language so that for
any predicate [ϕ](α) with exactly one free variable α, ια([ϕ](α)) is a closed
term. For any interpretation ℑ such that [ϕ](α) denotes a singleton in ℑ,
the term ια([ϕ](α)) denotes the element of that singleton, and is undefined
otherwise.
Besides this description operator we can introduce a uniqueness
operator ‘!’ that works as follows. For any predicate [ϕ](α1 , . . . , αn ), we
can get another predicate
(43) [!αi ϕ](α1 , . . . , αn )
meaning that αi is the only thing such that [ϕ](α1 , . . . , αn ). The occurrence
of the uniqueness operator can actually be eliminated in terms of quantification and identity, and (43) considered to be an abbreviation for
(44) [ϕ & ∀α(ϕ′ ⇒ α = αi )](α1 , . . . , αn ).
where ϕ′ is the result of substituting α for all free occurrences of αi in ϕ,
with α a variable that does not occur in ϕ8 .
Although this may seem at first sight an unusual operator, it is
not; we use it all the time. It’s the way we get predicates like ‘x is the
father of y’, ‘x is the husband of y’, ‘x is the square of y’, etc. The difference
is that we normally treat those predicates as functional expressions, and we
write them as ‘the father of (y) = x’, ‘the husband of (y) = x’, ‘the square
of (y) = x’, etc9 . In many occasions this makes no difference, but since one
can introduce functional variables and constants into the language nothing
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is lost by treating these “functional” predicates as predicates. Semantically,
of course, the denotation of these predicates requires uniqueness of the ith
member of any n-tuple that satisfies ϕ relative to the other members – otherwise the predicate does not apply to that n-tuple. Since these predicates
are functional predicates, this means that as “functions” they may be only
partially defined. The predicates ‘x is the brother of y’ and ‘x is the largest
even divisor of y’ are partially defined in this sense, and the predicate ‘x
is the positive multiple of y’ is undefined for every “argument” y – which
means, however, that it denotes the empty set, not that it does not denote.
We can also generalize the ordinary description operator for predicates [ϕ](α1 , . . . , αn ) with n free variables so that
[ιαi ([ϕ](α1 , . . . , αn ))](α1 , . . . , αi−1 , αi+1 , . . . , αn )
is a term that in any interpretation ℑ denotes a (possibly) partial function.
The syntactic difference between
[ιαi ([ϕ](α1 , . . . , αn ))](α1 , . . . , αi−1 , αi+1 , . . . , αn )
and
[!αi ϕ](α1 , . . . , αn ).
is that the first is a term and the second is a predicate, but for n > 1 there
is no difference in their denotation. Since [!αi ϕ](α1 , . . . , αn ) is a predicate
however, we can apply the uniqueness operator simultaneously to several
variables. Thus we can express the predicate ‘x is the brother of z and z is
the sister of y’ as
(45) [!x!z(x is a brother of z & z is a sister of y)](x, y, z),
and the predicate ‘x is the square of the even prime y’ as
(46) [!x!y(x is a square of y & y is even & y is prime)](x,y).
Using the description and uniqueness operators we can see that an
ordinary sentence of the form
(47) a is the F,
can have the two readings that I distinguished in Chapter 3; the predicate
reading
(48) [[!xF x](x)](a),
which means
(49) [[F x & ∀y(F y ⇒ y = x)](x)](a),
and the identity reading
(50) a = ιx([F x](x)),
which means
(51) [[x = y](x, y)](a, ιx([F x](x))).
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It is quite clear that (49) and (51) are not d-logically equivalent or even tvlogically equivalent – because if ‘[Fx](x)’ does not apply to a unique thing,
then (49) will be false and (51) will be truth valueless10 .
In Russell’s theory of descriptions (50) is supposed to be the reading of (47), but since the results of taking the description as a denoting term
are counter-intuitive as an analysis of (47), (47) is interpreted informally as
(49) and formally as
(52) ∃x(F x & ∀y(F y ⇒ y = x) & a = x).
The most natural way of formulating Russell’s analysis in terms of my notation is
(53) [[∃xZx](Z)]([F x & ∀y(F y ⇒ y = x) & a = x](x)),
although one can also formulate it as
(54) [[∃x(Zx & ∀y(Zy ⇒ y = x) & a = x)](Z)]([F x](x)),
or as
(55) [[∃x(Zx & ∀y(Zy ⇒ y = x) & w = x)](Z, w)]([F x](x), a),
or as
(56) [[∃x(Zx & ∀y(Zy ⇒ y = x) & W x)](Z, W )]([F x](x), [a = x](x)).
If ‘a’ and ‘F’ always denote, then (49) and (53)-(56) are tv-logically equivalent in the standard semantics, but are not d-logically equivalent. The
important sense in which all these sentences are logically equivalent, independently of whether or not ‘a’ and ‘F’ denote, is the sense of mutual logical
consequence; i.e., c-logical equivalence.
Also Frege’s analysis (51) is c-logically equivalent to Russell’s analysis (53) – and, hence, to (54)-(56) – because if either of these sentences is true
in an interpretation, then the other must be true in that interpretation11 .
This may account for a certain feeling that one has, at least sometimes,
that in some sense both Frege and Russell are right in their analyses of descriptions. What confuses the issue is that (51) and (53) are not tv-logically
equivalent, which seems to imply that they should not be logically equivalent at all. I think that the most reasonable thing to say is that the main
notion of logical equivalence is c-logical equivalence, and that this notion
preserves neither truth value nor denotation12 .
We must consider now what happens with the sentential connectives as connectives of sentences in the subject-predicate analysis of first
order logic.
Let us introduce sentential variables
p, q, r, . . . ,
into the language and define (pure) sentential formulas in the usual way.
We can then define sentential predicates:
(57) For n > 0, a sentential predicate is an expression of the form [ϕ](δ1 , . . . ,
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δn ), where ϕ is a sentential formula and δ1 , . . . , δn are the distinct sentential
variables occurring in ϕ.
And sentential sentences:
(58) A sentential sentence is an expression of the form [ϑ](σ1 , . . . , σn ), where
ϑ is a sentential predicate and σ1 , . . . , σn are (non-sentential) sentences (not
necessarily distinct).
The closed formula
(59) ¬F a,
for instance, has the first order reading
(60) [[¬F x](x)](a),
the second order readings
(61) [[¬Zx](Z, x)]([F x](x), a)
and
(62) [[¬Za](Z)]([F x](x)),
and the sentential reading
(63) [[¬p](p)]([[F x](x)](a)).
We thus have four different readings for (59), and the same holds for the
other connectives and for more complex closed formulas. The question now
is how to interpret sentential sentences semantically.
The standard interpretation is to take the sentential connectives
as truth functions that operate on the denotations of sentences, which are
conceived as two objects T and F. The nature of these objects is either left
unclear or is stipulated arbitrarily – as 1 and 0, or as the very letters ‘T’
and ‘F’. Since we are treating sentences as denoting states of affairs and as
possibly lacking denotation, we cannot take the connectives to be functions
of states of affairs, but a natural counterpart of the standard semantics is to
take sentential sentences as either denoting states of affairs or not denoting
anything, depending on the argument sentences of the sentential predicate.
Thus, we may take the denotation of (63) to be the denotation of (60), if
any, and the denotation of, say,
(64) [[p & q](p, q)]([[F x](x)](a), [[Gx](x)](a)),
to be the denotation of
(65) [[F x & Gx](x)](a).
We can easily generalize this to all sentential sentences, and what
we get is essentially what we already have; namely, the subject-predicate
reading of first order sentences. There is a notational advantage, though,
because the sentential sentences give us a general uniform way of expressing
the “composite” subject-predicate sentences in terms of their “components”
without actually having to write them down. To do this directly in terms of
the predicates of the “component” sentences is somewhat messy (because
variables may have to be changed and re-arranged), so we can do it once
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and for all schematically and then use the notation freely.
Even those who postulate truth values as the denotation of sentences often read the connectives in terms of expressions like ‘it is the case
that’ or ‘it is not the case that’, however. Thus, the negation of a sentence
S is often read as ‘it is not the case that S’, and the disjunction of two sentences S and S ′ is often read as ‘either S is the case or S ′ is the case’. This
can be interpreted in several ways, but one interpretation is that the ‘T’ and
‘F’ are really abbreviations for predicates of sentences rather than names of
objects. This is actually quite reasonable because it is hard to find anyone
who takes seriously the idea that truth values are objects in a significant
philosophical sense. What this suggests, though, is that sentential variables
are intuitively interpreted as ranging over sentences rather than truth values, and that sentential logic is a logic of sentences – or propositions, or
something like that; but right now I am sticking to sentences13 .
This is also relevant to the question of non-denoting terms because
there are rather strong intuitions that certain sentential combinations involving non-denoting terms should be true. An example may be
(66) Socrates is mortal or Sherlock Holmes is mortal.
The feeling is that the first disjunct, being true, is enough to make the whole
disjunction true. This feeling, or intuition, seems to me to be based on the
view that (66) says that either ‘Socrates is mortal’ is true, or that ‘Sherlock
Holmes is mortal’ is true. Since ‘Socrates is mortal’ is indeed true, it follows
that (66) is true. I quite agree; if that’s what (66) says, then it is true. It
is true because if I assert the disjunction, I am asserting something about
the sentences ‘Socrates is mortal’ and ‘Sherlock Holmes is mortal’.
We see, therefore, that with this reading of propositional assertions
the intuition that (66) is true is well justified even if ‘Sherlock Holmes’ does
not denote. We also see that this intuition is based on an interpretation
of the sentential connectives as predicate operators that operate on the
predicate ‘is true’ (or ‘is the case’).
I conclude from the preceding considerations that as part of predicate logic sentential logic is accounted for by the interpretation of sentential
sentences that I suggested earlier, but that there is room for a sentential
logic of sentences; in particular, for a sentential logic of the sentences of
first order logic. So let’s think of the sentential variables as ranging over
the sentences of the first order language. We can then allow quantification
of sentential variables in sentential formulas and re-define sentential predicates by adding the qualification that δ1 , . . . , δn are the distinct sentential
variables occurring free in ϕ. Sentential sentences are then defined as before.
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To do the semantics we can extend the notion of denotation in
an interpretation ℑ so that for each sentential variable δ, dℑ (δ) is a (nonsentential) sentence of the first order language. We can then say that:
(67) dℑ satisfies the sentential formula ϕ if and only if
(1) ϕ = δ, where δ is a sentential variable, and dℑ (δ) is true in ℑ;
(2) ϕ = ¬ψ and dℑ does not satisfy ψ;
(3) ϕ = (ψ ∨ χ) and dℑ satisfies ψ or dℑ satisfies χ;
(4) ϕ = (ψ & χ) and dℑ satisfies both ψ and χ;
(5) ϕ = (ψ ⇒ χ) and dℑ satisfies ψ only if dℑ satisfies χ;
(6) ϕ = (ψ ⇔ χ) and dℑ satisfies ψ iff dℑ satisfies χ;
(7) ϕ = ∃δψ and d′ℑ satisfies ψ, for some d′ℑ that differs from dℑ at most
on the assignment to δ;
(8) ϕ = ∀δψ and d′ℑ satisfies ψ, for all d′ℑ that differ from dℑ at most on
the assignment to δ.
We can now define:
(68) A sentential sentence [[ϕ](δ1 , . . . , δn )](σ1 , . . . , σn ) is true in an interpretation ℑ if, and only if, any dℑ such that dℑ (δi ) = σi (1 ≤ i ≤ n) satisfies
ϕ; otherwise [[ϕ](δ1 , . . . , δn )](σ1 , . . . , σn ) is false in ℑ.
With these definitions we have a standard “two-valued” classical
sentential logic where the sentential variables do not range over truth values
but over the sentences of the first order language. Since we have a predicate
‘is true’ defined for these sentences, we can use this predicate to define truth
for the sentential sentences. That is, we are treating sentential predicates as
complex “truth relations” among non-sentential sentences, which is pretty
much what one does in ordinary sentential logic if one does not mean the
truth values to be really objects.
The sentential reading of (59) is now different semantically from
the other readings, for whereas (60)-(62) may be truth-valueless in an interpretation ℑ, (63) is either true or false in ℑ; it is true in ℑ if (60) is false in
ℑ, and it is false in ℑ if (60) is either true or truth-valueless in ℑ. If every
sentence were either true or false, however, as in the standard semantics,
then all the different readings would be tv-logically equivalent.
We can also identify states of affairs denoted by the true sentential
sentences, although these states of affairs are different from the elementary
states of affairs that account for the truth or falsity of non-sentential sentences. Given the analysis of (66) in terms of truth, (66) should be interpreted as having the logical form
(69) [[p is true ∨ q is true] (p, q)] (‘Socrates is mortal’, ‘Sherlock Holmes is
mortal’),
where ‘∨’ is a predicate operator. Since the extension of the predicates
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(70) p is true,
and
(71) q is true,
is the set of sentences that denote a state of affairs, the extension of the
predicate
(72) [p is true ∨ q is true](p, q)
is the set of ordered pairs of sentences such that at least one of them denotes
a state of affairs. Therefore the state of affairs denoted by (66) is:
(73) <extension of (72), ‘Socrates is mortal’, ‘Sherlock Holmes is mortal’>.
Thus, in this interpretation the various sentential “connectives”
are really the corresponding predicate operators operating on the predicate
‘is true’, and sentential logic is a special kind of predicate logic. It is in
this sense that I referred to sentential predicates as truth relations, and to
emphasize this I will write these predicates explicitly in terms of a truth
predicate as ‘[Tp](p)’, ‘[¬ Tp](p)’, ‘[Tp ∨ Tq](p, q)’, etc., and refer to them
in general as T-predicates rather than sentential predicates. (The latter
terminology, which we can abbreviate to ‘S-predicate’, I will reserve for the
earlier interpretation of sentential logic.) When we apply these T- predicates
to sentences of a specific language we may want to use quotes to avoid
ambiguity, and we can consider that for any sentence of this language the
result of putting quotes around it is a closed term of the sentential logic. A
T-sentence is then something like
(74) [[Tp & Tq](p, q)](‘[[F x](x)](a)’, ‘[[Gx](x)](a)’).
We could also allow other designations for sentences, but I won’t get into
that right now.
One way to formalize the denotational semantics is to add to any
interpretation ℑ a second domain DSℑ consisting of all the sentences of the
first order language, and define:
(75) A sentential state of affairs of ℑ is an ordered sequence < R, σ1 , . . . ,
σn >, where R is an n-ary relation on DSℑ and < σ1 , . . . , σn > is an n-tuple
of elements of R.
Each n-ary T-predicate denotes such an n-ary relation in every interpretation ℑ, and a T-sentence is true in ℑ if, and only if, it denotes a sentential
state of affairs of ℑ, and is false in ℑ if, and only if, its predicate negation
denotes a sentential state of affairs of ℑ.
We may also want to iterate sentential discourse so that T- sentences can themselves be arguments of T-predicates. To do this we can
distinguish orders of sentences, with the original sentences of first order
logic having order 0, and the T-sentences involving only such arguments
having order 1. In general, T-sentences of order n(n ≥ 1) are T-sentences
whose arguments of highest order are of order n-1. Similarly, we can intro-
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duce a sequence of domains DSℑ0 ⊆ DSℑ1 ⊆ . . ., and a corresponding sequence
of sentential states of affairs of order 1, order 2, etc. In each case we can
define truth and falsity as before, and at the limit we get something like a
sentential logic of order ω associated to the original first order language14 .
These considerations are relevant to the distinction between the
material conditional and material implication, and to other similar distinctions. I think that the introduction of truth values as objects by Frege, and
their assimilation as a formal expedient by logicians, created the illusion
that one can hold both that propositional assertions are not about propositions (or sentences) and that they are not really about the objects the True
and the False either – or about any other entities. This leads to a rather
obscure explanation of the distinction between material implication and the
material conditional, material equivalence and the material biconditional,
etc., as the distinction between a relation that relates sentences and a connective that “connects” sentences into another sentence. Although formally
this seems coherent, I think that the explanation is quite unsatisfactory because the content of the connected sentence is explained in terms of truth
tables which do not have a clear interpretation if one denies that sentences
literally denote truth values15 .
In my notation the distinction between the material conditional
(76) If Mary went to the movies, then Peter went home,
and the material implication
(77) ‘Mary went to the movies’ implies ‘Peter went home’,
is the distinction between
(78) [[p ⇒ q](p, q)]([[x went to the movies](x)](Mary), [[x went home](x)]
(Peter))
and
(79) [[Tp ⇒ T q](p, q)](‘[[x went to the movies](x)](Mary)’, ‘[[x went home](x)]
(Peter)’).
The content of (78) is
(80) [[x went to the movies ⇒ y went home](x, y)](Mary,Peter),
which asserts a relation of Mary and Peter and is true if Mary did not go
to the movies or Peter went home. More precisely, the predicate of (80)
denotes a relation between objects, and the sentence is true if, and only
if, the names ‘Mary’ and ‘Peter’ denote objects having this relation. The
content of (79), on the other hand, is essentially (77), and it is true if, and
only if, ‘Mary went to the movies’ is not true or ‘Peter went home’ is true.
If the names ‘Mary’ and ‘Peter’ denote, then (76) and (77) are
materially equivalent, and this material equivalence can be expressed as the
second order T-sentence
(81) [[Tp ⇔ Tq](p, q)]((78), (79)),
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where in place of ‘(78)’ and ‘(79)’ one should have the corresponding sentences within quotes. But the sentences are not c-logically equivalent, and
hence not logically equivalent in any of the senses I distinguished, because
whereas (77) is a logical consequence of (76), (76) is not a logical consequence of (77) – for if the name ‘Mary’ does not denote, (77) is true and
(76) is not.
One thing that becomes quite clear with this treatment of sentential logic is the nature of the connection between a sentence σ and a Tsentence [[Tp](p)](σ) that asserts its truth. Such pairs are always c-logically
equivalent, but are only tv-logically equivalent if σ is a logical truth – and
even for logical truths we don’t get d-logical equivalence. If σ is not a logical
truth, then σ and [[Tp](p)](σ) are materially equivalent if, and only if, σ is
either true or false. Of course, if we assume that every sentence is either
true or false and that sentences denote truth values, then all distinctions
collapse.
I will conclude now by commenting briefly on the question of
whether a “logically perfect language” should allow for the generalized possibility of lack of denotation that I have been allowing in the semantics.
It certainly does not seem to be the proper business of logicians
to determine whether ‘Sherlock Holmes’, ‘Homer’, ‘Odysseus’, ‘unicorn’,
‘centaur’, ‘tiger’, etc., have denotation or not16 . Why should a logician care
about that? The standard view seems to be that unless these terms have
denotation, sentences involving them may lack truth value, and, therefore,
we either won’t be able to apply logic to them or we would have to give up
certain logical laws such as the law of excluded middle17 .
I have tried to show that this view is incorrect. We can apply logic
to sentences involving non-denoting terms without any difficulties whatsoever. What we cannot do is say that a sentence like
(82) Sherlock Holmes = Sherlock Holmes
is a logical truth, or that it is a logical consequence of the law of identity
(20); but this seems pretty obviously as it should be. We can always state
(82) in the form (38), i.e.
(83) Sherlock Holmes |= Sherlock Holmes = Sherlock Holmes.
And we don’t actually have to give up the principle of excluded
middle in its propositional form, because
(84) [p is true ∨ p is not true](p)
is true for any sentence p, whether p has denotation or not. Therefore, if
we distinguish between, say,
(85) [[F x ∨ ¬F x](x)](a),
which is neither true nor false if ‘a’ or ‘F’ do not denote, and
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(86) [[Tp ∨ ¬ Tp](p)](‘[[F x](x)](a)’),
which is true, and logically true, we have no failures of logic – and we can
always state (85) in the form (41). Moreover, even if ‘F’ fails to denote,
this is no objection to excluded middle for properties (or concepts, or sets),
because the proper formulation of excluded middle should be in terms of
variables; i.e., in second order logic,
(87) [[∀ZYZ](Y)]([∀x(Zx ∨ ¬Zx)](Z)),
or
(88) [[∀Z∀xYZx](Y)]([Zx ∨ ¬Zx](Z, x)).
In first order logic what we can say is that the logical predicates
(89) [∀x(Zx ∨ ¬Zx)](Z)
and
(90) [(Zx ∨ ¬Zx)](Z, x)
are universally applicable to all sets (or properties) and objects.
From the point of view of first order logic as a theory of logical
relations between sets (or properties) and objects, the introduction of nonlogical constants is a mistake; one should only use variables ranging over
the appropriate entities – a point that was often emphasized by Russell18 .
Since in first order logic one does not have quantification for predicate (or,
rather, property or set) variables, one should interpret the logical truths as
universally applicable logical predicates. If one has a description operator,
however, then the question of lack of denotation for singular terms still
arises, and there is no good reason I can see why one should make sure that
the descriptions always denote.
If, on the other hand, one is thinking of first order logic as a sort
of applied logic, then it makes sense to have the non-logical constants. And
it also makes sense to require that they always denote, but then one is
restricting the range of applications to those cases where this condition
holds19 . It is still a mistake, however, to characterize logical truth, logical
consequence and logical equivalence as if such applications were the only
ones that matter. And, in any case, this does not give an adequate treatment
of descriptions and of partially defined functional terms.
It is quite clear that one loses absolutely nothing by making the
various distinctions I made in this chapter, and one can certainly ignore
them when they are not relevant to the particular subject-matter. In fact,
I think that one loses by failing to make the distinctions, because this leads
to a series of confusions. Thus Frege’s confusion that to assert a sentence
is the same as asserting the truth of that sentence; Russell’s confusion that
when I say that the mailman came, I am not talking about the mailman
but about the property of being a mailman; etc. Since this permeates the
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entire treatment of logic, and language, at the end we never know what we
are really talking about.
From an epistemological point of view, the idea that we must always assure a denotation for everything, even if artificially, also seems to
me completely unjustified. It is true that in science, and in ordinary communication, we generally mean to refer with names and predicates. But we
cannot always be sure that we do refer, and there are many cases where we
know that we don’t. The mathematician knows full well that the description ‘the largest odd prime’ does not refer to anything. What is the point
of saying that it refers to (say) 0? Does this make mathematical language
more “perfect”? In every discourse there will be combinations of predicates
with predicates and with names that yield descriptions that do not refer
to anything. And there may even be combinations of predicates that refer
which yield predicates that do not refer – as, for example, in the expression
‘particle x has such an such momentum and such and such position’. It
makes no sense for logic to rule this out.
The problem of lack of denotation is not really a problem of ordinary language versus scientific language, but is an important ontological
and epistemological problem that logic cannot put aside by relegating it to
ordinary language. Although the various moves with which the problem is
avoided seem innocuous, they have very serious consequences. The result
are a number of confusions that are very hard to sort out – as we shall see
in later chapters.
Aside from the confusions, the effect of artificially assigning a denotation to every term is to get landed with a lot of strange “true” sentences;
for instance, that the largest odd prime is less than 5. One can avoid some
of this by means of Scott’s treatment of descriptions mentioned in Chapter
4 (note 18), but one doesn’t avoid all – the largest odd prime is identical
to the largest even number – and the treatment is still artificial. It seems
much more reasonable to me to acknowledge that in those cases we are not
talking about anything and that we are none the worse for that.
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Notes
1. This is an interim formulation in the extensional spirit discussed at the end of
Chapter 1. A more systematic treatment is given in the logic chapters of Part II.
2. See Mates Elementary Logic, Chapter 3. The only difference with Mates’
formulation is that I’m using a slightly different symbolism and that I don’t have
sentential variables (or constants); but I will introduce sentential variables later.
3. Although there are many logical predicates in first order logic, the only logical
sentences are sentences like (20) whose arguments are logical predicates built up
from identity and logical notions. But from second order logic on we do get a rich
stock of logical sentences.
4. See Enderton A Mathematical Introduction to Logic, §2.2.
5. The way I am drawing the distinction between predicates and sentences of
various orders is in terms of the arguments of the predicate. This means that an
ordinary second order closed formula may have first order, second order and third
order readings. For example, (25) has the second order reading
(26′ ) [[∀xW x](W )]([∀Z(Zx ⇒ Zx)](x))
as well as the alternative third order reading
(26′′ ) [[∀Z∀xYZx](Y)]([Zx ⇒ Zx](Z, x)),
all of which are logical sentences. The standard second order axiom of induction,
formulated as the closed formula
(a) ∀W ((W o & ∀u∀v((W u & Suv) ⇒ W v)) ⇒ ∀uWu),
has (among others) the third order reading
(b) [[∀W YW ](Y)]([(W o & ∀u∀v((W u & Suv) ⇒ W v)) ⇒ ∀uWu](W)),
the second order reading
(c) [[∀W (W x & ∀u∀v((W u & Zuv) ⇒ W v)) ⇒ ∀uWu)](Z,x)]([Sxy](x, y), o),
and the first order reading
(d) [[∀W (W x & ∀u∀v((W u & Suv) ⇒ W v)) ⇒ ∀uWu)](x)](o).
6. The same thing happens when one has to choose between Russell’s theory of
descriptions and a referential theory of descriptions, because the distinctions I’m
making parallel Russell’s distinctions of scope.
7. The general connection between the behavior of the various logical notions in
the new semantics with respect to their behavior in the earlier semantics is as
follows:
(a) A sentence σ is a logical truth if and only if σ is a logical sentence and is a
logical truth in the earlier semantics.
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(b) A sentence σ is a logical consequence of a set of sentences Γ if and only if either
Γ is consistent, σ is a logical consequence of Γ in the earlier semantics, and all
non-logical constants occurring in σ occur in sentences of Γ, or Γ is inconsistent,
where Γ is consistent if and only if there is an interpretation ℑ such that all
sentences of Γ are true in ℑ.
(c) Two sentences σ and σ ′ are c-logically equivalent if and only if either {σ} and
{σ ′ } are consistent, σ and σ ′ are c-logically equivalent in the earlier semantics, and
σ and σ ′ contain the same non-logical constants, or {σ} and {σ ′ } are inconsistent.
(d) Two sentences σ and σ ′ are tv-logically equivalent if and only if σ and σ ′ are
tv-logically equivalent in the earlier semantics and contain the same non-logical
constants.
8. This analysis of descriptions can also be generalized to higher order logic. In
fact, even in first order logic as I formulated it above (with predicate variables), the
two operators can be applied to predicate variables as well as individual variables
– although one cannot then eliminate such uses of the uniqueness operator in
terms of quantification and identity.
9. In Principia Mathematica *30, Whitehead and Russell call such functions
“descriptive functions” and give the following “general definition” (p. 232):
30.01. R ‘y = (ιx)(xRy) Df.
Since in any specific context the description is to be eliminated, we get that
(a) a = R ‘b
becomes
(b) ∃x(xRb & ∀z(zRb ⇒ z = x) & a = x),
which is tv-logically equivalent to
(c) [[!xRxb](x)](a),
in my notation.
10. There are other readings of (50); for instance:
(51′ ) [[a = x](x)](ιx([F x](x))),
(51′′ ) [[x = ιx([F x](x))](x)](a),
(51′′′ ) [[x = ιx([Zx](x))](Z, x)]([F x](x), a).
11. We can also state the usual condition that the description ‘ιx([F x](x))’ denotes
in an interpretation if, and only if, the predicate ‘[Fx](x)’ applies to exactly one
thing in that interpretation as the c-logical equivalence
ιx([F x](x)) |= =| [[∃x(Zx & ∀y(Zy ⇒ y = x))](Z)]([F x](x)).
12. In fact, this is the notion I generally have in mind when I speak of logical
equivalence.
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If we look again at Gödel’s argument (24P)-(30P) at the end of Chapter
4, we can see that the problematic steps (24P)-(25I), (26I)-(27P-1), (27P-1)-(27P2), (27P-2)-(28I), (29I)-(30P), are logically equivalent sentences both in the sense
of c-logical equivalence and of tv-logical equivalence. Similarly for the steps (24P)(25P), (26P)-(27P-1), (27P-2)-(28P), and (29P)-(30P). But of these only (26P)(27P-1) and (27P-2)-(28P) are d-logically equivalent sentences.
13. This is essentially Boole’s distinction between primary and secondary propositions (An Investigation of the Laws of Thought, pp. 52-53):
Every assertion that we make may be referred to one or the other of
the two following kinds. Either it expresses a relation among things,
or it expresses, or is equivalent to the expression of, a relation among
propositions. An assertion respecting the properties of things, or the
phaenomena which they manifest, or the circumstances in which
they are placed, is, properly speaking, the assertion of a relation
among things. To say that “snow is white,” is for the ends of logic
equivalent to saying, that “snow is a white thing.” An assertion
respecting facts or events, their mutual connexion and dependence,
is, for the same ends, generally equivalent to the assertion, that such
and such propositions concerning those events have a certain relation
to each other as respects their mutual truth or falsehood. The former
class of propositions, relating to things, I call “Primary;” the latter
class, relating to propositions, I call “Secondary.” The distinction
is in practice nearly but not quite co-extensive with the common
logical distinction of propositions as categorical or hypothetical.
Later in the book Boole remarks on the relation between a secondary proposition
and its subject propositions (p. 163):
Again, the relations among these subject propositions are relations
of coexistent truth or falsehood, not of substantive equivalence. We
do not say, when expressing the connexion of two distinct propositions, that the one is the other, but use some such forms of speech as
the following, according to the meaning which we desire to convey:
“Either the proposition X is true, or the proposition Y is true;” “If
the proposition X is true, the proposition Y is true;” “The propositions X and Y are jointly true;” and so on.
14. The systematic discussion of propositional logic begins in Chapter 16 and the
treatment there is somewhat different from the specific formulation in sentential
terms I just gave. There are also certain questions one can raise in connection
with uses of T-sentences that I discuss in the next chapter.
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15. That is, if sentences do not literally denote truth values, then the only reasonable interpretation for truth tables is in terms of the predicates ‘is true’ and ‘is
false’, which is the interpretation of the conditional as material implication, the
biconditional as material equivalence, etc.
16. A linguistic explanation as to why some of these terms don’t have denotation
is given in Chapter 11.
17. We have seen this claimed by Russell as an objection to Frege’s analysis of
ordinary language (Chapter 3, note 18), but the point was made forcefully by
Frege himself in many occasions. For example, in “Function and Concept”, pp.
32-33, he says:
It seems to be demanded by scientific rigour that we should have provisos against an expression’s possibly coming to have no reference;
we must see to it that we never perform calculations with empty
signs in the belief that we are dealing with objects. People have in
the past carried out invalid procedures with divergent infinite series.
It is thus necessary to lay down rules from which it follows, e.g.,
what
‘⊙ + 1’
stands for, if ‘⊙’ is to stand for the Sun. What rules we lay down
is a matter of comparative indifference; but it is essential that we
should do so – that ‘a + b’ should always have a reference, whatever
signs for definite objects may be inserted in place of ‘a’ and ‘b’.
This involves the requirement as regards concepts, that, for any
argument, they shall have a truth-value as their value; that it shall
be determinate, for any object, whether it falls under the concept or
not. In other words: as regards concepts we have a requirement of
sharp delimitation; if this were not satisfied it would be impossible
to set forth logical laws about them. For any argument x for which ‘x
+ 1’ were devoid of reference, the function x + 1 = 10 would likewise
have no value, and thus no truth-value either, so that the concept
‘what gives the result 10 when increased by 1’
would have no sharp boundaries. The requirement of the sharp delimitation of concepts thus carries along with it this requirement for
functions in general that they must have a value for each argument.
In his (second) manuscript “Logic”, pp. 129-130, Frege talks of proper names that
do not denote as “mock proper names” and of thoughts that are neither true nor
false as “mock thoughts”. He says:
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The sentence ‘Scylla has six heads’ is not true, but the sentence
‘Scylla does not have six heads’ is not true either; for it to be true
the proper name ‘Scylla’ would have to designate something. . . .
Names that fail to fulfill the usual role of a proper name, which is
to name something, may be called mock proper names. Although
the tale of William Tell is a legend and not history and the name
‘William Tell’ is a mock proper name, we cannot deny it a sense.
But the sense of the sentence ‘William Tell shot an apple off his son’s
head’ is no more true than is that of the sentence ‘William Tell did
not shoot an apple off his son’s head’. I do not say, however, that
this sense is false either, but I characterize it as fictitious. . . .
Instead of speaking of ‘fiction’, we could speak of ‘mock thoughts’.
Thus if the sense of an assertoric sentence is not true, it is either
false or fictitious, and will generally be the latter if it contains a
mock proper name. . . .
The logician does not have to bother with mock thoughts, just as
a physicist, who sets out to investigate thunder, will not pay any
attention to stage-thunder. When we speak of thoughts in what
follows we mean thoughts proper, thoughts that are either true or
false.
I quite agree that it is reasonable to distinguish cases which we know to be fictional
from others, but the problem is that we do not know in general which cases are
fictitious (in a broader sense) and which are not. Moreover, it would seem that
logic should apply to fictional discourse just as much as to non-fictional discourse,
so that the sentence ‘William Tell shot something off his son’s head’ should be a
logical consequence of the sentence ‘William Tell shot an apple off his son’s head’.
This is true in my semantics; i.e., lack of denotation has no significant effect on
logical consequence.
18. See, for instance, “Mathematical Logic as Based on the Theory of Types”, p.
171, Principia Mathematica 1, p. 93, and the quotation in the next note.
19. In Human Knowledge: Its Scope and Limits Russell says (pp. 88-89):
Pure logic has no occasion for names, since its propositions contain
only variables. But the logician may wonder, in his unprofessional
moments, what constants could be substituted for his variables. The
logician announces, as one of his principles, that, if “fx” is true for
every value of “x”, then “fa” is true, where “a” is any constant.
This principle does not mention a constant, because “any constant”
is a variable; but it is intended to justify those who want to apply
logic. Every application of logic or mathematics consists in the substitution of constants for variables; it is therefore essential, if logic or
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mathematics is to be applied, to know what sort of constants can
be substituted for what sort of variables.
I have been treating first order logic as an applied logic rather than as pure logic
partly because this is an important aspect of it, and partly because as pure logic
first order logic is restricted to the theory of identity, to the study of certain relations between logical predicates, and to the study of universally applicable logical
predicates. My view is that pure logic begins in earnest with second order logic
and with propositional logic as a logic of truth relations between propositions.
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Chapter 7
Tarski’s Semantic Conception of Truth
The idea of characterizing truth as denotation of states of affairs
is considered by Tarski in his paper “The Semantic Conception of Truth”.
After formulating what he calls “the classical Aristotelian conception of
truth”, encapsulated by the formula
To say of what is that it is not, or of what is not that it is, is false,
while to say of what is that it is, or of what is not that it is not, is
true,
he remarks:
If . . .we should decide to extend the popular usage of the term “designate” by applying it not only to names, but also to sentences, and
if we agreed to speak of the designata of sentences as “states of
affairs”, we could possibly use for the same purpose the following
phrase:
A sentence is true if it designates an existing state of affairs.
Although Tarski does not consider this suggestion to be very clear, it doesn’t
seem to be excluded by his initial intuitions about the concept of truth1 .
An interesting feature of the conception of truth as denotation is
the formulation of Tarski’s criterion of material adequacy for a definition of
truth in a given language. This criterion is based on the following schema:
(T) X is true if and only if S.
Particular instances of this schema are obtained replacing ‘S’ by any sentence of the language in question and replacing ‘X’ by a name or description
of this sentence. The criterion C(T) of material adequacy for a definition
of truth (for a given language) is that all the instances of (T) should follow
logically from the definition2 .
If we consider now the notion of denotation (designation, naming,
etc.), we can ask, in a way exactly parallel to Tarski’s, under what circumstances this notion can be defined for a given language. And we can
formulate an analogous criterion of material adequacy based on the schema
(D) the denotation of X = N .
Particular instances of (D) are obtained replacing ‘N’ by any name in the
language in question and replacing ‘X’ by a name or description of this
name. The corresponding criterion C(D) of material adequacy for a definition of denotation (for a given language) is that all the identities that are
instances of (D) should follow logically from the definition.
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If sentences denote states of affairs, or truth values, then Tarski’s
schema (T) becomes a special case of (D); namely
(DT) the denotation of X = S,
where particular instances are obtained replacing ‘S’ by a sentence and
replacing ‘X’ by a name or description of this sentence.
Unless we assume that every name in the language has a denotation, however, there will be instances of (D) that are neither true nor false.
Similarly, unless we assume that every sentence in the language has a denotation, there will be instances of (DT) that are neither true nor false. If
sentences denote truth values and the language contains truth-valueless sentences – such as sentences involving non-denoting terms – then instances of
(DT) for such sentences are neither true nor false. If sentences denote states
of affairs, then instances of (DT) for either false sentences or truth-valueless
sentences are neither true nor false. No instance of (D) or (DT) can be false,
but they needn’t all be true. This means that for such languages we cannot
require as a condition of material adequacy for a definition of denotation
or truth that all instances of (D) or (DT) must follow logically from the
definition. What we should require is that all true instances of (D) and
(DT) must follow logically from the definition.
With (T) the situation is similar. If ‘S’ is replaced by a sentence
that is neither true nor false and ‘X’ is replaced by a name or description
of this sentence, then the left hand side of such an instance of (T) will be
false, because it states that the sentence is true, and by all rights the biconditional itself should be neither true nor false. Of course, this depends on
what we mean by the ‘if and only if’, but I am assuming that it is a material
biconditional. Tarski supposes that the biconditionals are true and restricts
himself to languages that obey the principles of classical logic as usually
conceived, for which one could conclude that there are no denotationless
names and truth-valueless sentences. Hence he offers C(T) as a criterion for
the material adequacy of a definition of truth for these languages3 . I argued
in the last chapter that the laws of classical logic are perfectly compatible
with truth-valueless sentences and denotationless names, from which it follows that C(T) is too strong as a criterion of adequacy in general. What I
will argue in this chapter is that Tarski’s approach through (T) undermines
his work on truth both as an attempt to capture the classical conception of
truth and as an attempt to formulate a semantic account of truth4 .
Tarski’s general characterization of semantics in “The Semantic
Conception of Truth” is that semantics deals with the relations between the
expressions of a language and the objects referred to by these expressions.
He holds that the predicate ‘true’ is not literally a semantic notion in this
sense, because it is a classification of sentences, but that “the easiest and
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most natural way of obtaining an exact definition of truth is one which
involves the use of other semantic notions”5 . It seems, in fact, that the
reason Tarski considers truth to be a semantic notion is that one must
appeal to “other” semantic notions in order to define ‘true’. Presumably
then, if one could write down a definition of the predicate ‘true’ (for a given
language) using only the syntactic machinery of that language, then truth
would be a syntactic notion that classifies expressions.
Tarski begins his original paper on truth discussing whether one
can express (T) directly as a definition of truth. The main idea he examines is the possibility of doing this by means of a quotation function. A
quotation function is a syntactic operator that applied to any expression
of the language gives as a result another expression. Thus, applied to the
expression
Socrates
it gives as result the expression
‘Socrates’
which is a quotation name of the original expression. If we could define
truth by such means, without appealing to any semantic notions, then our
job would be done syntactically. Why can’t we?
Tarski considers the candidate definition:
(1) For all x, x is a true sentence if and only if, for a certain p, x is identical
with ‘p’ and p,
where we must understand the expression
‘p’
as a functional expression with the variable as argument, and not as a name
of the letter
p
which is within the quotes.
Tarski has an objection to quotation functions, however, in that
he claims that they are not extensional. He argues that the sentence
(2) For all p and q, if p if and only if q, then ‘p’ is identical to ‘q’,
“is in palpable contradiction to the customary way of using quotation
marks”6 . This is a rather strange claim, though, which plays on several
ambiguities, including an ambiguity on the notion of extensionality.
What does it mean to say that a function f defined over a certain
set A is extensional? According to the usual formulations, it means that the
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value of f for any element a of A depends only on a and not on the manner
in which a is individuated – where f is considered as a functional rule and
not as the graph of a functional rule (i.e., not as a function in extension).
If so, then in order to show that f is not extensional one has to show that
there is an a ∈ A, and expressions e and e′ that denote a, such that f gives
different results when operating on e and on e′ .
Since the range of the variables in (2), and in (1), seems to consist
of expressions, i.e. sentences, the quotation function is not extensional in
the previous sense only if by operating on two different individuations of the
same sentence it gives different results. But that’s not what (2) shows. What
Tarski’s appeal to (2) involves is a certain confusion of use and mention that
is common in propositional logic and is particularly apparent in quantified
propositional logic. On the one hand the sentential variables range over
sentences, and on the other hand they are substitutable by sentences.
If the variables range over sentences, then an appropriate instance
of (2) would read
(2′ ) If ‘Socrates is mortal’ if and only if ‘Plato is mortal’, then ‘ ‘Socrates
is mortal’ ’ is identical to ‘ ‘Plato is mortal’ ’,
which doesn’t make sense and is no counterexample to the extensionality of
the quotation function. If the variables are to be substituted by sentences,
then the quantifiers would have to be substitutional quantifiers – which they
are not intended to be.
What we are supposed to do with (2) is to forget about the quantifiers when we get the instances, and to treat the variables as substitutable
by sentences, thus:
(2′′ ) If Socrates is mortal if and only if Plato is mortal, then ‘Socrates is
mortal’ is identical to ‘Plato is mortal’.
The ambiguity on extensionality depends on this confusion. The
biconditional in the antecedent of (2) is a proper expression of extensionality in the above sense only if it is treated as an identity relation between
the denotations of sentences, which would be the case if sentences denote
truth values, as Frege held, or if they denote something else such as states
of affairs7 . But in this case the variables range over such things and the
problem with the quotation function is that it doesn’t make sense for things
that are not expressions.
As a matter of fact (1) involves the same confusion. If both quantifiers range over sentences, then we get an instance
(1′ ) ‘Socrates is mortal’ is a true sentence if and only if, for a certain p,
‘Socrates is mortal’ is identical to ‘p’ and p.
The natural choice is to take the very sentence ‘Socrates is mortal’ as a
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value of the existentially quantified variable to make good the existential
claim. But this doesn’t work because the claim becomes:
(1*) ‘Socrates is mortal’ is identical to ‘ ‘Socrates is mortal’ ’ and ‘Socrates
is mortal’.
In order for (1) to make sense one must take the variable ‘x’ as ranging over
sentences and the variable ‘p’ as substitutable by sentences. There is no
uniform interpretation of both quantifiers that makes sense of (1)8 .
The confusion is a confusion between using sentences and talking
about them. The last ‘p’ in (1) is meant as a use of a sentence substituted
in place of ‘p’, but the variable in the existential quantifier is supposed to
be ranging over sentences. This confusion is seen in its simplest form when
people, including Tarski, write such things as
∀p p
in quantified propositional logic. This is supposed to be false because it
means something like
∀p p holds.
But if this is what it means, then I think that one should write it explicitly
as
∀p p is true,
where the quantified variable is naturally interpreted as ranging over sentences. What else can be the point of the ‘holds’ ?9
With this interpretation (1) is quite obviously incorrect, and circular, because what it says is
(1′′ ) For all x, x is a true sentence if and only if, for a certain p, x is identical
with ‘p’ and p is true.
What makes it incorrect is the use of the quotation function. Without the
quotes it is correct, but trivially circular.
The confusion between using sentences and talking about them is
also quite apparent in (2), because the ‘p if and only if q’ is meant to be
used as part of an assertion. If one is quantifying over sentences, what one
should say is
(2′′′ ) For all p and q, if p is true if and only if q is true, then p is identical
to q,
which is quite false – but its falsity has nothing to do with the quotation
function.
One can interpret the quantifiers of (2) substitutionally, however,
and one can interpret (1) as involving a mixture of objectual quantification
(‘for all x’) and substitutional quantification (‘for a certain p’). In this case
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(1) is literally an expression of Tarski’s schema (T) as a single sentence
offered as a possible definition of ‘true’ (for a given language).
Although Tarski has several reservations to this general approach
to a definition of truth on the grounds that the quotation function is not
extensional, his main objection is that under certain natural assumptions
the use of a quotation function leads to contradiction10 .
Suppose that ‘c’ is a typographical abbreviation of the expression
‘the sentence printed on this page, line 10 from the top’, and take the
sentence:
For all p, if c is identical with the sentence ‘p’, then not p.
We can establish empirically, says Tarski, that
(3) The sentence ‘For all p, if c is identical with the sentence ‘p’, then not
p’ is identical with c.
And we assume that the quotation function has the following property:
(4) For all p and q, if the sentence ‘p’ is identical with the sentence ‘q’, then
p if and only if q.
To see how the argument goes, it is easier to symbolize (1), (3)
and (4) leaving the qualification to sentences implicit. We get:
(1#) ∀x(x is true ⇔ ∃p(x = ‘p’ & p)
(3#) c = ‘∀p(c = ‘p’ ⇒ ¬p)’
(4#) ∀p∀q( ‘p’= ‘q’ ⇒ (p ⇔ q)).
The formal argument is quite simple. Suppose that
(5) ∀p(c = ‘p’ ⇒ ¬p).
Instantiating the universal quantifier to (5) itself, we get
(6) c = ‘∀p(c = ‘p’ ⇒ ¬p)’ ⇒ ¬∀p(c = ‘p’ ⇒ ¬p),
from which, by (3#) and Modus Ponens, follows
(7) ¬∀p(c = ‘p’ ⇒ ¬p).
Therefore, the assumption (5) leads to its own negation.
Suppose (7) now. By the properties of the quantifiers and the
connectives this yields
(8) ∃p(c = ‘p’ & p).
By the technique of existential instantiation we get
(9) c = ‘t’ & t,
from which follows
(10) c = ‘t’.
Using (3#) and the substitutivity of identicals it follows that
(11) ‘∀p(c = ‘p’ ⇒ ¬p)’ = ‘t’,
and from (4#) and Modus Ponens
(12) ∀p(c = ‘p’ ⇒ ¬p) ⇔ t,
which, since we have the right hand side in (9), yields (5).
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So, from (3#), (4#), and logic, we get
(13) ∀p(c = ‘p’ ⇒ ¬p) ⇔ ¬∀p(c = ‘p’ ⇒ ¬p),
which is a contradiction.
Formally this seems correct, but whether it establishes anything
about the quotation function or about the possibility of defining truth by
some such means is open to question on at least three counts. First, if
we allow that sentences may be truth-valueless, then (13) needn’t be a
contradiction and what we may want to conclude from it is that c is truthvalueless – or that certain forms of the principle of excluded middle do
not hold. Second, it should be clear in the formulation of (3#) and (4#)
(and the argument as a whole) whether the quantifiers are interpreted as
objectual quantifiers ranging over sentences or as substitutional quantifiers.
Third, one must decide whether the negations that appear in various places
are sentential negations or predicate negations. In particular, for ‘Socrates
is mortal’ as a value of ‘p’, is ‘¬p’ to be interpreted as ‘Socrates is not mortal’
or as ‘it is not the case that Socrates is mortal’ ? If the proper interpretation
is the latter, then we should express the negation as: ‘Socrates is mortal’ is
not true.
If we interpret the quantifiers objectually and distinguish sentential
negation from predicate negation, then using ‘T’ to express ‘is true’, ‘¬’ to
express predicate negation, and ‘¬ T( )’ to express sentential negation, we
have two versions for c (which I formulate as versions of (3#)): the sentential
negation version
(3†) c = ‘∀p(c = p ⇒ ¬T(p))’,
and the predicate negation version
(3‡) c = ‘∀p(c = p ⇒ T(¬p))’.
If we interpret the quantifiers substitutionally, then the sentential
negation version is
(3†S) c = ‘∀p(c = ‘p’ ⇒ ¬T(‘p’))’,
and the predicate negation version is
(3‡S) c = ‘∀p(c = ‘p’ ⇒ ¬p))’,
which is (3#) itself.
Since (4#) does not involve negation we have only two versions for
it; the objectual version
(4†) ∀p∀q(p = q ⇒ (T(p) ⇔ T(q))),
and the substitutional version
(4†S) ∀p∀q( ‘p’ = ‘q’ ⇒ (T( ‘p’) ⇔ T ( ‘q’)).
If we allow for the possibility of truth-valueless sentences, then the
use of the ‘T’ predicate in this way is essential even in the substitutional
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version. For to interpret (4#) substitutionally is essentially to interpret it
as a schema which is correct if all its substitution-instances are true. But if
we take a sentence that is neither true nor false and substitute it for both
‘p’ and ‘q’, then we get an instance of (4#) whose antecedent is true but
whose consequent is neither true nor false – hence the conditional itself as
a material conditional would be neither true nor false.
These considerations are relevant to Tarski’s argument because
the point he is trying to make with the formulation of c using a quotation
function is that one can obtain the paradox of the liar without appealing to
the predicate ‘is true’. He has already given a version of the paradox using
a sentence c*, (introduced by the same technique used for introducing c)
such that
(3*) c∗ = ‘¬T(c∗)’.
From this and the instance of schema (T)
(Tc∗) T( ‘¬T(c∗)’) ⇔ ¬T(c∗)
one immediately gets the contradiction
(13*) T(c∗) ⇔ ¬T(c∗).
So what he is arguing now in connection with definition (1) is that the use
of the predicate ‘T’ is not essential if one has a quotation function.
It should be noticed that we cannot attempt to escape the previous
contradiction by claiming that c* may be truth-valueless, because if c* is
not true, then ‘¬T(c∗)’ (which is c*) is true. That is, the conditional
(13a*) ¬T(c∗) ⇒ T(c∗)
follows independently of (T) from (3*) and the special case of the principle
of excluded middle
(14a†) ∀p(T(p) ∨ T ( ‘¬T (p)’)).
Similarly, we can get
(13b*) T(c∗) ⇒ ¬T(c∗)
independently of (T) from (3*) and the special case of the principle of
contradiction
(14b†) ∀p¬(T(p) & T( ‘¬T(p)’)).
These principles (14a†) and (14b†) hold for all sentences, whether true or
false or neither, and can be jointly formulated as
(14c†) ∀p(¬T(p) ⇔ T( ‘¬T(p)’)).
The question I want to examine now is whether Tarski’s formulation using the quotation function establishes that one can obtain a contradiction without using ‘T’. The only version of c that satisfies this condition
is version (‡S) and I don’t think that this version is contradictory. In fact,
only versions (†) and (†S) are clearly contradictory. I will discuss the various versions below, but there are three more points I want to make before
I do.
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First, definition (1) (or schema (T)) is not used in the argument
(5)-(13), which supports the claim that the contradiction is independent of
the actual definition of truth that Tarski is examining – I will discuss later
the connection between (1) (or (T)) and Tarski’s argument. If one allows
that sentences may be neither true nor false, however, then this is not so
when the argument is formulated for the versions I distinguished above.
But I have already argued that (1) cannot be formulated as a definition
in any of the versions, although if one interprets the universal quantifier
objectually and the existential quantifier substitutionally, then one could
use instances of (1) in connection with version (‡S) – and one could use
instances of (T) in connection with any of the versions. But, in any case,
if one allows sentences that are neither true nor false, then (1) and (T) will
have instances that are not true. In order to avoid this I will appeal instead
to the following schemas of inference related to (T):
(T1-Inf) T(X) ⊢ S
(T2-Inf) S ⊢ T(X),
where specific instances are obtained as for (T). Although with such inferences one can go from a false sentence to a sentence that is neither true nor
false, and viceversa, the inferences are truth-preserving because from true
sentences one can only go to true sentences11 .
The second point is that if one has truth-valueless sentences, then
the rule of conditionalization (or the deduction theorem) cannot be used
with such sentences as antecedent and/or consequent; for otherwise one
could prove sentences that are not true. In particular, one cannot use instances of (T1-Inf) and (T2-Inf) to prove an instance of (T) unless the
sentence replacing ‘S’ is either true or false.
The third point is that my discussion is not meant to be a discussion of the Liar Paradox in general. As I have already pointed out in
connection with (3*), that version of the Liar cannot be escaped by claiming
that c* is truth-valueless. What I am going to discuss is Tarski’s formulation of the Liar using the quotation function, its relation to schema (T) and
its relation to Tarski’s conception of truth.
Let’s consider now version (†). If we follow the reasoning in (5)(12), the argument goes as follows. Suppose that
(5†) ∀p(c = p ⇒ ¬T(p)).
Then, by instantiation,
(6†) c = ‘∀p(c = p ⇒ ¬T(p))’ ⇒ ¬T( ‘∀p(c = p ⇒ ¬T(p))’),
and by Modus ponens
(6a†) ¬T( ‘∀p(c = p ⇒ ¬T(p))’).
But in order to get from this to
(7†) ¬∀p(c = p ⇒ ¬T(p)),
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where the negation is predicate negation (for the sentential negation of (5†)
is (6a†)), we must assume that (5†) is either true or false – i.e., that either
(5†) or its predicate negation (7†) is true. We could justify this by means
of the version of the principle of excluded middle
(15†) ∀p(T(p) ∨ T(¬p)),
though what we actually need is just the instance
(15†′ ) T( ‘∀p(c = p ⇒ ¬T(p))’) ∨ T( ‘¬∀p(c = p ⇒ ¬T(p))’).
This would license the passage from (6a†) to
(6b†) T( ‘¬∀p(c = p ⇒ ¬T(p))’),
from which we get to (7†) by (T1-Inf).
The first five steps of the argument starting from (7†) are:
(8†) ∃p(c = p & T(p))
(9†) c = t & T(t)
(10†) c = t
(11†) ‘∀p(c = p ⇒ ¬T(p))’ = t
(12†) T( ‘∀p(c = p ⇒ ¬T(p))’) ⇔ T(t).
From which it does follow that
(12a†) T( ‘∀p(c = p ⇒ ¬T(p))’)
and (5†), where the last step depends again on (T1-Inf). We can actually
avoid the use of (4†) in step (12†) by substituting the left hand side of (11†)
for ‘t’ in the second conjunct of (9†) – which is justified by the principle of
substitutivity of identicals – and going directly to (12a†).
So it appears that we can only get a contradiction if we assume
the principle of excluded middle (15†), or at least the special case (15†′ ).
And we can actually use the argument to refute (15†′ ), for by starting the
first part with (12a†) instead of (5†) we get
(16a†) T( ‘∀p(c = p ⇒ ¬T(p))’) ⇒ ¬T( ‘∀p(c = p ⇒ ¬T(p))’),
and by starting the second part with (6b†) instead of (7 †) we get
(16b†) T( ‘¬∀p(c = p ⇒ ¬T(p))’) ⇒ T( ‘∀p(c = p ⇒ ¬T(p))’),
from which we can conclude that neither (5†) nor (7†) are true – i.e., (6a†)
and the negation of (6b†).
This does not eliminate the contradiction, however, because from
(6a†) we can prove
(17†) ∀p( ‘¬∀p(c = p ⇒ ¬T(p))’ = p ⇒ ¬T(p))
by the laws of identity, and using (3†) we can prove (5†). We can then use
(T2-Inf) to get (12a†), which contradicts (6a†).
In version (‡) the first part of the argument is
(5‡) ∀p(c = p ⇒ T(¬p))
(6‡) c = ‘∀p(c = p ⇒ T(¬p))’ ⇒ T( ‘¬∀p(c = p ⇒ T(¬p))’)
(6a‡) T( ‘¬∀p(c = p ⇒ T(¬p))’)
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(7‡) ¬∀p(c = p ⇒ T(¬p)),
where the last step depends on (T1-Inf). In the second part what we get
from (7‡) is
(8‡) ∃p(c = p & ¬T(¬p))
(9‡) c = t & ¬T(¬t)
(10‡) c = t
(11‡) ‘∀p(c = p ⇒ T(¬p))’ = t
(12‡) T( ‘∀p(c = p ⇒ T(¬p))’) ⇔ T(t)
but we cannot go from ¬T(¬t) to T(t) unless we assume that either t is true
or ¬t is true; i.e., we need (15†) to go to
(12a‡) T( ‘∀p(c = p ⇒ T(¬p))’).
We can also appeal to
(15‡′ ) T( ‘∀p(c = p ⇒ T(¬p))’) ∨ T( ‘¬∀p(c = p ⇒ T(¬p))’)
however, because by substituting the left hand side of (11‡) for ‘t’ in the
second conjunct of (9‡) we can get the intermediate step
(6b‡) ¬T( ‘¬∀p(c = p ⇒ T(¬p))’),
from which we can go to (12a‡) by (15‡′ ). By (T1-Inf) we then infer (5‡).
So without (15‡′ ) we don’t get a contradiction directly here either,
and we can reformulate the argument to refute (15‡′ ) by deriving
(16a‡) T( ‘∀p(c = p ⇒ T(¬p))’) ⇒ T( ‘¬∀p(c = p ⇒ ¬T(p))’)
and
(16b‡) T( ‘¬∀p(c = p ⇒ T(¬p))’) ⇒ ¬T( ‘¬∀p(c = p ⇒ T(¬p))’),
from which we can conclude that neither (5‡) nor (7‡) are true – i.e., the
negation of (12a‡) and (6b‡).
But as opposed to the previous case there seems to be no way in
which we can derive an actual contradiction from these conclusions – at
least the obvious moves require (15†) or (15‡′ ).
For version (†S) the first part of the argument goes
(5†S) ∀p(c = ‘p’ ⇒ ¬T( ‘p’))
(6†S) c = ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’ ⇒ ¬T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’)
(6a†S) ¬T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’)
(6b†S) T( ‘¬∀p(c = p ⇒ ¬T(p))’)
(7†S) ¬∀p(c = ‘p’ ⇒ ¬T( ‘p’)),
where the passage from (6a†S) to (6b†S) would have to be justified either
by
(15†S) ∀p(T ( ‘p’) ∨ T( ‘¬p’))
or by its special case (15†S′ ) for (5†S), and the passage from (6b†S) to (7†S)
is justified by (T1-Inf).
The second part of the argument goes
(7†S) ¬∀p(c = ‘p’ ⇒ ¬T( ‘p’))
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(8†S) ∃p(c = ‘p’ & T( ‘p’))
(9†S) c = ‘t’ & T( ‘t’)
(10†S) c = ‘t’
(11†S) ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’ = ‘t’
(12†S) T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’) ⇔ T( ‘t’)
(12a†S) T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’),
from which we get (5†S) by (T1-Inf).
But as in version (†) we can refute (15†S′ ) by deriving
(16a†S) T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’) ⇒ ¬T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’)
and
(16b†S) T( ‘¬∀p(c = ‘p’ ⇒ ¬T( ‘p’))’) ⇒ T( ‘∀p(c = ‘p’ ⇒ ¬T( ‘p’))’)
and then derive a contradiction from (6a†S) using
(17†S) ∀p( ‘¬∀p(c = ‘p’ ⇒ ¬T( ‘p’))’ = ‘p’ ⇒ ¬T( ‘p’))
and (3†S) to prove (12a†S).
In version (‡S), finally, the argument is essentially (5)-(12), except
that due to the formulation of (4†S) we get
(12‡S) T( ‘∀p(c = ‘p’ ⇒ ¬p))’) ⇔ T( ‘t’),
from which using (T2-Inf) on the second conjunct of (9) follows
(12a‡S) T( ‘∀p(c = ‘p’ ⇒ ¬p))’),
and then (5) by (T1-Inf).
Because of the restriction on conditionalization we cannot derive
(13) though – unless we assume that (5) is either true or false. What we
can derive by modifying the argument is
(18‡S) T( ‘∀p(c = ‘p’ ⇒ ¬p)’) ⇒ T( ‘¬∀p(c = ‘p’ ⇒ ¬p)’),
from which we can conclude
(19‡S) ¬T( ‘∀p(c = ‘p’ ⇒ ¬p)’)
and
(20‡S) ¬T( ‘¬∀p(c = ‘p’ ⇒ ¬p)’)
using the version of the principle of contradiction
(21‡S) ∀p¬(T( ‘p’) & T( ‘¬p’)).
But as in version (‡) there doesn’t seem to be a way of deriving a contradiction from these conclusions.
It appears, therefore, that when negation is interpreted as predicate negation what we should conclude is that c is neither true nor false.
And it seems to me that the main reason for claiming that these versions
of c must be either true or false is the general assumption that every sentence must be either true or false, which I formulated as the corresponding
versions of the principle of excluded middle. But for negation as predicate
negation this is obviously false independently of questions concerning the
validity of the laws of classical logic.
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For if we distinguish sentential negation from predicate negation,
what we can conclude is that the principle of excluded middle for predicate
negation either in the form (15†) or in the substitutional forms (15†S) and
(22‡S) ∀p(p ∨ ¬p)
does not hold, but not that it doesn’t hold in the forms
(23†) ∀p(T(p) ∨ ¬T(p))
and
(24†S) ∀p(T( ‘p’) ∨ ¬T( ‘p’)).
The main reason for holding that (15†), (15†S) and (22‡S) are laws of classical logic depends on interpreting the negation as sentential negation and
thus confusing them with (23†) or (24†S). This confusion is particularly hard
to detect when excluded middle is formulated in the standard propositional
form (22‡S). For this to make sense the quantifier must be interpreted substitutionally and the negation as predicate negation, in which case there are
no good reasons to accept it. But the usual interpretation of (22‡S) is essentially (23†) (where the quantifier ranges over sentences or propositions), as
shown by the ordinary reading of (22‡S) as: For any sentence (proposition)
p, either p is the case or p is not the case12 .
One could argue, however, that to claim that these forms of c are
neither true nor false does not eliminate the paradox, because c still “says”
or “asserts” that c is not true, and if c is indeed not true, then what c
asserts is correct and c “should” be true. But if c is true, then one can
prove that c is not true. Hence c is both true and not true. Although I
agree that in versions (†) and (†S) it is reasonable to claim that c asserts
that c is not a true sentence, I think that the distinction between sentential
negation and predicate negation shows that this line of argumentation is
not convincing for versions (‡) and (‡S).
How should one understand the claim that c asserts that c is not
true in version (‡S)? Immediately after the line where c is first stated (in p.
169) Tarski comments that “if we accept [(1)] as a definition of truth, then
the above statement asserts that c is not a true sentence.” If we interpret
the universal quantifier of (1) objectually and instantiate it to c, then what
we get is
(1c) T(c) ⇔ ∃p(c = ‘p’ & p),
where the existential quantifier is interpreted substitutionally. Negating
both sides we get
(1c¬) ¬T(c) ⇔ ∀p(c = ‘p’ ⇒ ¬p),
and we can say that it follows from (1) that c (which is the right hand side)
asserts that c is not true in the sense that (1c¬) follows from (1) and logic.
But if c is neither true nor false, then the left hand side of (1c¬) is true and
the left hand side of (1c) is false, whereas the right hand sides are neither
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true nor false, from which I would conclude that neither (1c) nor (1c¬) are
true as material biconditionals. Since this conclusion is quite independent
of the peculiarities of c, because it holds for any instantiation of (1) to a
sentence that is neither true nor false, I would conclude further that the
fact that (1c¬) follows from (1) doesn’t really tell us anything about what
c asserts.
Nevertheless, Tarski’s idea suggests that if one can prove a biconditional like (1c¬) independently of (1), then one can claim that c asserts
that c is not a true sentence and recover the paradox independently of (1).
But unless there is an actual contradiction we cannot prove such a biconditional in versions (‡) and (‡S). What we can prove in version (‡S) is
(25‡S) (3‡S), (4†S), ∀p(c = ‘p’ ⇒ ¬p) ⊢ ¬T( ‘∀p(c = ‘p’ ⇒ ¬p)’).
Even if we grant, which I do grant, that (4†S), (T1-Inf) and (T2-Inf) have a
logical status (or are analytic for the ‘T’ predicate and the quotation function) and that (3‡S) is true by stipulation, all that can be claimed from
(25‡S) is that it follows logically from c and these analytic and stipulative
premisses and schemas of inference that c is not true. Using ‘entails’ in this
sense, we may say that c entails that c is not true. And since we can prove
(26‡S) (3‡S), (4†S), ∀p(c = ‘p’ ⇒ ¬p) ⊢ T( ‘¬∀p(c = ‘p’ ⇒ ¬p)’),
it can also be claimed that c entails that its predicate negation is true. But
it doesn’t follow that this is a sense in which c asserts that c is not true –
and from which it follows even intuitively that c should be true.
One can appeal to a notion of meaning, but without a specific
account of meaning it is hard to see how the argument would go. And, in
any case, the distinction between predicate negation and sentential negation
undermines the claim that c means that c is not true in any straightforward
sense – i.e., without establishing a biconditional like (1c¬) there is no real
basis for the claim.
It seems to me, therefore, that Tarski has not established that the
paradox of the liar can be recovered simply using the quotation function
and the assumptions (3) and (4). And since even the two contradictory
versions (†) and (†S) of c are essentially the same as version c*, what one
can claim is that such uses of the predicate ‘true’ lead to contradiction. But
the argument does show more. In fact, I think that rather than showing
that the quotation function leads to contradiction, Tarski’s argument comes
very close to refuting the claim that all instances of (T) are true even if one
assumes that all non-logical terms and predicates in the language denote.
One can certainly argue that Tarski’s formulation of c refutes the
combination of (1), (T) and (3‡S), because if we take (1c¬) together with
the instance of (T)
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(Tc) T(c) ⇔ ∀p(c = ‘p’ ⇒ ¬p),
justified by (3‡S), and assume that all instances of (1) and (T) are true, then
we get a contradiction directly. And to the extent that (1) is a reformulation
of (T) as a definition of truth, it seems that to interpret this contradiction
as a reductio ad absurdum of (1) is really to interpret it as a reductio ad
absurdum of (T). I think that if one accepts c (in version (‡S)) as a legitimate
sentence, then this is the most natural conclusion to draw from Tarski’s
argument.
One can deny that c is a legitimate sentence by denying that substitutional quantification is a legitimate (logical) operation and holding that
substitutional quantification is really shorthand for the formulation of a
schema. If we interpret c as a schema, then the inconsistency disapears because c is no longer a sentence that can be used to obtain instances of (T).
Also the argument (5)-(12) doesn’t make sense with this interpretation because it depends on instantiating the “quantifier” of c to c itself and hence
on treating c as a sentence rather than as a schema. But, of course, on this
view Tarski hasn’t shown anything about the quotation function as such
or about obtaining the paradox of the liar without the use of the predicate
‘true’.
There is still version (‡) to be considered, however. This version
involves neither a quotation function nor substitutional quantification, although it does involve the predicate ‘true’. I have argued that it is not
contradictory either and that what one can prove for it is that c is neither
true nor false. But from (T) we get
(27‡) T(c) ⇔ ∀p(c = p ⇒ T(¬p)),
and from (3#) and the laws of identity we get
(28‡) ∀p(c = p ⇒ T(¬p)) ⇔ T(¬c).
Hence,
(29‡) T(c) ⇔ T(¬c)
follows directly from (T) and logic, and if we assume that all instances of
(T) are true and that the principle
(30‡) ∀p∀q(T(p ⇔ q) ⇒ ((T(p) & T(q)) ∨ (T(¬p) & T(¬q))))
holds for material biconditionals, then we get a contradiction. We can
conclude, therefore, that either (T) leads to contradiction or that not all
instances of (T) are true. In either case (T) is refuted.
One may argue that the problem lies with the use of the predicate
‘true’ rather than with (T), and there is no question that by means of the
predicate ‘true’ and Tarski’s technique for introducing such sentences as c
one can formulate contradictory sentences such as c* and the two contradictory versions (†) and (†S) of c – which are essentially the same as version
c*. But there is a very important difference between these three versions
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and version (‡); namely, that in these versions we get a contradiction from
the obviously correct principles of inference (T1-Inf) and (T2-Inf) whereas
in version (‡) we don’t. I haven’t actually shown this for version c* but the
argument is quite simple.
Suppose
(31) T(c∗).
By (3*) and substitution of identicals we get
(32) T( ‘¬T (c∗)’),
and by (T1-Inf)
(33) ¬T(c∗).
And, conversely, from (33) we can infer (32) by (T2-Inf), and then (31) by
(3*) and substitution of identicals.
The immediate conclusion that Tarski draws from his arguments
is that the approach through quotation functions won’t do, and that one
must seek other methods. He then says13 :
I will draw attention here to only one such attempt, namely the
attempt to construct a structural definition. The general scheme
of this definition would be somewhat as follows: a true sentence is
a sentence which posseses such and such structural properties (i.e.
properties concerning the form and arrangement in sequence of the
single parts of the expression) or which can be obtained from such
and such structurally described expressions by means of such and
such structural transformations.
The idea is again purely syntactic, and it couldn’t be more clearly expressed,
but Tarski goes on to remark that for ordinary language such an attempt
would be “almost hopeless” because we don’t have a precise structural specification of the sentences of the language. His general conclusion is then that
it is the universality of ordinary language that is responsible for the semantic paradoxes and that any language L such that:
(I) L contains a name for every sentence of L,
(II) every instance of (T) for L is a true sentence of L,
(III) (3*) can be formulated as a true sentence of L,
(IV) the usual laws of logic hold for L,
must be inconsistent14 .
Tarski then turns his attention to formalized languages and proceeds to show that one can use the structural approach to define truth for
certain formalized languages such as the calculus of classes. The way he
does it is by first introducing his definition of satisfaction and truth, and
then showing (Theorem 28) that the class of true sentences in this sense
coincides with the class of theorems that can be proved from the axioms.
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He then argues15 :
This sentence [Theorem 28] could, in its form, obviously be regarded
as a definition of true sentence. It would then be a purely structural
definition, completely analogous to Def. 17 of provable theorem. But
it must be strongly emphasized that the possibility of constructing
a definition of such a kind is purely accidental. We owe it to the
specific peculiarities of the science in question . . . as well as – in
some degree – to the strong existential assumptions adopted in the
metatheory.
What if it wasn’t accidental? Suppose, counterfactually, that one
could construct such a structural definition for every formalized language.
It would follow that ‘true’ is a predicate that expresses a purely syntactic
notion. At least I presume that this would be Tarski’s conclusion, for what
Tarski is claiming above is that the structural definition of ‘true’ is purely
syntactic because it only appeals to logical syntax, the syntax of the given
language, and the syntax of the metalanguage. So if this kind of definition
could be given in all cases, it would follow that the notion of truth is really
a syntactic notion. But since it is accidental, Tarski is lead to his classic
work on truth whose central idea is to define ‘true’ for a given language in
a (higher order) metalanguage.
At the end of the paper, Tarski summarizes his achievement as
follows16 :
For every formalized language a formally correct and materially adequate definition of true sentence can be constructed in the metalanguage with the help only of general logical expressions, expressions
of the language itself, and of terms from the morphology of language
– but under the condition that the metalanguage posseses a higher
order than the language which is the object of investigation.
What this claim would lead us to believe is that Tarski has achieved a
purely syntactic definition of truth. And this is reinforced by his more
general conclusion17 :
The semantics of any formalized language can be established as a
part of the morphology of language based on suitably constructed
definitions, provided, however, that the language in which the morphology is carried out has a higher order than the language whose
morphology it is.
This suggests that we have been mislead into thinking that Tarski’s
semantics is semantics by the usual inductive definitions of satisfaction and
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truth in an interpretation. There it seems that we have something like
the world, and denotation, and that we are establishing a relation between
sentences and the world. It appears however that this is only a mirage, and
that truth is part of the morphology of language18 .
One can hold that the connection to the world comes with the
language and with the metalanguage through the range of the quantifiers
and the denotation of the non-logical constants. But since the language and
metalanguage can be treated purely syntactically, the definitions go through
even if there is no connection to the world, as long as the sentences of the
metalanguage obey the laws of classical logic in the usual sense. One can
define ‘true’ for the metalanguage in the same way, but that just pushes the
question back to the meta-metalanguage; and so on, indefinitely. One ends
up with an indefinite sequence of languages which can be treated purely
syntactically and which need not have any relation to anything except to
each other. That’s what it means to say that semantics is part of the
morphology of language.
Even the idea that when we do semantics with interpretations we
are relating syntax to something non-syntactic that is somewhat like the
world is basically an illusion, because the interpretations are part of our
informal metatheory for which we assume a notion of ‘true’ whose fundamental characteristic is to obey the principles of classical logic as usually
formulated. The sort of “reality” that is involved here is more like a syntactic feature of the metalanguage, and semantic interpretations may be better
viewed as syntactic translations19 .
It seems to me, therefore, that Tarski’s semantic conception of
truth is really a syntactic conception of truth. The initial syntactic moves
didn’t seem to work, but nevertheless Tarski manages to recover the syntactic idea through the appeal to an indefinite sequence of metalanguages
within each of which the definition is purely syntactic.
The main reason for this is the idea that the problem of truth is
to define a predicate ‘true’ that classifies sentences. Tarski emphasizes that
the sentences of a formalized language are supposed to be meaningful, but
it is a central aspect of the semantic conception of truth that the problem
of truth is basically to classify sentences in accordance with (T)20 .
Whatever truth may be, however, it makes perfectly good sense to
ask about the extension of the predicate ‘true’; which sentences are true and
which are not. In particular, it makes perfectly good sense to ask whether
the extension of ‘true’ can be defined for a given language. That’s what
Tarski is about in “The Concept of Truth for Formalized Languages”; he
wants to see if, and in which way, the extension of ‘true’ can be defined
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for a formalized language. The problem of truth for formalized languages
is not so much to give an account of truth, but to give a definition of the
extension of ‘true’ for such languages. This depends on stronger claims,
however, concerning the adequacy of the semantic conception of truth as an
expression of the classical conception of truth.
When he introduces schema (T), Tarski talks about its instances
as “partial definitions” of truth, or as “partial explanations” of the phrase
‘x is a true sentence’21 . His first example is:
(34) ‘it is snowing’ is a true sentence if and only if it is snowing.
If our language consists of this single solitary sentence, then (34) wouldn’t
be partial; it would be a full definition of the phrase ‘x is a true sentence’
for that language.
Evidently, if our language consists of finitely many sentences, no
matter how many, the conjunction of all the partial definitions that can be
obtained in this way would again be a full definition of the phrase ‘x is a
true sentence’ for that language22 . Moreover, even if our language consists
of infinitely many sentences, we could write down the infinite conjunction
of all the partial definitions and obtain a full definition of the phrase ‘x is a
true sentence’ for that language. In fact, Tarski says that what he gets “is
in some intuitive sense equivalent to the imaginary infinite conjunction”23 .
Why don’t we do this and be done with it? The immediate response is that one can’t write down an infinite conjunction. But this is
plainly false, because mathematicians, and logicians, do it all the time.
They write down such things as
A1 ∪ A2 ∪ . . . ∪ An ∪ . . .,
or
S
{Ai : i ∈ ω},
or
1
1
1
1
+ 2!
+ 3!
+ . . . + n!
+ ...
e = 1 + 1!
or
e = lim an (n → ∞),
where an is the partial sum of the first n terms in the previous expression.
And some logicians do write infinite conjunctions in the same way; in fact,
it was Tarski himself who gave the main impetus for the contemporary
treatment of infinitary languages24 .
The problem with infinite conjunctions is not so much that one
can’t write them down, though obviously one cannot literally write then
down, but that by going at it that way it seems that the whole enterprise
becomes trivial. And the character of Tarski’s definition seems quite different from the infinite conjunction of all the (T) biconditionals25 .
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The philosophical question, however, is whether Tarski’s work throws
any light on truth. It does. By showing how the truth of a sentence depends
on its structural features and on the truth of simpler component sentences,
or on the satisfaction of component predicates, his work does throw light
on truth. How much light?
Tarski held that there is no specifically philosophical problem of
truth, but that there are various problems concerning the notion of truth
that “can be exactly formulated and possibly solved only on the basis of
a precise conception of this notion”26 . He also held that there are some
specific intuitions about truth, that Aristotle and other philosophers had or
have, that he was trying to capture more precisely with his own work.
The basic intuition behind the Aristotelian formulation that Tarski
cites is that truth is an expression of reality. There cannot be any doubt
that the Platonic and Aristotelian intuition was that a sentence that asserts
something of something is true because reality is as the sentence states it to
be, and false because reality is different from what the sentence states it to
be. No doubt there are serious problems in formulating precisely what this
means, and philosophers have been going at them ever since, but it seems to
me that the intuition is quite clear. Is this the intuition that Tarski wants
to capture with his biconditionals (T)27 ?
One sense in which Tarski’s biconditionals do not capture the original intuition has to do with the ‘because’. Aristotle would basically agree
that
(35) ‘Snow is white’ is true if and only if snow is white,
but would add that
(36) ‘Snow is white’ is true because snow is white,
and that
(37) It is not the case that snow is white because ‘Snow is white’ is true.
Thus the passage in Metaphysics 1051b7: “It is not because we think truly
that you are pale, that you are pale, but because you are pale we who say
this have the truth.”
With this proviso Tarski’s biconditionals do seem to capture the
basic intuition. Why do they? Because the biconditionals are supposed
to be grounded in reality. This is particularly clear in Tarski’s example
(34). ‘It is snowing’ is true because it is snowing; not in the metalanguage
but out there, on your backyard. It is precisely because of this that the
biconditionals seem to provide a good adequacy condition for an extensional
definition of ‘x is a true sentence’.
This would appear to give the lie to a perverse argument that has
been making the rounds for years now, that Tarski’s schema (T) – and
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the criterion of material adequacy C(T), and Tarski’s work in general – is
neutral as to various conceptions of truth: realist, coherentist, pragmatist,
and so on28 . I have always thought that this is nonsense, but have had
a very hard time arguing convincingly for my view. The problem is that
Tarski’s position on this issue is somewhat ambiguous. If one reads Tarski
as trying to capture the Platonic and Aristotelian intuitions about truth,
with the biconditionals grounded in reality, then the claim is nonsense. But
it is not clear that this is what Tarski is up to, because the claim that
the semantic conception of truth is philosophically neutral originates with
him. Not with respect to other conceptions of truth, but with respect to
epistemological or ontological issues29 . To understand what this may mean
we must backtrack a bit.
Tarski started out considering the possibility of formulating a purely
syntactic definition of ‘true sentence’. His formulation (1) was an attempt
to express schema (T) as a definition by means of a single sentence. He
then went on to the semantic definition, but the idea that each instance
of (T) is a partial definition of truth, and that his semantic definition is
in some sense equivalent to the infinite conjunction of all the instances, is
just another way to express the syntactic idea in (1). If we have infinite
conjunctions, we can formulate the syntactic definition as a single sentence;
it is essentially schema (T) used as a definition. Thus by claiming that his
semantic definition in terms of satisfaction is in some sense equivalent to
the infinite conjunction, and that it is part of the morphology of language,
Tarski is claiming that his definition recovers the purely syntactic initial
idea. So we must examine this issue more thoroughly.
Tarski objects to some criticisms of (T) that take it as a definition
and that qualify the right hand side with ‘is true’ or ‘is the case’ on the
grounds that “the phrase “if, and only if” (in opposition to such phrases
as “are equivalent” or “is equivalent to”) expresses no relation between sentences at all since it does not combine names of sentences”, and that to
equate these different forms of expression “is an obvious confusion between
sentences and their names”30 . I have argued that Tarski himself makes a
similar confusion by treating quantifiers both as standard quantifiers ranging over sentences and as something like substitutional quantifiers. With
the infinite conjunction this problem is eliminated, because the quantifiers
give way to all instances of (T). But there is still the problem with negation,
because the distinction between predicate negation and sentential negation
parallels the distinction between the material biconditional and material
equivalence. And to confuse the two uses of negation is also to confuse sentences and their names – or at least to confuse what a sentence states with
a statement about what the sentence states.
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Now suppose that instead of (34) we take
(38) ‘Sherlock Holmes is tall’ is a true sentence if and only if Sherlock Holmes
is tall.
In what sense is this adequate as a partial definition of ‘true’ ? If one takes
the ‘if and only if’ as a material biconditional, then the natural thing to say
is that (38) is neither true nor false, because the left hand side is false and
the right hand side is truth-valueless. It would certainly be odd to say that
(38) is true.
One could argue, perhaps, that the ‘if and only if’ is a definitional
‘if and only if’ and that (38) is something like a meaning postulate that
gives the meaning of the word ‘true’ in this very limited context. In fact,
in his discussion of methodology Tarski says that “the task of clarifying the
meaning of the word ‘true’ is left to, and fulfilled by, semantics”31 , from
which one could infer that the biconditionals have the function of meaning
postulates for the word ‘true’. Although there is a stipulative sense in which
this may be unassailable as a convention, it does have consequences that
one can discuss.
For one thing, it doesn’t make much sense anymore to interpret
the biconditional truth-functionally. If we say that the sentence
(39) Sherlock Holmes is tall
is neither true nor false, the only reasonable choice we have for the sentence
(40) ‘Sherlock Holmes is tall’ is true
is that it is false. Since the biconditional (38) is supposed to be true, it
follows that biconditionals connecting false sentences and truth-valueless
sentences must always be true. And it follows from this that if the negation
of (39),
(41) Sherlock Holmes is not tall,
is also truth-valueless, then the biconditional
(42) ‘Sherlock Holmes is tall’ is true if and only if Sherlock Holmes is not
tall
is true as well. If we say that (39) is false because the name ‘Sherlock
Holmes’ does not denote, then we should also hold that (41) is false. And
in this case we get that the biconditionals (42) and
(43) Sherlock Holmes is tall if and only if Sherlock Holmes is not tall
are also true.
The only way to avoid these conclusions is to deny that negation
can be interpreted as predicate negation. But this is the confusion I pointed
out before, for it depends on holding that the very notion of negation involves the name of the sentence as well as the predicate ‘is true’. Moreover,
if we interpret (41) as
(44) ‘Sherlock Holmes is tall’ is not true,
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then we must interpret (44) as
(45) ‘ ‘Sherlock Holmes is tall’ is true’ is not true,
and so on, indefinitely.
I think, therefore, that the only reasonable choice is to hold that
(39) and (41) are truth-valueless, that (40) is false, and that (38) is truthvalueless; which goes against the stipulative account of (38) as true.
It may seem that these considerations are futile because Tarski
never meant to be dealing with truth-valueless sentences. Indeed, he didn’t;
but how can one formulate this assumption without begging the question?
In other words, how can one delimit the sentences (or languages) to which
Tarski’s semantic account of truth applies without presupposing the notion
of truth?
Although Tarski does not address this question explicitly, the solution I would attribute to him is to appeal to classical logic. The account
applies to languages that obey the principles of classical logic, and classical logic rules out truth-valueless sentences. Moreover, classical logic is
formulated as a deductive syntax and does not presuppose the notion of
truth32 .
I have already argued in Chapter 6 that classical logic does not
force the exclusion of truth-valueless sentences, because what characterizes
classical logic are certain general principles such as the principle of contradiction and the principle of excluded middle which can be formulated
ontologically independently of bivalence. So I don’t think that the appeal
to classical logic settles the issue.
In any case, if Tarski were to appeal to a deductive formulation of
classical logic in order to avoid any talk of truth, so that he won’t beg the
question, then there is no reason not to apply his account to such sentences
as (39); because the account is supposed to clarify the meaning of ‘true’
for arbitrary sentences and for formalized languages in general. What we
have to assume is that such languages obey the principles of classical logic
as usually formulated. What happens then? For a language that includes
sentence (39) and is closed under the laws of classical logic, we take the
infinite conjunction of the biconditional (38) and all the other biconditionals
for the language. We now have a full definition of ‘true’ for this language,
which is a formalized language in the sense that Tarski is considering – see
note 20. Does this in any way clarify the meaning of the word ‘true’ or
under what circumstances (39) is true? In fact, for any sentence or totality
of sentences of a language that obeys the laws of classical logic take the
infinite conjunction of the corresponding biconditionals. Are we any wiser
about the meaning of ‘true’ ? What is the connection between sentences
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and reality? This is the most central aspect of the classical problem of
truth (or of clarifying the meaning of the word ‘true’, if one wants to put
it this way), and it seems to be simply swept away by this move. The
idea that the semantic conception of truth is philosophically neutral derives
essentially from this attempt to eliminate the question of the connection
between sentences and reality.
Both Tarski and many supporters of Tarski’s semantic conception
of truth appeal to the world and make it appear that the (T) biconditionals
do have something to say about the connection between a sentence and
reality. Mates, for instance, claims that the biconditionals “say just what
needs to be said about the relation between a true sentence and the world,
and they say it neat.” Yet this appeal to the world is not supposed to involve
us with metaphysics. Thus, referring to the biconditional
(M1) ‘Socrates taught Plato’ is true if and only if Socrates taught Plato,
Mates continues33 :
All additions with which philosophers have encumbered them are
excess baggage, doing no good and possibly some harm. Compare,
for example, a Carnapian version of (M1):
(M1′ ) ‘Socrates taught Plato’ is true if and only if the individual
denoted by ‘Socrates’ stands in the relation denoted by ‘taught’ to
the individual denoted by ‘Plato.’
Here the metaphysics of individuals, attributes, and relations threatens to enter the scene. If it is taken seriously, then if we do not
believe in the existence of such things as attributes and relations,
we shall have to conclude from (M1′ ) that ‘Socrates taught Plato’
is not true. If the metaphysical verbiage is not to be taken seriously
in (M1′ ), then surely it is better omitted.
Similar remarks will apply to a Fregean version:
(M1′′ ) ‘Socrates taught Plato’ is true if and only if the object,
Socrates, falls under the concept, taught Plato
and to “state of affairs” talk:
(M1′′′ ) ‘Socrates taught Plato’ is true if and only if the state of affairs that Socrates taught Plato exists
and to all other versions obtained by pumping metaphysical terminology into the right hand side of (M1). If there is the slightest
difference in content between the right hand sides of these versions
and the right hand side of (M1), then they do not serve as well
as it. No sentence does a better job than “Socrates taught Plato”
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in describing precisely the conditions under which “Socrates taught
Plato” is true.
If we generalize the last point, however, then we should say that no sentence
does a better job than (39) in describing precisely the conditions under
which (39) is true. So what are these conditions? That Sherlock Holmes is
tall?
Mates’ point applies only to sentences that are either true or false,
as in his example, because they do indeed describe the world – correctly
or incorrectly – but it certainly does not apply to truth-valueless sentences,
which, not being connected to the world, do not give conditions for truth
or for anything else. The aim of the metaphysical analyses that Mates
finds objectionable, when they are in fact interpreted metaphysically, is
not to formulate biconditionals for true or false sentences but to analyse
the connections between sentences and the world in terms of an analysis
of the structure of the world. The formulation of such biconditionals, for
any sentence, is a consequence of this. But the Tarski biconditionals are
not such an analysis, and the appeal to the world in connection with them
is entirely vacuous. Whether one likes it or not, if one does appeal to the
world, then one must engage in metaphysics – in whatever terms one finds
acceptable. What the allegedly non-metaphysical talk does is to beg the
question through a veiled appeal to truth.
An instance of (T) can be used as a partial definition of ‘true’
only if it is true. A finite or infinite set of instances of (T) can be used
as a definition of ‘true’ only if they are all true. That’s why the infinite
conjunction idea is no solution even for a purely extensional definition of
truth.
Unless Tarski’s approach to “truth” is interpreted purely syntactically, and is therefore completely divorced from any questions concerning
truth and reality – which would render it meaningless according to Tarski
himself (see note 20 again) – it is simply question-begging. And this is not
an issue of ordinary language versus formalized languages (in Tarski’s not
purely syntactic sense).
For as I already pointed out, to say that one only wants to deal
with languages whose sentences are either true or false clearly begs the question. How does one express this assumption without appealing to truth?
The standard move is to appeal to the laws of classical logic formulated as
a syntactic formalism, but even aside from the fact that this does not settle
the issue, it also seems to me to beg the question. For, as I argued in the
Introduction, if we are dealing with a mere syntactic formalism, then why
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accept these laws and not others? And why should these laws be particularly
relevant to the question of truth?
An alternative would be to claim that truth-valueless sentences
are meaningless, and that the account of truth is only meant to apply to
languages consisting of meaningful sentences. I think that it is very hard
to make a convincing case that truth-valueless sentences are meaningless
– Russell held this position, but his view depended on very specific philosophical assumptions (as I discussed in Chapter 3). In any case, this would
not be a way out for the problem I am raising in connection with Tarski,
for the semantic account of truth would then depend on a prior account of
meaning that rules out truth-valueless sentences as meaningless. Although
it might be possible to give such an account of meaning, Tarski certainly
didn’t – and many people dealing with meaning nowadays go the other way
around and appeal to Tarski’s account of truth in order to give an account
of meaning.
To say that one only wants to deal with languages whose singular
and general terms denote is a much more plausible tack, and is consistent
with the idea that the fundamental notion of semantics is the notion of denotation. But Tarski’s argument with the liar paradox, as I analysed it above,
undercuts the claim that the restriction to languages whose terms denote
guarantees that every sentence of the language is either true or false – and
this is actually independent of having a quotation function in the language
because there are other techniques that can be used instead. Moreover,
aside from restricting the account of truth to languages all whose terms denote (and that do not allow the formulation of such sentences), the appeal
to denotation would force the semantic conception of truth to come to terms
with this notion, which Tarski does not analyse at all34 . In particular, since
the main languages to which Tarski’s account is applied are the languages
used in mathematics, one would have to give an account of denotation for
mathematical languages – which may be an important consideration underlying Tarski’s attempt to bypass the notion of denotation.
To say that one only wants to deal with languages for which the
biconditionals are true also clearly begs the question. For each biconditional
its truth must be analysed either by means of a partial definition of ‘true’
for that biconditional, which is just another instance of (T) and leads to
an infinite regress35 , or in terms of the truth of the components, which is
circular. One can, of course, reduce the account of truth to denotation, as in
the usual inductive definition of truth in an interpretation, but this brings
us back to the problem of accounting for denotation.
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One can stipulate the meaning of ‘true’ by means of the biconditionals, though this has the consequences I discussed earlier. In particular,
it has the consequence that one shouldn’t make any claims about the biconditionals saying something about the connection between a sentence and the
world. It is interesting, in fact, that many people who claim that mathematical sentences are not connected to reality also claim that the biconditionals
are true for mathematical sentences. This suggests that either the biconditionals are held to be analytic in general, or that they are held to be analytic
for any language which obeys the laws of classical logic, independently of
any connections with reality. But this does undercut the claim that the
biconditionals say something about the connection between a sentence and
reality, which is the point of the conception of truth that they are meant to
capture.
As far as I can see this leaves us either with a purely syntactic
theory of truth based on a convention that is not justified by more general
considerations, or else with the view that the biconditionals give conditions
for asserting (or denying) sentences qualified by the words ‘is a true sentence’. It is clear that this is not what Tarski meant by the biconditionals,
and it doesn’t clarify the classical conception of truth, but it is an interpretation. On this view the semantic conception of truth would be a sort of
recommendation for the use of the word ‘true’ when making assertions. But
even as a recommendation it is problematic. If one states it in the strongest
form – namely, that one should be prepared to assert or deny one of the
sides of a (T) biconditional if and only if one is prepared to assert or deny
the other side, – then it is only a good recommendation if by the denial
of a sentence one means the assertion that it is not true. In this case the
‘denial’ part of the recommendation amounts to saying that one should be
prepared to assert
‘X is a true sentence’ is not a true sentence
if and only if one is prepared to assert
X is not a true sentence.
If by the denial of a sentence one means the assertion of its predicate negation, which is how the left hand side of a (T) biconditional is usually denied,
then the recommendation is a bad recommendation, because as I have argued one should be prepared to assert the negation of the left hand side of
(38) without being prepared to assert the negation of its right hand side.
So either one states the recommendation for assertion and denial (in the
sense of predicate negation) but only from right to left, as Tarski does, or
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one states it in both directions but only for assertion, as Putnam does –
see notes 28 and 29. In either case all we get are partial recommendations
which can hardly be claimed to clarify the meaning of the word ‘true’ even
in the context of assertions. What does it mean to assert
‘Sherlock Holmes is tall’ is not a true sentence,
for example?
If one interprets the semantic conception of truth as such partial
recommendations for using the word ‘true’, then one can reasonably claim
that it is philosophically neutral. In fact, I think that this is the only sense
in which it may be philosophically neutral, and what it amounts to is the
recognition of the principles (T1-Inf) and (T2-Inf) as valid principles of
inference for the predicate ‘true’. But to the extent that this recognition is
philosophically neutral, it just goes to show that in this interpretation the
semantic conception of truth does not capture the classical conception of
truth36 .
Tarski’s work on truth created the illusion that one can separate
the problem of truth from metaphysics. Thus the problem of truth became
the problem of ‘true’, and the problem of ‘true’ became the problem of
characterizing this predicate as part of the morphology of language. It
is precisely for this reason that Tarski’s semantic conception of truth is
inadequate as an expression of the classical conception of truth deriving
from Plato and Aristotle. What is true for them is what is real, and this is
certainly not a structural feature of language. No classification of sentences
by syntactic means, be it in the language or in the metalanguage, could
possibly constitute an account of truth or a definition of truth for either
Plato or Aristotle. So if that’s the conception one wants to capture, then it
is essential to preserve the connection with reality; because to lose that is
to lose the whole idea.
The merit of Tarski’s semantic definition of truth in terms of satisfaction lies in that it can, and in my view should, be interpreted realistically
in terms of an analysis of the structure of reality. The intuitions behind
Tarski’s initial conception of truth are the Platonic and Aristotelian realist
intuitions about truth, but he was misled by his formulation of C(T) as
a criterion of adequacy. If each instance of (T) is a partial definition of
truth, then the aim of a full definition of truth is to capture the totality of
instances of (T), and this is precisely what guided Tarski’s approach – as he
says explicitly in the passage I quoted in note 23. But as shown by instances
of (T) for truth-valueless sentences, C(T) isn’t a good criterion of adequacy.
Moreover, the non-contradictory version (‡S) of the liar shows that C(T)
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can’t be a good criterion of adequacy even if the terms and predicates of
the language denote. Hence Tarski’s approach is only correct for languages
whose sentences are either true or false, and what he achieved is a definition
of the extension of the predicate ‘is true’ for formalized languages satisfying
this condition. Since this presupposes the notion of truth, it seems to me
that the claim that the semantic conception of truth is a general account
of truth (or clarifies the meaning of the word ‘true’ in any broader sense) is
incorrect.
The conception of truth as denotation of states of affairs does
preserve the connection with reality in a very strong sense, close to the
original sense that truth had for Plato and for Aristotle. I would say, in
fact, that it is very close to Tarski’s initial intuition as to what a semantic
characterization of truth should be – as shown by the various remarks I
quoted throughout the chapter concerning these initial intuitions.
The account of truth as denotation yields an account of ‘true’ and
‘false’ by classifying sentences, say, as true or false, or neither, according
to their relation to reality. True sentences are those that denote a state of
affairs; false sentences are those whose predicate negation denotes a state
of affairs; the rest are neither true nor false. This combines Plato’s view
concerning the truth and falsity of sentences with Frege’s denotational view
and with the view derived from Russell that the proper candidates for denotation are states of affairs rather than truth values. It is a general and
simple semantic characterization of truth (in Tarski’s sense) which we can
formulate more formally on a par with his syntactic formulation of (T) in
(1) as follows:
(46-T) ∀p(p is true if and only if ∃s(p denotes s)
(46-F) ∀p(p is false if and only if ∃s(¬p denotes s),
where the variable ‘p’ ranges over sentences and the variable ‘s’ ranges over
states of affairs.
As an explicit characterization of ‘true’ and ‘false’, (46) allows the
elimination of these predicates in any (reasonable) context. For example,
for such sentences as
(47) The first sentence written by Plato is true
and
(48) All consequences of true sentences are true37 .
An important feature of the analysis in terms of states of affairs
is that it is rather immaterial whether the variable ‘p’ in (46) ranges over
sentences, or thoughts, or propositions, or statements, or beliefs, or any
other reasonable candidates for the qualifications ‘is true’ and ‘is false’.
Of course, whether or not one can apply Tarski’s structural approach to
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such things will depend on having some sort of analysis of their nature and
structure. One cannot expect that a Tarski-type account that works for
sentences, based on a certain analysis of their structure, will work in the
same way for everything else. To the extent that those other things may
have structural expressions, Tarski’s structural account is very useful, but
it may only be a partial account of the connections that these things have
to reality.
As to the philosophical problem of truth, I have already argued
to some extent, and will argue more later, that the conception of truth
as denotation of states of affairs allows us to see the way out of various
problems associated to the problem of non-being which have been central
to the discussion of truth ever since Plato. Moreover, it seems to me to be
very close to the Platonic solution to the problem of false predication. So
even as developed thus far I think that there is a basis for claiming that this
conception of truth throws some light on traditional philosophical problems
surrounding this notion.
Also, by analysing truth in terms of states of affairs one makes it
into a local affair; which is as it should be, it seems to me. It is simply not
plausible that the whole of reality is involved in the truth of each specific
sentence. This is clear from Tarski’s definition as well, of course, although
it does not identify a specific bit of reality that is responsible for the truth
of a given sentence. And it doesn’t even stick to the “components” of that
bit, because if the sentence involves a description that is used to identify
one of these components, Tarski’s definition will have to take into account
whatever other bits of reality are used in the identification. Something like
this also happens with my formulations in Chapter 6, but the analysis of
logical structure makes a clear distinction between what is relevant for the
truth of the sentence and what is relevant for the identification of the state
of affairs in question.
Although a definition such as (46) is trivial, because it reduces
truth and falsity to denotation of states of affairs and to the subjectpredicate analysis of sentences, I have tried to show (to some extent) that
this reduction is not trivial and that the task of clarification of the classical
conception of truth is essentially metaphysical, not syntactic – which again
agrees with Tarski’s initial intuitions, at least as far as the last point is
concerned.
But there is still the notion of denotation. Doesn’t the problem of
truth reduce also to the problem of giving an account of this notion? Well,
yes; in part it does – and if one interprets Tarski’s definition as based on a
primitive notion of denotation, it does for him also. We must describe how
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denotation works in language, and we must try to account for the general
principles that it obeys – such as Frege’s principle of substitutivity. We can
also define the denotation of an expression in terms of its structural features,
as Tarski did for truth. In fact, (D) and (T) are completely symmetrical,
but it would be ridiculous to claim that the infinite conjunction of all the
instances of (D) can be used to define denotation, or to explain the meaning
of ‘denotes’.
Denotation is an essentially relational realistic notion, which can
certainly not be defined syntactically. If I take the totality of true instances
of (D) for a given language, then I get the extension of the denotation
relation for that language. Similarly, if I take the totality of true instances
of (DT) for a given language, then I get the extension of the denotation
relation for the sentences of that language. This yields a truth predicate,
or a classification of sentences, that is essentially relational as well. But
neither of these is a definition, not even in the purely extensional sense,
because of the appeal to truth.
This is nothing to be upset about, however, because although there
is a need for a logical account of denotation, there is no need for a definition
of denotation that somehow legitimizes the connection between language
and the world. The real problem is not to define truth and denotation for
one or another language, but to characterize and articulate these notions in
relation to a general account of such entities as senses, propositions, states of
affairs, properties, sets, etc., which I begin to discuss in a more systematic
way in chapters 9-12. Before that, however, I want to go back to truth
values once again and examine some aspects of Frege’s logic.
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Notes
1. Op. Cit., pp. 342-343. Tarski’s original paper “The Concept of Truth in
Formalized Languages” is technical and somewhat guarded concerning philosophical issues, whereas “The Semantic Conception of Truth” is an expository and
polemical paper containing a more explicit discussion of philosophical issues.
Tarski first expresses Aristotle’s conception by “means of the familiar
formula:
The truth of a sentence consists in its agreement with (or correspondence to)
reality,”
and then suggests the formulation in terms of designation of states of affairs.
Afterwards he comments (p. 343):
However, all these formulations can lead to various misunderstandings, for none of them is sufficiently precise and clear (though this
applies much less to the original Aristotelian formulation than to
either of the others); at any rate, none of them can be considered
a satisfactory definition of truth. It is up to us to look for a more
precise expression of our intuitions.
And in “Truth and Proof”, p. 63, he says: “. . . it is my feeling that the new
formulations, when analysed more closely, prove to be less clear and unequivocal
than the one put forward by Aristotle.” Although Tarski considers Aristotle’s
formulation to be intuitively clear he does not consider it to be very precise. He
remarks (Ibid., p. 63):
The intuitive content of Aristotle’s formulation appears to be rather
clear. Nevertheless the formulation leaves much to be desired from
the point of view of precision and formal correctness. For one thing,
it is not general enough; it refers only to sentences that “say” about
something “that it is” or “that it is not”; in most cases it would
hardly be possible to cast a sentence in this mold without slanting
the sense of the sentence and forcing the spirit of the language.
This objection seems to be based on a misunderstanding, however, because the
passage in which Aristotle gives the formulation to which Tarski is referring is
part of a discussion of the principle of excluded middle, and Aristotle is talking
about affirming or denying a predicate of a subject. The ‘what is’ can refer to
something like Socrates’ whiteness, or to Socrates being white, not just to the
individual Socrates. The passage goes as follows (Metaphysics 1011b23):
But on the other hand there cannot be an intermediate between
contradictories, but of one subject we must either affirm or deny
any one predicate. This is clear, in the first place, if we define what
the true and the false are. To say of what is that it is not, or of
what is not that it is, is false, while to say of what is that it is, and
of what is not that it is not, is true; so that he who says of anything
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that it is, or that it is not, will say either what is true or what is
false; but neither what is nor what is not is said to be or not to be.
To which Ross adds as a continuation in a footnote: “by those who say that there
is an intermediate between contradictories. Hence such a statement is neither true
nor false, which is absurd.”
2. “The Semantic Conception of Truth”, pp. 343-345. Tarski emphasizes that (T)
is not offered as a definition of truth but as part of the formulation of C(T), which
is a necessary condition that must be met by a definition of truth that accords
with the initial intuitions. If we have such a definition and an instance of (T)
that does not follow logically from the definition, then the definition is materially
inadequate. (This is all supposed to be relative to a given language, but I shall
mostly skip the qualification whenever no confusion can arise.)
3. He says (Op. Cit., p. 349): “It would be superfluous to stress here the consequences of rejecting the assumption (II), that is, of changing our logic (supposing
this were possible) even in its more elementary and fundamental parts.” The
assumption that there are no denotationless names and predicates is the standard assumption in formal semantics even independently of classical logic. Tarski
doesn’t put it this way because of his assumptions about classical logic, among
other reasons, so to say that he assumes that there are no denotationless terms
and truth-valueless sentences is strictly speaking incorrect.
4. There are many discussions of (and objections to) Tarski’s semantic conception
of truth. A recent critical survey can be found in Kirkham Theories of Truth,
chapters 5 and 6.
5. Tarski says (Op. Cit., p. 345):
I should like to propose the name “the semantic conception of truth”
for the conception of truth which has just been discussed.
Semantics is a discipline which, speaking loosely, deals with certain
relationships between expressions of a language and the objects (or
“states of affairs”) “referred to” by those expressions. As typical
examples of semantic concepts we may mention the concepts of designation, satisfaction, and definition as these occur in the following
examples:
the expression “the father of his country” designates (denotes) George
Washington;
snow satisfies the sentential function (condition) “x is white”;
the equation “2 · x = 1” defines (uniquely determines) the number
1/2.
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While the words “designates”, “satisfies”, and “defines” express relations (between certain expressions and the objects “referred to” by
these expressions), the word “true” is of a different logical nature:
it expresses a property (or denotes a class) of certain expressions,
viz., of sentences. However, it is easily seen that all the formulations
which were given earlier and which aimed to explain the meaning
of this word . . . referred not only to sentences themselves, but also
to objects talked about by these sentences, or possibly to “states
of affairs” described by them. And, moreover, it turns out that
the simplest and more natural way of obtaining an exact definition
of truth is one which involves the use of other semantic notions,
e.g., the notion of satisfaction. It is for this reason that we count
the concept of truth which is discussed here among the concepts of
semantics, and the problem of defining truth proves to be closely
related to the more general problem of setting up the foundations of
general semantics.
In “The Concept of Truth in Formalized Languages” Tarski does not give this general characterization of semantics, but introduces the idea of a semantic definition
of truth as follows (p. 155):
Amongst the manifold efforts which the construction of a correct
definition of truth for the sentences of colloquial language has called
forth, perhaps the most natural is the search for a semantical definition. By this I mean a definition which we can express in the
following words:
(1) a true sentence is one that says that the state of affairs is so and
so, and the state of affairs is indeed so and so.
From the point of view of formal correctness, clarity, and freedom
from ambiguity of the expressions occurring in it, the above formulation obviously leaves much to be desired. Nevertheless its intuitive
meaning and general intention seem to be quite clear and intelligible. To make this intention more definite, and to give it a correct
form, is precisely the task of a semantical definition.
6. “The Concept of Truth in Formalized Languages”, p. 161.
7. The usual notion of extensionality for contexts involving sentences can be formulated as the schema
(SE) If S if and only if S ′ , then . . . S . . . if and only if . . . S ′ . . .,
where the letters ‘S’ and ‘S ′ ’ can be replaced by any sentences and ‘. . . S . . .’ and
‘. . . S ′ . . .’ can be replaced by appropriate sentential contexts. A context ‘. . . S . . .’
is extensional if it satisfies (SE). For any specific context we can formulate (SE)
as a single sentence using substitutional quantifiers. Thus (2) interpreted substitutionally can be obtained from
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(SE/‘S’= ‘S’) For all p and q, if p if and only if q, then ‘p’= ‘p’ if and only if ‘p’
= ‘q’,
and logic.
On his use of the terms ‘extensional’ and ‘intensional’ Tarski remarks
(Op. Cit., p. 161 note 2): “It should be noted that usually the terms ‘extensional’
and ‘intensional’ are applied to sentence-forming functors, whilst in the text they
are applied to quotation marks and thus to name-forming functors.”
8. If one changes (1) to
(1S) For all x, ‘x’ is a true sentence if and only if for a certain p, ‘x’ is identical
with ‘p’ and p,
then one can interpret the quantifiers uniformly as substitutional quantifiers. An
instance would be:
(1S′ )‘Socrates is mortal’ is a true sentence if and only if for a certain p, ‘Socrates
is mortal’ is identical with ‘p’ and p.
But this gives only one choice of substitution for the existentially quantified variable; namely, the very sentence
Socrates is mortal.
Which means that (1S) reduces to:
(1S*) For all p, ‘p’ is a true sentence if and only if p,
where the quantifier is still interpreted substitutionally.
Tarski has already considered this formulation, however, and rejected it
on the grounds that “. . . the above sentence could not serve as a general definition
of the expression ‘x is a true sentence’ because the totality of possible substitutions
for the symbol ‘x’ is here restricted to quotation-mark names.” Ibid., p. 159. It is
this objection that leads him to formulate (1). The same objection would apply
to any formulation of (1) that interprets the universal quantifier substitutionally,
for example the following alternative formulation:
(1S**) For all x, ‘x’ is a true sentence if and only if for a certain p, x if and only
if p, and p.
Of course, Tarski is not interpreting the quantifier of (1S*) substitutionally, at least explicitly, but if the variable ranges over sentences, then (1S*)
doesn’t make sense either. An instance would be:
(1S*′ ) ‘ ‘Socrates is mortal’ ’ is a true sentence if and only if ‘Socrates is mortal’.
9. I discuss quantified propositional logic in Chapter 16. A classic paper is
Lukasiewicz and Tarski “Investigations into the Sentential Calculus”, where the
confusion I’m pointing out in the interpretation of quantification is quite evident.
The universal quantifier is introduced as follows (p. 54): “For the universal quantifier Lukasiewicz uses the sign ‘Π’ which was introduced by Peirce. With this
notation the formula ‘Πpq’ is the symbolic expression of the sentence ‘for all p, q
(holds)’.” For a discussion of the issue of propositional quantification in connection
with redundancy theories of truth see Haack Philosophy of Logics, pp. 130-133.
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10. Op. Cit., pp. 161-162. After remarking on the intensionality of the quotation
function Tarski comments:
For this reason alone definition [(1)] would be unacceptable to anyone who wishes consistently to avoid intensional functors and is even
of the opinion that a deeper analysis shows it to be impossible to
give any precise meaning to such functors. Moreover, the use of the
quotation functor exposes us to the danger of becoming involved
in various semantical antinomies, such as the antinomy of the liar.
This would be so if – taking every care – we make use only of those
properties of quotation-functions which seem almost evident.
And after his formulation of the antinomy (which I discuss in the text) he says
(p. 162):
I should like to draw attention, in passing, to other dangers to which
the consistent use of the above interpretation of quotation marks
exposes us, namely to the ambiguity of certain expressions (for example, the quotation expressions that occur in [(1S*)] and [(1)] must
be regarded in certain situations as a function with variable argument, whereas in others it is a constant name which denotes a letter
of the alphabet). Further, I would point out the necessity of admitting certain linguistic constructions whose agreement with the
fundamental laws of syntax is at least doubtful, e.g. meaningful expressions which contain meaningless expressions as syntactical parts
(every quotation-name of a meaningless expression will serve as an
example). For all these reasons the correctness of definition [(1)],
even with the new interpretation of quotation marks, seems to be
extremely doubtful.
These objections are not very convincing. To take care of the first objection it is
enough to distinguish notationally the quotation function from quotation names –
using something like Quine’s quasi-quotes, for instance. If the second objection is
taken seriously, then syntax would become impossible because meaningful expressions generally contain meaningless syntactic parts. Thus, the name ‘Socrates’
contains the meaningless part ‘oc’.
11. That is, as I pointed out in Chapter 6, although a sentence S need not be
materially equivalent to a sentence T(X) that asserts the truth of S, and, hence,
such pairs are not tv-logically equivalent, they are c-logically equivalent. And it
is precisely because S and T(X) are logical consequences of each other that the
principles (T1-Inf) and (T2-Inf) are valid principles of inference.
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12. Tarski distinguishes the propositional formulations such as (24‡S) from formulations in terms of truth. After outlining his definition of truth in “The Semantic
Conception of Truth” he comments (p. 354):
In particular, we can prove with its help the laws of contradiction
and of excluded middle, which are so characteristic of the Aristotelian
conception of truth; i.e., we can show that one and only one of any
two contradictory sentences is true. These semantic laws should
not be identified with the related logical laws of contradiction and
excluded middle; the latter belong to the sentential calculus, i.e., to
the most elementary part of logic, and do not involve the term ‘true’
at all.
13. Op. Cit., p. 163.
14. Ibid., pp. 164-165:
A characteristic feature of colloquial language (in contrast to various
scientific languages) is its universality. It would not be in harmony
with the spirit of this language if in some other language a word
occurred which could not be translated into it; it could be claimed
that ‘if we can speak meaningfully about anything at all, we can
also speak about it in colloquial language’. If we are to maintain
this universality of everyday language in connexion with semantical investigations, we must, to be consistent, admit into the language, in addition to its sentences and other expressions, also the
names of these sentences and expressions, and sentences containing
these names, as well as such semantic expressions as ‘true sentence’,
‘name’, ‘denote’, etc. But it is presumably just this universality
of everyday language which is the primary source of all semantical
antinomies, like the antinomies of the liar or of heterological words.
These antinomies seem to provide a proof that every language which
is universal in the above sense, and for which the normal laws of logic
hold, must be inconsistent. This applies especially to the formulation [(3*)] of the antinomy of the liar which I have given on pages
157 and 158, and which contains no quotation-function with variable
argument. If we analyse this antinomy in the above formulation we
reach the conviction that no consistent language can exist for which
the usual laws of logic hold and which at the same time satisfies the
following conditions: (I) for any sentence which occurs in the language a definite name of this sentence also belongs to the language;
(II) every expression formed from [(T)] by replacing the symbol [‘S’]
by any sentence of the language and the symbol [‘X’] by a name of
this sentence is to be regarded as a true sentence of this language;
(III) in the language in question an empirically established premiss
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having the same meaning as [(3*)] can be formulated and accepted
as a true sentence.
If these observations are correct, then the very possibility of a consistent use of the expression ‘true sentence’ which is in harmony
with the laws of logic and the spirit of everyday language seems to
be very questionable, and consequently the same doubt attaches to
constructing a correct definition of this expression.
15. Ibid., p. 208.
16. Ibid., p. 273. This formulation is from the Postscript to the English translation. The original version (p. 265) is very similar except that Tarski restricts his
claim to languages of finite order, and is not explicit about the higher order of the
metalanguage. He also adds there that by terms belonging to the morphology of
language he means “names of linguistic expressions and of the structural relations
existing between them”.
17. Ibid., p. 273.
18. It was precisely this syntactic feature of Tarski’s definition that made it so
dear to the logical positivists. Any theoretical appeal to the world smelled of an
appeal to metaphysics, and the only acceptable metaphysics is the metaphysics of
linguistic syntax. Theoretical semantics became acceptable because it was seen as
a part of the general syntax of language – or as being analytic in a closely related
sense. In Introduction to Semantics, for example, Carnap distinguishes between
descriptive semantics, which is
. . . the description and analysis of the semantical features either of
some particular historically given language, e.g. French, or of all
historically given languages in general . . .
and pure semantics (p. 12):
The construction and analysis of semantical systems is called pure
semantics. The rules of a semantical system S constitute, as we
shall see, nothing else than a definition of certain semantical concepts
with respect to S, e.g. ‘designation in S’ or ‘true in S’. Pure semantics
consists of definitions of this kind and their consequences; therefore,
in contradistinction to descriptive semantics, it is entirely analytic
and without factual content.
19. See Enderton A Mathematical Introduction to Logic, pp. 154-163. This is also
shown by the “interpretations” that are used in elementary logic texts, which are
typically fictional (“Mary”, “Peter”, “John”, etc.).
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20. “The Semantic Conception of Truth”, p. 342. In “The Concept of Truth in
Formalized Languages”, pp. 166-167, he says:
. . . we are not interested here in ‘formal’ languages and sciences
in one special sense of the word ‘formal’, namely sciences to the
signs and expressions of which no material sense is attached. For
such sciences the problem here discussed has no relevance, it is not
even meaningful. We shall always ascribe quite concrete and, for us,
intelligible meanings to the signs which occur in the languages we
shall consider. The expressions which we call sentences still remain
sentences after the signs which occur in them have been translated
into colloquial language. The sentences which are distinguished as
axioms seem to us to be materially true, and in choosing rules of
inference we are always guided by the principle that when such rules
are applied to true sentences the sentences obtained by their use
should also be true.
21. Op. Cit., pp. 155-156.
22. See “Truth and Proof”, p. 65.
23. “Truth and Proof”, p. 69. In “The Semantic Conception of Truth”, pp.
344-345, Tarski says:
It should be emphasized that neither the expression (T) itself (which
is not a sentence, but only a schema of a sentence) nor any particular instance of the form (T) can be regarded as a definition of truth.
We can only say that every equivalence of the form (T) obtained by
replacing [‘S’] by a particular sentence, and ‘X’ by a name of this
sentence, may be considered a partial definition of truth, which explains wherein the truth of this one individual sentence consists. The
general definition has to be, in a certain sense, a logical conjunction
of all these partial definitions.
(The last remark calls for some comments. A language may admit
the construction of infinitely many sentences; and thus the number
of partial definitions of truth referring to sentences of such a language will also be infinite. Hence to give our remark a precise sense
we should have to explain what is meant by a “logical conjunction
of infinitely many sentences”; but this would lead us too far into
technical problems of modern logic.)
24. See Tarski and Scott “The Sentential Calculus with Infinitely Long Expressions”, and Tarski “Remarks on Predicate Logic with Infinitely Long Expressions”.
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If one defines truth using infinite conjunctions, one can still go on to define truth
for the metalanguage in the same way, except that now the object language for
which the definition is given contains infinitary sentences and the metalanguage
must be able to handle them as well. Infinitary sentences actually played an important role in the early development of modern logic – see Moore “The Emergence
of First-Order Logic”.
25. This is particularly clear in the definition of truth in an interpretation in terms
of the notion of denotation, which does not appeal directly to the biconditionals.
In this definition one still appeals to infinitary conditions, however, though not
to infinitary sentences. For what one wants to do is to reduce the truth of a
quantified sentence to the truth of its instances, so that the definition of truth
for quantified sentences would be like the definition of truth for conjunctions and
disjunctions in terms of the truth or falsity of the components. But there is the
additional problem that there may not be enough names in the language to make
up all the necessary instances even separately from each other. One solution to
this problem, which is very much like the solution in terms of infinitary sentences,
is to add enough names to the language to handle the particular interpretation
in question. This is what Shoenfield does in Mathematical Logic (pp. 18-19).
Another solution is to get the instances by fiddling with the interpretation. If
one has at least one name (or individual constant) which does not appear in
the quantified sentence, then one can get one instance such that by varying the
interpretation of that name through all the objects in the interpretation that one
instance can play the role of each and every instance. So one defines the truth
of a universally quantified sentence in terms of the truth of this one instance in
all interpretations that differ from the given one at most in that respect; i.e.,
in that the name one selected may denote a different object. (And similarly for
existentially quantified sentences.) This is what Mates does in Elementary Logic
(p. 60). One can do the same thing using a variable instead of a constant, but in
either case the constant or the variable is treated as a variable name and the idea
is essentially equivalent to taking something like an infinite conjunction.
What mathematicians like Shoenfield and others realized is that from
the point of view of defining the extension of ‘true’ it makes no difference whatever whether one adds a whole bunch of constants to the language or works with
the infinite sequences used in the definition of satisfaction. A definition such as
Mates’, based on satisfaction, gives the impression that one is explaining truth in
a more direct fashion, but this is quite illusory. There is no explanatory difference
between Mates’ definition and Shoenfield’s. The explanatory power of the definition of truth derives from reducing the definition of complex sentential structures
to simpler parts in a finite process of decomposition, but for quantified sentences
one must appeal to infinitely many clauses (or part-like sentences, or variant interpretations, or whatever), which is essentially an appeal to infinite unions and/or
intersections.
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26. In “The Semantic Conception of Truth”, p. 361, he says:
I have heard it remarked that the formal definition of truth has
nothing to do with “the philosophical problem of truth”. However,
nobody has ever pointed out to me in an intelligible way just what
this problem is. I have been informed in this connection that my
definition, though it states necessary and sufficient conditions for a
sentence to be true, does not really grasp the “essence” of this concept. Since I have never been able to understand what the “essence”
of a concept is, I must be excused from discussing this point any
longer.
In general, I do not believe that there is such a thing as “the philosophical problem of truth.” I do believe that there are various intelligible and interesting (but not necessarily philosophical) problems
concerning the notion of truth, but I also believe that they can be
exactly formulated and possibly solved only on the basis of a precise
conception of this notion.
27. After discussing the question of the extension of the term ‘true’ in “The
Semantic Conception of Truth”, Tarski has a section on the meaning of the term
‘true’ in which he says (p. 342):
Much more serious difficulties are connected with the problem of the
meaning (or the intension) of the concept of truth.
The word “true”, like other words from our everyday language, is
certainly not unambiguous. And it does not seem to me that the
philosophers who have discussed this concept have helped to diminish its ambiguity. In works and discussions of philosophers we meet
many different conceptions of truth and falsity, and we must indicate
which conception will be the basis of our discussion.
We should like our definition to do justice to the intuitions which
adhere to the classical Aristotelian conception of truth – intuitions
which find their expression in the well-known words of Aristotle’s
Metaphysics: . . .
If we wished to adapt ourselves to modern philosophical terminology,
we could perhaps express this conception by means of the familiar
formula:
The truth of a sentence consists in its agreement with (or correspondence to) reality.
(For a theory of truth which is to be based upon the latter formulation the term “correspondence theory” has been suggested.)
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28. In Reason, Truth and History, pp. 128-129, Putnam remarks:
Some philosophers have appealed to the equivalence principle, that is
the principle that to say of a statement that it is true is equivalent to
asserting the statement, to argue that there are no real philosophical
problems about truth. Others appeal to the work of Alfred Tarski . . .
Tarski’s work was itself based on the equivalence principle: in fact
his criterion for a successful definition of ‘true’ was that it should
yield all sentences of the form ‘P’ is true if and only if P, e.g. [(28)] as
theorems of the meta-language (where P is a sentence of the formal
notation in question).
But the equivalence principle is philosophically neutral, and so is
Tarski’s work. On any theory of truth, ‘Snow is white’ is equivalent
to ‘ “Snow is white” is true’.
I argued against the claim (Frege’s) that to say
‘Snow is white’ is true
is to say the same thing as to say
Snow is white
in Chapter 2. I also argued in chapters 2 and 6 that although these two sentences
are c-logically equivalent, they are not tv-logically equivalent. Since Putnam does
not make these distinctions, it is not clear exactly what he means by “equivalence”,
but his next paragraph suggests that he may agree that the two sentences have
the same meaning:
Positivist philosophers would reply that if you know (T) above, you
know what ‘ “Snow is white” is true’ means: it means snow is white.
And if you don’t understand ‘snow’ and ‘white’, they would add,
you are in trouble indeed! But the problem is not that we don’t
understand ‘Snow is white’; the problem is that we don’t understand
what it is to understand ‘Snow is white’. This is the philosophical
problem. About this (T) says nothing.
Schema (T) is essentially a generalization of Frege’s claim that the content of a sentence is the same as the content of a sentence asserting the truth of
that sentence. Although it doesn’t follow from Frege’s view that all instances of
schema (T) are analytic, because this depends on the way in which the name or
description substituted for ‘X’ identifies the sentence substituted for ‘S’, it does
seem to follow that any instance of (T) obtained replacing ‘X’ by a quotation name
of the sentence replacing ‘S’ is analytic. (And from this it follows that all instances
of (T) are true, because the substitution of a quotation name of a sentence by
another name of the same sentence must preserve truth value.) The apparent
plausibility of this view depends partly on the recognition that the principles (T1-
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Inf) and (T2-Inf) are analytic for the predicate ‘true’ and quotation names when
‘X’ is replaced by a quotation name of the sentence replacing ‘S’. Since these
principles are indeed analytic, it is quite natural to conclude incorrectly that such
instances of schema (T) are also analytic.
29. In “The Semantic Conception of Truth”, pp. 361-362, Tarski says:
It has been claimed that – due to the fact that a sentence like “snow
is white” is taken to be semantically true if snow is in fact white
(italics by the critic) – logic finds itself involved in a most uncritical
realism.
If there were an opportunity to discuss the objection with its author,
I should raise two points. First, I should ask him to drop the words
“in fact,” which do not occur in the original formulation and which
are misleading, even if they do not affect the content. For these
words convey the impression that the semantic conception of truth
is intended to establish the conditions under which we are warranted
in asserting any given sentence, and in particular any empirical sentence. However, a moment’s reflection shows that this impression is
merely an illusion; and I think that the author of the objection falls
victim to the illusion which he himself created.
In fact, the semantic definition of truth implies nothing regarding
the conditions under which a sentence like (1):
(1) snow is white
can be asserted. It implies only that, whenever we assert or reject
this sentence, we must be ready to assert or reject the correlated
sentence (2):
(2) the sentence “snow is white” is true.
Thus, we may accept the semantic conception of truth without giving up any epistemological attitude we may have had; we may remain naive realists, critical realists or idealists, empiricists or metaphysicians – whatever we were before. The semantic conception is
completely neutral toward all these issues.
It should be noticed that Tarski’s claim about assertion is different from
Putnam’s because he is not saying that the two assertions are equivalent. Moreover, Tarski is only going from the assertion or rejection of (1) to the assertion or
rejection of (2).
The question I am examining in the text, however, is what to make
of Tarski’s conclusion. Unlike Putnam, Tarski is not claiming that the semantic
conception is neutral in relation to other conceptions of truth. In fact, I think
that he explicitly (or implicitly) denies this in the passage I quoted in note 27.
And he explicitly denies it in “The Concept of Truth in Formalized Languages”,
p. 153:
255
A thorough analysis of the meaning current in everyday life of the
term ‘true’ is not intended here. Every reader possesses in greater or
less degree an intuitive knowledge of the concept of truth and he can
find detailed discussions on it in works on the theory of knowledge.
I would only mention that throughout this work I shall be concerned
exclusively with grasping the intentions which are contained in the
so-called classical conception of truth (‘true – corresponding with
reality’) in contrast, for example, with the utilitarian conception
(‘true – in a certain respect useful’).
But, evidently, this raises an important point of interpretation concerning Tarski’s
claim of neutrality. See also Kirkham Theories of Truth, pp. 193-196.
30. “The Semantic Conception of Truth”, pp. 358-359:
The author of this objection mistakenly regards scheme (T) . . . as
a definition of truth. He charges this alleged definition with “inadmissible brevity, i.e., incompleteness,” which “does not give us the
means of deciding whether by ‘equivalence’ is meant a logical-formal,
or a non-logical and also structurally non-describable relation.” To
remove this “defect” he suggests supplementing (T) in one of the
two following ways:
(T′ ) X is true if, and only if, p is true,
or
(T′′ ) X is true if, and only if, p is the case (i.e., if what p states is
the case).
Then he discusses these two new “definitions,” which are supposedly free from the old, formal “defect,” but which turn out to be
unsatisfactory for other non-formal reasons.
This new objection seems to arise from a misunderstanding concerning the nature of sentential connectives . . . . The author of the
objection does not seem to realize that the phrase “if, and only if”
(in opposition to such phrases as “are equivalent” or “is equivalent
to”) expresses no relation between sentences at all since it does not
combine names of sentences.
In general, the whole argument is based upon an obvious confusion
between sentences and their names. It suffices to point out that – in
contradistinction to (T) – schemata (T′ ) and (T′′ ) do not give any
meaningful expressions if we replace in them ‘p’ by a sentence; for
the phrases “p is true” and “p is the case” (i.e., “what p states is
the case”) become meaningless if ‘p’ is replaced by a sentence, and
not by the name of a sentence . . . .
It follows from Tarski’s remarks that Putnam’s interpretation of the
biconditional in terms of assertion, or even material equivalence, is not Tarski’s.
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There is an implication about assertion, but that’s not what the biconditionals
are supposed to mean. It should be noticed, however, both in connection with
this point and in connection with Tarski’s remarks above, that in “The Semantic
Conception of Truth” he generally refers to instances of (T) as “equivalences”.
Thus, when he introduces schema (T) he says (p. 344):
Let us consider an arbitrary sentence; we shall replace it by the letter
‘p.’ We form the name of this sentence and we replace it by another
letter, say ‘X.’ We ask now what is the logical relation between the
two sentences “X is true” and ‘p.’ It is clear that from the point of
view of our basic conception of truth these sentences are equivalent.
In other words, the following equivalence holds:
(T) X is true if, and only if, p.
We shall call any such equivalence (with ‘p’ replaced by any sentence
of the language to which the word “true” refers, and ‘X’ replaced by
a name of this sentence) an “equivalence of the form (T).”
31. “The Semantic Conception of Truth”, p. 366. I am not saying that Tarski
meant the instances of schema (T) as meaning postulates in an explicit sense of
this term, but it is certainly hard to know what he meant. What he says is that
any adequate definition of ‘true sentence’ for a given language must have all the
biconditionals for sentences of that language as consequences. Which can either
be interpreted as a purely syntactic condition or as somehow suggesting that the
biconditionals are stipulative of the meaning of ‘true sentence’ for that language.
32. Tarski makes this point in “The Semantic Conception of Truth” in response
to the objection that in defining ‘true’ he uses connectives and that the meaning
of the connectives is usually explained in terms of ‘true’ and ‘false’. He says (p.
357):
It is undoubtedly the case that a strictly deductive development of
logic is often preceded by certain statements explaining the conditions under which sentences of the form “if p, then q”, etc., are
considered true or false. (Such explanations are often given schematically, by means of the so-called truth-tables.) However, these statements are outside of the system of logic and should not be regarded
as definitions of the terms involved. They are not formulated in the
language of the system, but constitute rather special consequences of
the definition of truth given in the meta-language. Moreover, these
statements do not influence the deductive development of logic in
any way. For in such a development we do not discuss the question whether a given sentence is true, we are only interested in the
problem whether it is provable.
On the other hand, the moment we find ourselves within the deductive system of logic – or of any discipline based upon logic, e.g., of
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semantics – we either treat sentential connectives as undefined terms,
or else we define them by means of other sentential connectives, but
never by means of semantic terms like “true” or “false”. . . . a vicious
circle in definition arises when the definiens contains either the term
to be defined itself, or other terms defined with its help. Thus we
clearly see that the use of sentential connectives in defining the semantic term “true” does not involve any circle.
33. Mates “Austin, Strawson, Tarski, and Truth”, p. 396. I have added an ‘M’
to Mates’ numbering to avoid confusion with my own earlier numbering.
Tarski is also quite worried in “The Semantic Conception of Truth” to
rebate charges that he is engaged in metaphysics, and he concludes (p. 364):
. . .I can summarize the arguments given above by stating that in no
interpretation of the term “metaphysical” which is familiar and more
or less intelligible to me does semantics involve any metaphysical
elements peculiar to itself.
34. See Hartry Field “Tarski’s Theory of Truth” where this question is discussed
from a somewhat different perspective.
35. This is essentially Frege’s regress argument that I discussed briefly in Chapter
2 note 18. There are also other aspects of my argument that are related to Frege’s
argument.
36. I think that it is quite questionable whether Tarski’s work on truth has any
bearing on various conceptions of truth that are usually mentioned in connection
with this issue of neutrality. Take the coherentist conception, for instance. If
one looks at a classic account of the coherence theory such as Joachim’s in The
Nature of Truth, one realizes that it is very far indeed from anything that Tarski
does. To begin with, truth is a feature of judgements, not of sentences. Second,
there are degrees of truth; no judgement is wholly true or wholly false. Third, the
truth of a judgement does not derive from its structure in terms of satisfaction,
but derives from its multiple connections to the entire body of judgements.
Joachim starts his presentation as follows (Op. Cit., pp. 66, 68):
We may start with the following as a provisional and rough formulation of the coherence-notion. ‘Anything is true which can be
conceived. It is true because, and in so far as, it can be conceived.
Conceivability is the essential nature of truth’. . . . To ‘conceive’
means for us to think out clearly and logically, to hold many elements together in a connexion necessitated by their several contents.
And to be ‘conceivable’ means to be a ‘significant whole’, or a whole
possessed of meaning for thought. A ‘significant whole’ is such that
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all its constituent elements reciprocally involve one another, or reciprocally determine one another’s being as contributory features in a
single concrete meaning. The elements thus cohering constitute a
whole which may be said to control the reciprocal adjustment of its
elements, as an end controls its constituent means. . . .
Thus ‘conceivability’ means for us systematic coherence, and is the
determining characteristic of a ‘significant whole’. The systematic
coherence of such a whole is expressed most adequately and explicitly in the system of reasoned knowledge which we call a science or
a branch of philosophy. Any element of such a whole shares in this
characteristic to a greater or less degree – i.e., is more or less ‘conceivable’ – in proportion as the whole, with its determinate inner
articulation, shines more or less clearly through that element; or in
proportion as the element, in manifesting itself, manifests also with
more or less clearness and fullness the remaining elements in their
reciprocal adjustment.
Another interesting statement of the view is given in section 26 (pp. 76-78).
Joachim proceeds to elaborate these ideas in various ways, and to discuss the
question of degrees of truth, but nothing he does in the book seems to me to bear
any connection whatsoever to what Tarski does. Judging from Tarski’s remarks
concerning the unintelligibility of philosophical notions such as ‘essence’, I should
think that Joachim’s account (and other classic accounts of the coherence theory)
would have been completely unintelligible to him. If so, it’s hard to see on what he
could have based his claim of philosophical neutrality if not on some very general
considerations concerning language, meaning, and extension which he assumed to
be independent of philosophical issues. I see no clear grounds for this assumption,
however.
It’s true, as I pointed out before, that Tarski doesn’t talk about neutrality in relation to other theories of truth, and refers only to idealism, realism, etc.,
but the views I quoted above are part of a certain version of idealism, and it is
not at all clear that one can separate the conception of truth from the philosophical position. Of course, somebody may develop (or hold) a coherence theory of
truth that has connections with Tarski’s work, and for which C(T) is a criterion
of adequacy, but this seems to me a different issue. (For a critical discussion of
Joachim’s views see Russell “The Monistic Theory of Truth”.)
37. See “The Semantic Conception of Truth”, pp. 358-359, and “The Concept of
Truth in Formalized Languages”, pp. 162-163. This is the objection I mentioned
in note 8 against using (1S*) as a definition of ‘true’.
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Chapter 8
The True and the False Revisited: Frege’s Logic
Frege is held to be largely responsible for debunking the traditional
conception that all sentences are of subject-predicate form. Partly because
of his relational analysis of sentences, but mostly because of his analysis
of quantification. The combination of quantifiers and connectives is not
supposed to be a subject-predicate combination. Although it is true that
Frege explicitly rejects the traditional subject-predicate distinction in Begriffsschrift, and does not separate notationally non-logical predicates from
non-logical subjects, his logical analysis actually involves a sharpening of
the traditional subject-predicate analysis. It is quite clear from Frege’s ontological analysis that the combinations of quantifiers and connectives are
properties, or concepts, or functions. And it is also clear from his notation,
for he separates the logical properties from the non-logical subjects.
Frege’s logical notation was very badly misunderstood, and people
have since felt that linearizing it was an improvement1 . Yet this linearization
had the effect of eliminating the careful separation between logical property
and non-logical subjects. As a result the distinction between what is said
and what it is said about became completely muddled2 . Nevertheless, the
problems with aboutness do indeed originate with Frege’s treatment of the
traditional subject-predicate distinction in Begriffsschrift, and he can be
blamed for some of the confusion – although it may be misleading to say
that he eliminated the distinction. To substantiate this, as well as other
points I want to make later, I will outline my view of the development of
some main aspects of Frege’s logical ideas from Begriffsschrift to The Basic
Laws of Arithmetic.
Frege argued in Begriffsschrift, as I did in Chapters 1 and 6 following him, that a relational sentence has different readings in terms of
subject-predicate structure. He argued further that this makes no difference for the consequences of the sentence and that all that matters for this
is the conceptual content, which is the same in all readings. Frege’s notion
of conceptual content is characterized in terms of deductive (logical) relations. For although Frege does not define ‘conceptual content’, he gives as a
criterion for two sentences having the same conceptual content that they be
intersubstitutable preserving deducibility (or logical consequence)3 . With
this move he eliminates the traditional subject-predicate distinction from
his conceptual script as an explicit distinction. The fundamental units are
conceptual contents, which Frege indicates by means of the content stroke
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‘—’ prefixed to a sentence, and the only “predicate” is the judgment sign
‘|—’; which can also be read, Frege suggests, as the predicate ‘is a fact’4 .
He concludes the section with the observation (p. 13):
In the first draft of my formula language I allowed myself to be mislead by the example of ordinary language into constructing judgments out of subject and predicate. But I soon became convinced
that this was an obstacle to my specific goal and led only to useless
prolixity.
Considered as a predicate the judgment sign together with Frege’s
notation for negation, conditionality, and universal quantification gives raise
to an infinite number of what we may reasonably call ‘logical predicates’.
A judgment of conditionality
|——–A
|−−B
can be thought of as a logical predicate that asserts a certain relation between the conceptual contents — A and — B. And this is precisely how
Frege explains it. He distinguishes the four possibilities
(1) A is affirmed and B is affirmed,
(2) A is affirmed and B is denied,
(3) A is denied and B is affirmed,
(4) A is denied and B is denied,
and says that the above formula “stands for the judgment that the third
of these possibilities does not take place, but one of the three others does”
(p. 14). Of course, what follows the vertical stroke is the content of the
judgment of conditionality, which is treated as a unit, but we can still
separate very clearly the logical predicate from the non-logical arguments5 .
We can also make separations within the logical predicates themselves, for Frege distinguishes the content stroke of A, the content stroke of
B, and the content stroke of the conditionality relation (pp. 14-15). Similarly, when he introduces negation
———–
A
|
he says (p. 18): “The part of the horizontal stroke to the right of the
negation stroke is the content stroke of A; the part to the left of the negation
stroke is the content stroke of the negation of A”.
Let’s consider now the ontological content of Frege’s conceptual
script. It is quite clear from the preface and from the development of Begriffsschrift that Frege’s main concern is not with ontology but with proof,
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inference, and the deductive relations between propositions6 . He shows little concern for ontological matters, and from an ontological point of view
his presentation is somewhat problematic. The ontological notions of truth,
concept, and object, central to his later philosophy and logic, play a different
role in Begriffsschrift.
The general notion of function, under which he subsumes the notion of concept, does originate in Begriffsschrift, but not clearly as an ontological notion. He introduces it linguistically. A sentence can be divided
into argument(s) and function in various ways, and in each case we have
the same conceptual content. The arguments of a function are also expressions, which can be functions as well. Thus Frege’s initial distinction
between function and argument does not have the same emphasis as the
later distinction between function and object7 .
This distinction between function and argument, together with the
claim that the conceptual content is the same in the different analyses, is the
basis for Frege’s elimination of the traditional subject-predicate distinction
as an explicit notational feature of his system. The important point, though,
is that the distinction itself is not eliminated but enriched; even if not in
ontological terms. Frege can make many of the distinctions I made in earlier
chapters, and he used them in his proofs. He emphasized this repeatedly in
his debate with the supporters of the Boolean formulation of logic.
He argued that by taking judgeable contents as his fundamental
units, together with the logical notions and the function-argument distinction, his system was more parsimonious, more natural, and more powerful
than theirs. He argued, to begin with, that either they had to take both
primary propositions – which are the equations involving classes (or extensions of concepts) – and secondary propositions (such as hypotheticals)
as basic, with the connection between the two becoming unclear, or they
had to express general hypothetical propositions in a very unnatural way
in terms of classes of moments of time. Second, he argued that he could
express the traditional distinctions in his notation but that it wasn’t clear
that they could – as for example in connection with existential statements.
Finally, he argued that his two-dimensional notation separated clearly the
logical relations from the non-logical contents, which theirs didn’t8 .
An interesting feature of Begriffsschrift is that the notion of truth
is not used explicitly in the characterization of the logical notions – we
have seen already that conditionality is introduced in terms of affirmation
and denial rather than truth and falsity. But the notion of truth does play
an important role in that Frege also uses the expressions ‘is a fact’, ‘takes
place’, ‘does not take place’, to translate various combinations that are first
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expressed in terms of affirmation and denial9 . He also uses this qualification
‘is a fact’ when he introduces the universal quantifier (p. 24):
In the expression of a judgment we can always regard the combination of signs to the right of |— as a function of one of the signs
occurring in it. If we replace this argument by a German letter and
if in the content stroke we introduce a concavity with this German
letter in it, as in
a−−Φ(a),
|−−∪
this stands for the judgment that, whatever we may take for its argument, the function is a fact.
Given Frege’s characterization of argument and function, it seems that the
quantifier should not be interpreted objectually but substitutionally – whatever substitution-instance we take, the content is a fact. If the predicate ‘is
a fact’ is distinguished from the judgment sign, then Frege’s formula indicates the judgment that every substitution-instance is a fact, or is true, or
is the case. If we read the judgment sign itself as the predicate ‘is a fact’,
then the formula means that it is a fact that every substitution-instance is
a fact.
We see, therefore, that Frege’s ontology in Begriffsschrift is fundamentally an ontology of conceptual contents that can be split up in various
ways and can be affirmed or denied. If we conflate affirmation with truth,
as Frege seems to do in various places, then we have a single predicate ‘is
true’ (or ‘is a fact’) that is used to assert conceptual contents. Of course,
the truth or falsity of a conceptual content will depend on reality – for example, on certain objects having a certain relation to each other – but the
system itself deals with these objects and relations only indirectly through
the conceptual contents10 .
In The Foundations of Arithmetic Frege states as the second of
three fundamental principles that guided his enquiry the following (p. x):
never to ask for the meaning [Bedeutung] of a word in isolation, but
only in the context of a proposition.
This has become known as “the context principle” and has played a very
important role in twentieth-century analytic philosophy11 . One of the many
puzzles about this dictum is what becomes of it after The Foundations of
Arithmetic, since Frege never mentions it again in this form. Another puzzle
is what Frege meant by it.
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In my view Frege’s principle is a direct expression of the formulation of logic in Begriffsschrift, that he defended so vehemently against the
Boolean logicians, and the use to which he puts it in various places in The
Foundations of Arithmetic is largely meant to preserve the conceptual contents as the basic units of the conceptual script. Nevertheless, his conception
was beginning to give way to a more explicit ontological conception.
That The Foundations of Arithmetic is more directly concerned
with ontological matters than Begriffsschrift is quite clear. In it Frege begins
his lifelong attack on psychologism, formalism, and empiricism. It is a
vicious attack, with no holds barred, and Frege is very good at it. The
other two fundamental principles that Frege states, namely
always to separate sharply the psychological from the logical, the
subjective from the objective;
and
never to lose sight of the distinction between concept and object,
are related to this attack12 .
These principles are not clearly formulated as ontological principles, however. Even in The Foundations of Arithmetic Frege does not draw
an explicit ontological distinction between concept and object, but uses a
formulation that is very close to his formulation in Begriffsschrift of the
distinction between function and argument, appealing again to the splitting
of judgment contents. The difference is that in characterizing concept and
object he takes singular judgment contents13 . Similarly, Frege’s claim that
numbers are self-subsistent objects leaves the ontological question somewhat
up in the air. He says (p. 72):
The self-subsistence which I am claiming for number is not to be
taken to mean that a number word signifies [bedeuten] something
when removed from the context of a proposition, but only to preclude the use of such words as predicates or attributes, which appreciably alters their meaning.
Although the first principle does seem to call for a more explicit
ontological interpretation, this is partly undercut by the context principle
and by the conception of objects as subjects of a judgeable content. Nevertheless, we begin to get some rather strong whiffs of a somewhat Platonic
ontological view. Frege emphasizes in no uncertain terms the objectivity of
numbers, and dismisses as irrelevant objections to the effect that numbers
cannot be perceived, or imagined, or spatially located14 . But how is this
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objectivity to be understood? This is where Frege makes a crucial appeal
to the context principle. He says (p. 73):
How, then, are numbers to be given to us, if we cannot have any
ideas or intuitions of them? Since it is only in the context of a
proposition that words have any meaning, our problem becomes
this: To define the sense of a proposition in which a number word
occurs. That, obviously, leaves us still a very wide choice. But we
have already settled that number words are to be understood as
standing for self-subsistent objects. And that is enough to give us a
class of propositions which must have a sense, namely those which
express our recognition of a number as the same again. If we are
to use the symbol a to signify an object, we must have a criterion
for deciding in all cases whether b is the same as a, even if it is not
always in our power to apply this criterion. In our present case, we
have to define the sense of the proposition
“the number which belongs to the concept F is the same as that
which belongs to the concept G”;
that is to say, we must reproduce the content of this proposition in
other terms, avoiding the use of the expression
“the number which belongs to the concept F”.
In doing this, we shall be giving a general criterion for the identity
of numbers. When we have thus acquired a means of arriving at a
determinate number and of recognizing it again as the same, we can
assign it a number word as its proper name.
We have already seen a special version of this idea in Frege’s characterization of sameness of conceptual content in Begriffsschrift, and he
states it several times in The Foundations of Arithmetic – pp. 116 and 119,
for instance. It is one of the central ideas of his definition of numbers. Yet,
with this use of the context principle he paints himself into a corner.
To define
(1) the number which belongs to the concept F is the same as that which
belongs to the concept G
simply in terms of one-one correlation between F and G will not do according
to Frege, because this will only settle the question of identity for numbers
that are presented in the form ‘the number which belongs to the concept Z’.
It will not settle the question whether Julius Caesar is the number which
belongs to the concept ‘is a moon of Mars’, for instance. Similarly, to define
(2) the direction of line a is the same as the direction of line b if, and only
if, line a is parallel to line b,
will not “decide for us whether England is the direction of the Earth’s axis”
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(p. 78). The criterion of sameness must apply in all cases, independently
of how the object is given.
It is because of this that Frege appeals to the extensions of concepts. He then defines directly
(3) the direction of line a is the extension of the concept ‘parallel to line a’.
And a similar move gives him the definition of numbers (pp. 78-79). But
this only pushes the problem one step further. Since extensions are objects for Frege, there must be a general criterion of identity that will settle
whether Julius Caesar, or England, is the extension of this or that concept;
and this Frege does not have. All he has is a criterion for settling questions
of identity of the form
(4) the extension of the concept F is the same as the extension of the concept G;
namely, that F and G are mutually subordinate.
We see, therefore, that in spite of the achievements Frege’s contextual conception was beginning to leak. The move through extensions would
not require further justification if one assumes that extensions are objects
in an ontological transcendental sense and that reality settles the identity
questions. But as it is formulated in terms of a criterion of identity, Frege
does not have a general contextual solution for his problem15 .
In Frege’s fairly detailed discussion of the example of directions we
begin to see a more explicit ontological treatment of objects, however. He
distinguishes the way in which an object is introduced (i.e., the definition of
an object), which “lays down the meaning [Bedeutung] of a symbol”, from
properties of the object and from assertions about the object (pp. 78-79):
If we were to try saying: q is a direction if it is introduced by means
of the definition set out above [i.e., q is a direction, if there is a
line b whose direction is q], then we should be treating the way in
which the object q is introduced as a property of q, which it is not.
The definition of an object does not, as such, really assert anything
about the object, but only lays down the meaning [Bedeutung] of a
symbol. After this has been done, the definition transforms itself
into a judgment, which does assert about the object; but now it
no longer introduces the object, it is exactly on a level with other
assertions made about it. If, moreover, we were to adopt this way
out, we should have to be presupposing that an object can only be
given in one single way; for otherwise it would not follow, from the
fact that q was not introduced by means of our definition, that it
could not have been introduced by means of it. All identities would
then amount simply to this, that whatever is given to us in the same
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way is to be reckoned as the same. This, however, is a principle so
obvious and so sterile as not to be worth stating. We could not, in
fact, draw from it any conclusion which was not the same as one of
our premisses. Why is it, after all, that we are able to make use of
identities with such significant results in such diverse fields? Surely
it is rather because we are able to recognize something as the same
again even although it is given in a different way.
It is with this problem that Frege begins “On Sense and Reference”.
One clear infirmity of Begriffsschrift is the treatment of identity
in section 8, where in spite of the characterization of identity as a relation
between signs, the ambiguity of Frege’s notion of content in relation to the
later distinction of sense and reference is quite apparent. In “On Sense and
Reference” the distinction between the sign, its sense, and its denotation is
crystal clear; the sign expresses a sense, the sense is a manner of presentation
of the denotation, and the denotation (if any) is something independent
given in reality. Senses are more general than the old contents in that
they are not restricted to “propositional” senses, and are now characterized
as manners of presentation rather than in terms of intersubstitutability in
deductive contexts. Intersubstitutability preserving consequence gives way
to intersubstitutability preserving truth value as a criterion for sameness of
sense16 .
In my view “On Sense and Reference” marks an important ontological turn in Frege’s thought – and the companion papers “Function
and Concept” and “On Concept and Object” draw out some of the consequences of this turn17 . We now have two fundamental ontological categories: functions and objects. Both are taken as primitives, as are the
notions of ‘course-of-values’, ‘sign’, ‘sense’18 . The fundamental use of contents in Begriffsschrift is as judgment contents, however; so what becomes
of the concept script? This is where Frege’s work on truth comes in. One
can only speculate on what led Frege to his ideas about truth, but I will
make some connections with the development of his views that I suggested
so far.
Frege’s main concern in Begriffsschrift is with proof – a concern
that is also clearly present and emphasized in The Foundations of Arithmetic and in The Basic Laws of Arithmetic – but this is not merely a concern
with deductive relations between judgment contents. Proof is a means for
the justification and apprehension of truth. So truth was always Frege’s
most central concern. The problem is that the treatment of truth in Begriffsschrift is somewhat ambiguous, as is the treatment of the judgment
sign. On the one hand Frege seems to suggest that the judgment sign is
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like a truth predicate, translatable by ‘is a fact’. But, on the other hand,
he also expresses explicitly the view of judgment as an acknowledgement of
truth by a subject, as when he says (p. 11):
If we omit the small vertical stroke at the left end of the horizontal
one, the judgment will be transformed into a mere combination of
ideas, of which the writer does not state whether he acknowledges
it to be true or not.
In “The Aim of “Conceptual Notation”” he is very emphatic about this
point and claims that with the distinction between the content stroke and
the judgment stroke he “meant to have a very clear distinction between the
act of judging and the formation of a mere assertible content”19 . If one
takes this interpretation literally, however, then it is not clear that Frege’s
characterization of the logical notions is independent of acts of judging.
These notions would be characterized in terms of assertibility conditions, not
truth conditions. To avoid this Frege would have to distinguish notationally
the truth of a content from the act of acknowledging its truth. Is Frege
talking about contents or using them? Or both? And what about the signs?
What is their role?
The treatment of identity in Begriffsschrift is quite pertinent to
this. Frege says (p. 21):
Now let
|— (A ≡ B)
mean that the sign A and the sign B have the same conceptual content, so that we can everywhere put B for A and conversely.
Here we see again that Frege seems to confer to the combination of symbols
the status of a statement of fact (the sameness of the two contents) rather
than that of an act. But, aside from this, Frege’s characterization of identity
involves serious ambiguities.
As opposed to the characterization of conditionality, where Frege
says that “A and B stand for contents” (p. 13), now ‘A’ and ‘B’ stand for
signs. In fact, there is an ambiguity in connection with conditionality as
well, because Frege also says that “[t]he part of the horizontal stroke between A and the condition stroke is the content stroke of A” (p. 15), which
suggests that ‘A’ stands for a sign rather than a content. Frege points out
the ambiguity between sign and content in connection with identity (pp.
20-21), but this involves another ambiguity in relation to the notion of ‘content’. For in Frege’s explanation of the need for an identity sign, which
is rather similar to the considerations at the beginning of “On Sense and
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Reference”, ‘content’ seems to mean the thing “denoted” by the sign. One
can’t interpret identity uniformly in this way, however, because the identity
sign is also used in connection with sentential contents in definitions and in
theorems; and what do these “denote”? Moreover, at the end of the preface (p. 8) Frege says that he could have combined the two laws of double
negation into the single formula
(5) |— (——
a ≡ a),
||
which suggests that identity can also be used to express something like logical equivalence. Given Frege’s conventions on the use of variables (p. 25),
(5) is a universally quantified formula that is judged true for all conceptual
contents. So for each specific sentence A, the conceptual contents ——
A
||
and — A are the same. But, obviously, this does not hold for conditionality
in general; i.e., from Frege’s characterization of conditionality one cannot
infer that if the relation of conditionality holds between the contents (of) A
and B in both directions, then A ≡ B. Since Frege does not introduce notions of logical implication and logical equivalence, we also have a question
about the relation between identity and biconditionality.
So when Frege distinguishes clearly between sign, sense, and denotation, it is not only the problem of identity for objects that has to be
straightened out; there are problems with the formulation of his concept
script in general. And in particular there is a very pressing problem about
what to do with sentential signs. Do they denote? What is the denotation
of a sentence? Given the formulations in Begriffsschrift, the most natural
solution would have been to take conceptual contents (now thoughts) as
their denotation. And many later logicians have indeed taken something
like propositions as the denotation of sentences. But this wouldn’t do, because although it may take care of the propositional logic, it doesn’t give
a solution to the other problems that Frege was facing. The problem of
identity, for instance. Given his solution to this problem, the substitutivity
argument with which he eliminates thoughts as the denotation of sentences
is inevitable. So Frege concluded that thoughts (the old conceptual contents) are the senses of sentences. But it now became imperative that he
should find denotations for sentences. Why? For several reasons, mostly
connected to the development of his views on truth.
In Frege’s Posthumous Writings there is a rather remarkable manuscript fragment titled “Logic” (the first of two with this title) which the
editors date between 1879 and 1891 – on the basis that there is an implicit
reference to Begriffsschrift and that Frege still uses the notion of ‘content of
possible judgment’ which he abandoned in 1891. What makes this manuscript remarkable is that it contains the basic views on truth that Frege will
express in very similar words in the preface to The Basic Laws of Arithmetic
270
and in “Thoughts”. We find nothing like this in any of Frege’s publications
up to and including The Foundations of Arithmetic. The discussion of
psychologism, and its relation to logic, is also much sharper than that in
The Foundations of Arithmetic, and resembles a lot more the discussion in
Frege’s later work. Yet there is no explicit distinction between sense and
reference. This suggests that this manuscript may have been written after
The Foundations of Arithmetic and that it reflects an important aspect of
the development that led to the ideas in “On Sense and Reference”.
The distinction between truth, judgment (as the inward recognition of truth), and assertion (as the expression of judgment) is made clear
in Frege’s very first sentence, as is the independence of truth from judgment
in the next few paragraphs. The laws of logic are the laws of truth, but the
laws of logic are only concerned with those truths that can be justified by
inference from other truths. The laws of logic are laws of valid inference
which unfold the content of the word ‘true’. Logic studies the property
‘true’20 .
The distinction between sense and reference adds a new perspective
to this. If logic is tied up to truth but truth is independent of judgment,
then logic cannot be formulated simply in terms of acts of judgment of
thoughts. Acts of judgment do not add truth; so truth must be already in
the connection of thought and reality. Acts of judgment must be literally
a recognition of this objective connection. Moreover, logic is not merely a
hypothetical calculus of truth, so there must be starting points for which
the recognition of truth is justified independently of logic. The problem
is how to make these two aspects, judgment and truth, fit together. The
introduction of the objects the True and the False seems to me to be a
solution to this problem.
In “Function and Concept” the content stroke becomes a function,
the horizontal, which is purely referential:
—x
is the True if x is the True, and is the False for any other object x. Conditionality is also a function
——–x
| −y
−
which is the False if y is the True and x is any object that is not the True,
and is the True otherwise. Negation is the function
———
x
|
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which is the True if — x is the False, and is the False if — x is the True.
Finally,
−−a∪−−f (a),
denotes the True if the function f(x) has the value the True for all arguments
(which may be objects or functions), and denotes the False otherwise. If
one takes ‘f’ as a variable for functions, the universal quantifier is also a
logical function; a second-level function whose value is the True or the False
for each (appropriate) first-level function as argument.
Thus the “content” part of the logic is now wholly referential; signs
denote objects or functions. Any “judgeable content” denotes a truth value,
and to judge a judgeable content is to recognize its truth value as the True.
But the judgment sign is not a function nor does it denote anything21 .
Frege’s characterization of conditionality in Begriffsschrift, if interpreted in terms of truth and falsity, was a characterization of material
implication, whereas his new characterization defines a very general function
which includes the usual truth-function with which the material conditional
is interpreted. And something similar applies to negation and to universal
quantification. Could Frege have effected this passage from sense to reference with any other choice as the denotation of sentences? Propositions,
thoughts, etc., are out, so the only candidates are entities like states of
affairs. But one can’t get a referential logic of states of affairs that will
be general enough for Frege’s purposes. What will correspond to negation
and conditionality? If a sentence denotes a state of affairs, its negation
does not denote a state of affairs. If a sentence does not denote a state of
affairs, what will a conditional with that sentence as consequent denote?
Even if one allows possible states of affairs, what is the possible state of
affairs denoted by such conditionals? Besides, Frege’s view that functions
must be everywhere defined, which is independent of these questions about
logical functions, requires that the conditional function with Julius Caesar
and England as arguments must have a value. Which state of affairs could
be the value of the conditional function for those arguments? If the logic
is to be formulated referentially in accordance with these constraints, then
the True and the False seem the best (and maybe only) choice.
But there is another very important reason for the choice of truth
values. Suppose one tries to do the propositional logic with thoughts as
the denotation of sentences. This not only creates difficulties in relation to
Frege’s solution to the problem of identity, but also creates a great deal of
tension between the propositional logic and the predicate logic if the latter
is formulated directly in terms of objects and functions. Frege’s objection
272
to the Boolean logicians’ distinction between primary and secondary propositions could now be turned against him. Moreover, if thoughts are the
denotation of sentences, then there is the crucial problem of what to do
about the laws of classical logic; because thoughts need not be either true
or false. In fact, this is also a consideration against the formulation in Begriffsschrift, which, though uniform, has variables for conceptual contents.
With the introduction of the True and the False as objects, Frege obtains
a uniform formulation of the logic where variables range only over functions and objects. As long as he can guarantee that the senses expressed
by the signs of the system (which are logical senses) always denote – and
therefore that the thoughts expressed by the signs of the system (which are
logical thoughts) always have a definite truth value – the existence of truthvalueless thoughts does not affect the logic as such. Thus, he could either
(re)interpret the logic of Begriffsschrift strictly as a logic of judgment or he
could reformulate it as a logic of truth.
Whether Frege arrived at the truth values through these considerations or through the arguments in “On Sense and Reference”, or through
a combination of both, is hard to tell, but there cannot be any doubt that
the room for maneuvering was getting very tight indeed. The choice of the
True and the False seems unnatural, but it was basically forced upon Frege
by the development of his ontological views. The arguments in “On Sense
and Reference” seem largely designed to motivate and to justify this choice.
Nevertheless, he acknowledges that the unnaturalness is not eliminated by
his arguments22 . It is interesting, however, in this connection, that in the
“Logic” manuscript Frege argues explicitly against the requirement that
the formulation of logic should reflect the natural course of thinking. In the
middle of his argumentation he says (p. 7):
There is no reproach the logician need fear less than the reproach
that his way of formulating things is unnatural, that the actual
process of thinking follows a different course.
Frege still had problems, though, because by turning the logic
referential he seems to leave the old conceptual contents pretty much out of
it. The connection between judgment, sentence, and truth value must play
an important role. Truth values obliterate the specific articulation of each
sentence, but the recognition of truth must be based on this articulation.
The thought contains the articulation, but if truth is not a property of the
thought, at least not in any intrinsic sense, then the recognition of truth
cannot be based on the thought alone; it must be based on how the various
parts of the sentence relate to reality. A sentence communicates something
or says something independently of its being true or false. The parts of the
273
sentence may or may not refer to reality, and if some don’t, then it is neither
true nor false. What happens then? If some parts of the sentence do not
refer to reality, then the judging or asserting won’t connect up to reality
either. There is no way for reality to fit or not fit what we are asserting,
so to speak; we are not really talking about it. If all the parts refer on the
other hand, then reality is accountable for fit or lack of fit. It is this fit
or lack of fit that the act of judging analyses in its advance from thought
to reality. In order for this advance to be possible, however, it must be
quite clear what the sentence is about. There can be various readings of
a sentence, but for each reading we have a specific ontological analysis in
terms of functions and objects23 .
Thus, with his account of judgment Frege recovers the diversity
of connections between sentences and reality from the unities that are the
truth values. It is very close to an account in terms of states of affairs,
except that these states of affairs are not the denotations of sentences but
are rather a division of the denotation into those parts which the parts of the
sentence denote in each specific analysis of its structure. Moreover, Frege’s
account of the logical notions as functions of various levels is an account of
an ontological category of logical functions which give raise to divisions of
the True and the False into purely logical parts, or into logical parts and
non-logical parts.
It seems to me, therefore, that the view that Frege’s logic eliminated aboutness and subject-predicate structures disregards important aspects of the analysis of truth and denotation that Frege presented in “On
Sense and Reference”. But Frege’s attempt in The Basic Laws to combine
his new views on sense, reference and truth values with his earlier contextual doctrine, especially the doctrine concerning the “introduction” of
logical objects embodied in the identity version of the context principle, has
disastrous consequences even aside from the paradoxes. I will now try to
substantiate this point.
Just as Frege does not have a general criterion of identity for extensions, he does not have a general criterion of identity for senses. Is Julius
Caesar identical to the sense expressed by the name ‘Aristotle’, for instance?
How does one settle such questions24 ? And, what’s worse, he doesn’t have
a general criterion of identity for the two objects he introduced as the True
and the False either. Since in order to develop the logic and his definition of
number Frege needs something like extensions and the two truth values, his
solution is to identify the True and the False with two specific courses-ofvalues and to try to make sure that he can always settle identity questions
within the system. To achieve this he views the system of The Basic Laws
as a sort of “constructive” system in which objects and functions are in-
274
troduced step by step in a dependent way. This makes the system both
open-ended and closed into itself.
Frege begins by introducing the truth values as objects and using
them to define the horizontal function, the negation function, the identity
function, and the generality function for functions of objects. He then introduces the courses-of-values as objects subject to the condition that
(6) ǫ’ Φ(ǫ) = α’ Ψ(α)
’
has the same denotation as
(7) −−a−− Φ(a) = Ψ(a),
∪
But this cannot be done without further ado, for (6)-(7) does not fix the
denotation of ‘ǫ’ Φ(ǫ)’. As Frege puts it (p. 16):
If we assume that
X(ξ)
is a function that never takes on the same value for different arguments, then for objects whose names are of the form
“X(ǫ’ Φ(ǫ))”
just the same distinguishing mark for recognition holds, as for objects signs for which are of the form “ǫ’ Φ(ǫ)”. To wit,
“X(ǫ’ Φ(ǫ)) = X(α’ Φ(α))”
then also has the same denotation as [(7)]. From this it follows that
by identifying the denotation of [(6)] with that of [(7)], we have by
no means fully determined the denotation of a name like “ǫ’ Φ(ǫ)” –
at least if there does exist such a function X(ξ) whose value for a
course-of-values is not always the same as the course-of-values itself.
How may this indefiniteness be overcome? By its being determined
for every function when it is introduced what values it takes on for
courses-of-values as arguments, just as for all other arguments. Let
us do this for the functions considered up to this point. There are
the following:
ξ = ζ , —– ξ, —–—–
ξ.
|
Frege goes on to argue that the negation function can be left “out
of account, since it can be considered always to take a truth-value as argument,” that the horizontal function can be reduced to the identity function
as the function ξ = (ξ = ξ)25 , and that since the only objects other than
courses-of-values introduced so far are the truth values, the question of the
criterion of identity for courses-of-values reduces to the question of whether
the truth values are courses-of-values or not. He argues further that this
275
also cannot be settled by the stipulation that (6) is to have the same denotation as (7), but that it is consistent with this stipulation that “it is always
possible to stipulate that an arbitrary course-of-values is to be the True and
another the False.” He therefore stipulates that the True is
(8) ǫ’(— ǫ),
and the False is
(9) ǫ’(ǫ = −−−a−− a = a))26 .
|
∪
This does not settle the issue, however, because Frege’s system cannot guarantee that the True is different from the False.
If the True is the same as the False, then every function of objects
that Frege has introduced so far has the True as value for any object as
argument, and the generality function has the True as value for any such
function as argument. Thus unless Frege can distinguish the True from the
False, the system does not even get off the ground. But this cannot be done
within the system, because within the system the identity function denotes
the True if the arguments are the same and the False if they are not the
same27 . To postulate in the system that
(10) |—|— the True = the False
doesn’t help, because if the True is the False, then both
(11) — the True = the False
and
(12) —|— the True = the False
denote the True. And to identify the True with (8) and the False with (9)
doesn’t help either, because (7) cannot determine whether any courses-ofvalues are the same or different unless something determines whether the
True and the False, upon which are based the logical functions involved in
(7), are the same or different.
The problem is that Frege’s system is trivially consistent (or inconsistent) in the sense that there is an “interpretation” of the system that
consists of only one object and such that every function is essentially an
identity function. This consistency (or inconsistency) of Frege’s system is
completely independent of the axioms and principles of inference that he
postulates. What Russell’s paradox shows is that the trivial interpretation is the only interpretation of Frege’s axioms and principles of inference.
With other axioms there may be non-trivial interpretations, but what my
argument shows is that no choice of axioms can eliminate the trivial interpretation. It is in this sense that we may say that the trivial interpretation
reveals an “inconsistency” in the system. Another way of putting the point
is to say that Frege’s system cannot guarantee the distinction between negation and assertion (because it cannot guarantee that the True is different
276
from the False). This is clear from the definition of the negation function
in terms of truth values28 .
It is quite obvious that Frege assumes that the two truth values
are different objects, and one could argue that this is a presupposition of the
system that is guaranteed outside the system. I think, however, that as long
as one adheres consistently to the identity version of the context principle
it cannot be guaranteed even outside the system. For in order to legitimize
the introduction of the True and the False as logical objects Frege needs
a criterion of identity that determines all identities involving these objects
and any other objects. In particular, the criterion must determine whether
the identity
(13) the True = Julius Caesar
is true or not. Now where is the criterion that determines this? Since truth
values are introduced in “On Sense and Reference” as the denotation of
sentences, one could try arguing that Julius Caesar is not the denotation of
a sentence and falsify the identity. But what guarantees that Julius Caesar
is not the denotation of a sentence? Is there a criterion for that?
But can one at least settle the question whether the identity
(14) the True = the False
is true or not? Here the problem is essentially the problem that Frege points
out in his regress argument against the definability of truth. For how are the
True and the False introduced? They are introduced as the denotation of
sentences. And when do two sentences S and S ′ have the same denotation?
When the biconditional
(15) S if and only if S ′
is true. But either (15) is true because S and S ′ have the same denotation,
which is circular, or because it is a sentence that denotes the True. In this
case we have the problem of figuring out the denotation of (15), and to
appeal to another sentence just pushes the question one step further. This
shows that we cannot introduce the True and the False as the denotations
of any particular sentences S and S ′ and guarantee without circularity that
they are different.
Could we introduce the True and the False as some arbitrary objects? We can’t use such logical objects as the numbers 1 and 0 because
the whole point of Frege’s work is to introduce the numbers as logical objects – and, besides, we would need a criterion of identity to distinguish 1
from 0. Could we use Julius Caesar and England, say? This would be very
odd, because the True and the False are supposed to be logical objects, not
persons or countries. But, in any case, this doesn’t help us with a criterion
of identity, because we still need a criterion that determines whether Julius
Caesar is the denotation of some sentence or other.
277
The natural solution is to reduce the problem of the denotation
of a sentence to the denotation of its parts, as in the Platonic and Fregean
analysis of
(16) Theaetetus is sitting
in terms of the object denoted by ‘Theaetetus’, the property denoted by
‘is sitting’ and the relation of instantiation. This seems to require a noncontextual approach, however. There may be nothing fundamentally wrong
with the postulation of the True and the False as the denotation of sentences,
but one must take the ontology seriously and give some account of these
objects, as well as of senses, extensions, courses-of-values, etc. The idea
that the whole logic can be done denotationally and contextually seems
to me essentially incoherent, as evidenced by Frege’s attempt to account
denotationally and contextually for truth values and courses-of-values.
Nobody has ever been able to figure out what Frege meant by a
course-of-values, in fact. Well, by his own arguments as to the indefiniteness
of (6)-(7) as a criterion of identity for courses-of-values, it is impossible
to figure it out. By closing the system into itself Frege thought that this
indefiniteness could be left aside. But since the very formulation of the logic
and of (6)-(7) as a criterion of identity requires the True and the False as
objects, it must at least be clear whether these objects are courses-of-values
or not. And by Frege’s own arguments again, this cannot be determined
from (6)-(7). Thus, to overcome the indefiniteness of (6)-(7) also requires
identifying the True and the False with some specific courses-of-values. But,
as I argued before, (7) cannot determine whether any courses-of-values are
the same or different unless something determines whether the True and
the False, upon which are based the logical functions involved in (7), are
the same or different. That’s why the constructive approach that involves
closing the system into itself does not overcome the indefiniteness of (6)-(7).
Moreover, the closing of the system into itself takes all its generality away. The very next function that Frege introduces is the function
\ξ that substitutes for the definite article. He characterizes it as follows (p.
19):
1. If to the argument there corresponds an object ∆ such that the
argument is ǫ’(∆ = ǫ), then let the value of the function \ξ be ∆
itself;
2. if to the argument there does not correspond an object ∆ such
that the argument is ǫ’(∆ = ǫ), then let the value of the function be
the argument itself.
This function should be defined for every object as argument, so it should be
clear what \England is. The obvious answer would seem to be that \England
278
is England itself, either on the grounds that England is not a course-ofvalues, and hence that to England there doesn’t correspond an object ∆
such that England is the same object as the course-of-values ǫ’(∆ = ǫ),
or on the grounds that England is the same object as the course-of-values
ǫ’(England= ǫ). But since there doesn’t seem to be anything that determines
whether England is a course-of-values or not, \ξ cannot be considered to be
defined for England as argument. One could try to stipulate that ∆ is the
same object as the course-of-values ǫ’(∆ = ǫ) for any object ∆, but Frege
himself argues that this cannot be done29 . Thus the function \ξ can be
considered to be well-defined for arguments that are given as a course-ofvalues, but cannot be considered to be well-defined in general because the
condition “to the argument there corresponds an object ∆ such that the
argument is ǫ’(∆ = ǫ)” is not definite enough30 .
Another oddity of the system is that whereas first order quantification is characterized objectually, second order quantification is characterized
substitutionally31 . This is forced upon Frege by the view that the functions
are introduced gradually into the system. First order quantification is not
a second-level function defined for any function of objects as argument, but
only for such functions as have already been introduced into the system. So
also in this respect the system has no generality.
It appears, therefore, that Frege’s attempt to solve the conflict
between subjectivity and objectivity was not successful. To interpret the
logic of Begriffsschrift as a logic of the judging acts of a subject would
have created an unbridgeable gap between logic and truth, if truth is conceived objectively as independent of the judging acts of the subject. To
transform the logic of Begriffsschrift into an objective logic of truth was
therefore necessary for Frege’s project of an objective foundation for mathematics. The judging subject is not eliminated by this transformation, for
it is him that sets up the system, recognizes thoughts as true, formulates
principles of proof and definition, chooses definitions, carries out proofs, etc.
He can make mistakes, though, and he did, which raises difficult problems
for Frege’s view that logic can be separated from more general epistemological considerations. If one could localize these errors as errors of judgment
concerning one or another fundamental law, this may not be so serious,
especially since even before the paradoxes the subject (Frege) did not consider basic law V to have the same degree of self-evidence as the other basic
laws32 .
The main problem, however, is that the very conception of objectivity seems to be inadequate as a resolution of the conflict between
subjectivity and objectivity. The logic of The Basic Laws becomes entirely
dependent on the constructional activity of the subject because both ob-
279
jects and functions must be “introduced” in such a way that guarantees
the objectivity of the system in accordance with the identity version of the
context principle. The subject must continuously go back to the beginning
and “verify” that the introduction of new objects and functions does not
disrupt the objectivity of the system as developed so far. In particular, no
law can be judged true simpliciter, but only true so far. For how can a law
be judged to be true for all objects, say, if new objects can be introduced
into the system? To restrict objects to courses-of-values doesn’t avoid the
problem, because the courses-of-values are not completely determined by
the identity condition (7). As Frege himself says, the introduction of a new
function can “be regarded as much as a further determination of the coursesof-values as of that function” (see note 26). It would seem to follow from
this that at any one time the subject can only judge a law for courses-ofvalues as determined so far.
In my view the identity version of the context principle undermines
every aspect of the ideas that Frege presented in “On Sense and Reference”.
By keeping this principle Frege undercuts the objectivity he sought for logic
and truth in the notions of object, function, concept, sense, etc. Many
of the ontological intuitions that begin to emerge in The Foundations of
Arithmetic can be fairly characterized as basically Platonic intuitions, but
these intuitions do not mesh with the contextual intuitions that played such
a large role in Begriffsschrift and that led Frege to the formulation of the
context principle. The problems of the system of The Basic Laws seem to
me to derive much more from the “idealistic” contextual intuitions than
from the Platonic elements as such. From a straightforwardly Platonistic
point of view the ontology should be approached directly, not through the
contextual device of a criterion of identity.
This is one of the main conclusions I want to draw from my brief
examination of Frege’s views in this chapter. It is also related to my conclusions in the last chapter regarding Tarski’s approach to the problem of truth
via schema (T). In the next few chapters I explore this direct approach to
the ontology while still maintaining a generally Fregean outlook.
280
Notes
1. Even when we read Frege today we tend to “translate” (and interpret) his
symbolism in terms of the modern linear symbolism.
2. There have been some attempts to define aboutness, but since the primary
instrument has been the formalism of logic itself, it couldn’t really work. If one
starts from an ontological analysis, ‘about’ must be a primitive notion, at least
in a restricted form. One can represent aboutness, but cannot define it in any
reasonable sense.
For an early attempt to define aboutness see Goodman’s “About”.
There are a number of interesting ideas and distinctions in this paper – e.g. the
distinction between absolutely about, relatively about and rhetorically about –
which by and large are quite compatible with my approach.
3. Op. Cit., p. 12:
A distinction between subject and predicate does not occur in my
way of representing a judgment. In order to justify this I remark
that the contents of two judgments may differ in two ways: either
the consequences derivable from the first, when it is combined with
certain other judgments, always follow also from the second, when
it is combined with these same judgments, [and conversely,] or this
is not the case. The two propositions “The Greeks defeated the Persians at Plataea” and “The Persians were defeated by the Greeks
at Plataea” differ in the first way. Even if one can detect a slight
difference in meaning, the agreement outweighs it. Now I call that
part of the content that is the same in both the conceptual content. Since it alone is of significance for our ideography, we need
not introduce any distinction between propositions having the same
conceptual content. If one says of the subject that it “is the concept with which the judgment is concerned”, this is equally true
of the object. We can therefore only say that the subject “is the
concept with which the judgment is chiefly concerned”. In ordinary
language, the place of the subject in the sequence of words has the
significance of a distinguished place, where we put that to which we
wish especially to direct the attention of the listener. This may,
for example, have the purpose of pointing out a certain relation of
the given judgment to others and thereby making it easier for the
listener to grasp the entire context. Now, all those peculiarities of
ordinary language that result only from the interaction of speaker
and listener – as when, for example, the speaker takes the expectations of the listener into account and seeks to put them on the
right track even before the complete sentence is enunciated – have
281
nothing that answers to them in my formula language, since in a
judgment I consider only that which influences its possible consequences. Everything necessary for a correct inference is expressed
in full, but what is not necessary is generally not indicated; nothing
is left to guesswork. In this I faithfully follow the example of the
formula language of mathematics, a language to which one would
do violence if he were to distinguish between subject and predicate
in it.
(The bracketed addition is the translator’s.)
4. Frege continues (pp. 12-13):
We can imagine a language in which the proposition “Archimedes
perished at the capture of Syracuse” would be expressed thus: “The
violent death of Archimedes at the capture of Syracuse is a fact”. To
be sure, one can distinguish between subject and predicate here, too,
if one wishes to do so, but the subject contains the whole content,
and the predicate serves only to turn the content into a judgment.
Such a language would have only a single predicate for all judgments,
namely, “is a fact”. We see that there cannot be any question here
of subject and predicate in the ordinary sense. Our ideography is a
language of this sort, and in it the sign |— is the common predicate
for all judgments.
It is important to notice that the point that the predicate ‘is a fact’ turns the
content into a judgment is precisely what Frege rejects later in “On Sense and
Reference” with ‘is true’ in place of ‘is a fact’. From which it follows also that
the judgment sign has an altogether different nature than a predicate. The view
of judgment as an act is clearly present in Begriffsschrift (p. 11), but Frege’s
treatment is somewhat ambiguous. I will discuss this ambiguity later.
It is also interesting to compare this conception of Frege’s with Tarski’s
criticism of Aristotle that I briefly discussed at the end of note 1 in Chapter 7.
5. When he introduces the judgment and the content strokes Frege remarks (p.
12):
The horizontal stroke that is part of the sign |— combines the signs
that follow it into a totality, and the affirmation expressed by the
vertical stroke at the left end of the horizontal one refers to this
totality. Let us call the horizontal stroke the content stroke and
the vertical stroke the judgment stroke. The content stroke will in
general serve to relate any sign to the totality of the signs that follow
the stroke. Whatever follows the content stroke must have a content
that can become a judgment.
282
6. In the preface he says (pp. 5-6):
The most reliable way of carrying out a proof, obviously, is to follow
pure logic, a way that, disregarding the particular characteristics
of objects, depends solely on those laws upon which all knowledge
rests. Accordingly we divide all truths that require justification into
two kinds, those for which the proof can be carried out purely by
means of logic and those for which it must be supported by facts of
experience. . . . Now, when I came to consider the question to which
of these two kinds the judgments of arithmetic belong, I first had
to ascertain how far one could proceed in arithmetic by means of
inferences alone, with the sole support of those laws of thought that
transcend all particulars. . . . To prevent anything intuitive from
penetrating here unnoticed, I had to bend every effort to keep the
chain of inferences free of gaps. In attempting to comply with this
requirement in the strictest possible way I found the inadequacy of
language to be an obstacle; no matter how unwieldy the expressions
I was ready to accept, I was less and less able, as the relations
became more and more complex, to attain the precision that my
purpose required. This deficiency led me to the idea of the present
ideography. Its first purpose, therefore, is to provide us with the
most reliable test of the validity of a chain of inferences and to point
out every presupposition that tries to sneak in unnoticed, so that its
origin can be investigated. That is why I decided to forgo expressing
anything that is without significance for the inferential sequence. In
§3 I called what alone mattered to me the conceptual content. Hence
this definition must always be kept in mind if one wishes to gain a
proper understanding of what my formula language is.
I quoted the whole section 3 in notes 3, 4, and the text.
7. He says (pp. 22-23):
Let us assume that the circumstance that hydrogen is lighter than
carbon dioxide is expressed in our formula language; we can then
replace the sign for hydrogen by the sign for oxygen or that for
nitrogen. This changes the meaning in such a way that “oxygen”
or “nitrogen” enters into the relations in which “hydrogen” stood
before. If we imagine that an expression can thus be altered, it
decomposes into a stable component, representing the totality of relations, and the sign, regarded as replaceable by others, that denotes
the object standing in these relations. The former component I call
a function, the latter its argument. The distinction has nothing
to do with the conceptual content; it comes about only because we
view the expression in a particular way. According to the conception
sketched above, “hydrogen” is the argument and “being lighter than
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carbon dioxide” the function; but we can also conceive of the same
conceptual content in such a way that “carbon dioxide” becomes the
argument and “being heavier than hydrogen” the function. We then
need only regard “carbon dioxide” as replaceable by other ideas, such
as “hydrochloric acid” or “ammonia”.
8. See the papers “On the Scientific Justification of a Conceptual Notation”, “On
the Aim of the “Conceptual Notation””, “Applications of the “Conceptual Notation””, “Boole’s Logical Calculus and the Concept-script”, and “Boole’s Logical
Formula-language and my Concept-script”. In the second of these he says (p. 91):
Boole distinguishes primary propositions from secondary propositions. The former compare the extensions of concepts, the latter
express relations among assertible contents. This division is insufficient since existential judgments fail to find a place.
Let us consider first primary propositions. Here the letters denote
the extensions of concepts. Particular things as such are not signified; and this is an important deficiency in the Boolean formula
language, for even if a concept covers only a single thing, a great
difference still remains between it and this thing.
(See also the quotation from Boole in note 13 of Chapter 6 for this distinction.)
After a brief examination of Boole’s treatment of primary propositions – about
which he comments that “[e]verything thus far is already found with only superficial divergencies in Leibniz, whose works in this area I dare say were unknown
to Boole” (p. 92) – he continues (p. 93):
Boole reduces secondary propositions – for example, hypothetical
and disjunctive judgments – to primary propositions in a very artificial way. He interprets the judgment “if x = 2, then x2 = 4”
this way: the class of moments of time in which x = 2 is subordinate to the class of moments of time in which x2 = 4. Thus,
here again the matter amounts to the comparison of the extensions
of concepts; only here these concepts are fixed more precisely as
classes of moments of time in which a sentence is true. This conception has the disadvantage that time becomes involved where it
should remain completely out of the matter. MacColl explains the
expressions for secondary propositions independently of the primary
ones. In this way the intermingling of time is certainly avoided; but
as a result, every interconnection is severed between the two parts
which, according to Boole, compose logic. . . .
When we view the Boolean formula language as a whole, we discover
that it is a clothing of abstract logic in the dress of algebraic symbols.
It is not suited for the rendering of a content, and that was also
not its purpose. But this is exactly my intention. I wish to blend
together the few symbols which I introduce and the symbols already
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available in mathematics to form a single formula language. In it, the
existing symbols correspond to the word-stems of language; while the
symbols I add to them are comparable to the suffixes and formwords
that logically interrelate the contents embedded in the stems.
And as to the charge that his notation was “a monstrous waste of space”, he adds
(p. 97):
The “conceptual notation” makes the most of the two-dimensionality
of the writing surface by allowing the assertible contents to follow
one below the other while each of these extend from left to right.
Thus, the separate contents are clearly separated from each other,
and yet their logical relations are easily visible at a glance. For
Boole, a single line, often excessively long, would result.
In the various papers I cited Frege gives many examples to show both
the power and the subtlety of his formula language, and concludes “Boole’s Logical
Calculus and the Concept-script” as follows (p. 46):
I believe in this essay I have shown:
(1) My concept-script has a more far-reaching aim than Boolean
logic, in that it strives to make it possible to present a content when
combined with arithmetical and geometrical signs.
(2) Disregarding content, within the domain of pure logic it also,
thanks to the notation for generality, commands a somewhat wider
domain than Boole’s formula-language.
(3) It avoids the division in Boolean logic into two parts (primary
and secondary propositions) by construing judgments as prior to
concept formation.
(4) It is in a position to represent the formations of the concepts
actually needed in science, in contrast to the relatively sterile multiplicative and additive combinations we find in Boole.
(5) It needs fewer primitive signs for logical relations and hence fewer
primitive laws.
(6) It can be used to solve the sort of problems Boole tackles, and
even to do so with fewer preliminary rules for computation. This is
the point to which I attach least importance, since such problems
will seldom, if ever, occur in science.
9. Frege says, for instance (p. 19):
[X] means . . . “The case in which both A and B are affirmed occurs”. . . .
Accordingly, we can translate [X] by “Both A and B are facts”.
A little later (p. 20):
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[X] means “Of the four possibilities the third, namely, that A is
denied and B is affirmed, occurs”. We can therefore translate it as
“B takes place and (but) A does not”.
And a little later in the same page:
[X] means “The case in which both A and B are denied occurs”.
Hence we can translate it as “Neither A nor B is a fact”.
Where I have used ‘X’ to indicate the formula that Frege is considering in each
case.
In “The Aim of “Conceptual Notation””, p. 95, Frege formulates the
possibilities (1)-(4) for conditionality simply in terms of ‘A’, ‘B’, ‘not A’, and
‘not B’. For example, (2) is stated as: A and not B. He talks about contents
as correct [richtig] but continues to use ‘affirmed’ and ‘denied’ to translate some
of the formulas. It is also interesting that he characterizes conditionality as “the
negation of the third case” and does not add the clause that one of the other three
cases takes place. This is significant because it suggests that in Begriffsschrift
affirmation and denial may have been meant as such – or at least that there may
have been some ambiguity in Frege’s conception. This, however, is denied by van
Heijenoort in his introduction to the translation of Begriffsschrift (p. 4):
A few words should be said about Frege’s use of the term “Verneinung”. In a first use “Verneinung” is opposed to “Bejahung”, “verneinen” to “bejahen”, and what these words express is, in fact, the
ascription of truth values to contents of judgments; they are translated, respectively, by “denial” and “affirmation”, “to deny” and “to
affirm”. The second use of “Verneinung” is for the connective, and
when so used it is translated by “negation”.
10. In “Boole’s Logical Calculus and the Concept-script”, Frege remarks (p. 17):
And so instead of putting a judgment together out of an individual as
subject∗ and an already previously formed concept as predicate, we
do the opposite and arrive at a concept by splitting up the content
of possible judgment.∗∗ Of course, if the expression of the content of
possible judgment is to be analyzable in this way, it must already be
itself articulated. We may infer from this that at least the properties
and relations that are not further analyzable must have their own
simple designations. But it doesn’t follow from this that the ideas
of these properties and relations are formed apart from objects: on
the contrary they arise simultaneously with the first judgment in
which they are ascribed to things. Hence in the concept-script their
designations never occur on their own, but always in combinations
which express contents of possible judgment. I could compare this
with the behavior of the atom: we suppose an atom never to be
found on its own, but only combined with others, moving out of one
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combination only to enter immediately into another.∗∗∗ A sign for
a property never appears without a thing to which it may belong
being at least indicated, a designation of a relation never without
indication of the things which may stand in it.
[Note*] The cases where the subject is not an individual are completely different from these and are here left out of consideration.
[Note**] A great deal of tedious discussion about negative concepts
such as ‘not-triangle’ will, as I hope, be rendered redundant by the
conception of the relation of judgment and concept outlined here. In
such a case one simply doesn’t have anything complete, but only the
predicate of a judgment which as yet lacks a subject. The difficulties
arise when people treat such a fragment as something whole.
In this connection, I find it [noteworthy] that some linguists have
recently viewed a ‘Satzwort’ (sentence-word), a word expressing a
whole judgment, as the primitive form of speech and ascribe no independent existence to the roots, as mere abstractions. . . .
[Note***] As I have since seen, Wundt makes a similar use of this
image in his Logik.
I have followed Sluga’s suggestion in “Frege Against the Booleans” (p. 96 n. 9)
that in the second paragraph of note ** one should translate ‘bemerkenswert’ as
‘noteworthy’ rather than the translators ‘extraordinary’. The issue is whether
Frege is expressing approval or disapproval for the view. The problem is that
there are two different views involved here. One is that whole judgments are
basic, with which Frege cannot but agree; the other is that the roots have no
independent existence, or are mere abstractions. It seems to me that Frege agrees
with this insofar as negative concepts are concerned, which is the main point
he is making in the note, but it doesn’t necessarily follow that he extends his
agreement to concepts and objects in general in any straightforward sense; and
such an extension does not seem to be supported by Frege’s main text following the
note. I think that Dummett – who uses the translation ‘remarkable’, and whose
view that “what Frege found remarkable was Sayce’s unwarranted exaggeration”
Sluga is opposing – could agree with this interpretation, and that the translation
‘noteworthy’ does not settle the issue in favor of Sluga’s interpretation (Op. Cit.,
pp. 86-87).
11. Wittgenstein states versions of it both in Tractatus Logico-Philosophicus 3.3
and in Philosophical Investigations §49, among other places, and they are quite
central to his philosophy. Also Quine makes it a centerpiece of his philosophy
by tracing to Frege “an important reorientation in semantics – the reorientation
whereby the primary vehicle of meaning came to be seen no longer in the term
but in the statement” (“Two Dogmas of Empiricism”, p. 39). He then broadens
this formulation to what is to become one of his main doctrines (p. 42):
The idea of defining a symbol in use was, as remarked, an advance
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over the impossible term-by-term empiricism of Locke and Hume.
The statement, rather than the term, came with Frege to be recognized as the unit accountable to an empiricist critique. But what I
am now urging is that even in taking the statement as unit we have
drawn our grid too finely. The unit of empirical significance is the
whole of science.
12. After introducing the three principles, Frege comments:
In compliance with the first principle, I have used the word “idea”
always in the psychological sense, and have distinguished ideas from
concepts and from objects. If the second principle is not observed,
one is almost forced to take as the meanings [Bedeutung] of words
mental pictures or acts of the individual mind, and so to offend
against the first principle as well. As to the third point, it is a mere
illusion to suppose that a concept can be made an object without
altering it. From this it follows that a widely held formalist theory
of fractional, negative, etc., numbers is untenable. How I propose
to improve upon it can be no more than indicated in the present
work. With numbers of all these types, as with the positive whole
numbers, it is a matter of fixing the sense of an identity.
13. Op. Cit., p. 77:
In the proposition
“the direction of a is identical with the direction of b”
the direction of a plays the part of an object,2 . . .
Note 2. This is shown by the definite article. A concept is for
me that which can be predicate of a singular judgment-content, an
object that which can be subject of the same. If in the proposition
“the direction of the axis of the telescope is identical with the direction of the Earth’s axis”
we take the direction of the axis of the telescope as subject, then
the predicate is “is identical with the direction of the Earth’s axis”.
This is a concept. But the direction of the Earth’s axis is only an
element in the predicate; it, since it can also be made the subject,
is an object.
14. After the remarks I quoted in the text Frege says:
But, it will perhaps be objected, even if the Earth is really not
imaginable, it is at any rate an external thing, occupying a definite
place; but where is the number 4? It is neither outside us nor within
us. And, taking those words in their spatial sense, that is quite
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correct. To give spatial co-ordinates for the number 4 makes no
sense; but the only conclusion to be drawn from that is, that 4 is
not a spatial object, not that it is not an object at all. Not every
object has a place. Even our ideas are in this sense not within
us (beneath our skin); beneath the skin are nerve-ganglia, blood
corpuscles and things of that sort, but not ideas. Spatial predicates
are not applicable to them: an idea is neither to the right nor to
the left of another idea; we cannot give distances between ideas in
millimitres. If we still say they are within us, then we intend by this
to signify that they are subjective.
Yet even granted that what is subjective has no position in space,
how is it possible for the number 4, which is objective, not to be
anywhere? Now I contend that there is no contradiction in this
whatever. It is a fact that the number 4 is exactly the same for
everyone who deals with it; but that has nothing to do with being
spatial. Not every objective object [objektives Gegenstand] has a
place.
There are several other places in The Foundations of Arithmetic where
we find these strong claims of objectivity which suggest a Platonic interpretation –
another main passage occurring in pp. 34-35. Yet, there are also several passages
which suggest that the contrast between objectivity and subjectivity is not to be
interpreted in a transcendental ontological way; for example (p. 36):
Often, therefore, a colour word does not signify our subjective sensation, which we cannot know to agree with anyone else’s (for obviously
our calling things by the same name does not guarantee as much),
but rather an objective quality. It is in this way that I understand
objective to mean what is independent of our sensation, intuition
and imagination, and of all construction of mental pictures out of
memories of earlier sensations, but not what is independent of the
reason, – for what are things independent of the reason? To answer
that would be as much as to judge without judging, or to wash the
fur without wetting it.
15. This is an important problem for Frege’s system in The Basic Laws of Arithmetic, for although Frege discarded the context principle in its general formulation,
he held on to the idea that there must be a general criterion of identity. He points
out the difficulty I indicated in the text at the beginning of §10 (p. 16):
Although we have laid it down that the combination of signs
“ǫ’ Φ(ǫ) = α’ Ψ(α)”
has the same denotation as
a−− Φ(a) = Ψ(a),
−−∪
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this by no means fixes completely the denotation of a name like
“ǫ’Φ(ǫ)”. We have only a means of always recognizing a course-ofvalues if it is designated by a name like “ǫ’Φ(ǫ)”, by which it is already
recognizable as a course-of-values. But we can neither decide, so
far, whether an object is a course-of-values that is not given us as
such, and to what function it may correspond, nor decide in general
whether a given course-of-values has a given property unless we know
that this property is connected with a property of the corresponding
function.
I see no clear evidence that Frege was aware of this problem when he wrote The
Foundations of Arithmetic. I will examine his solution in The Basic Laws later.
16. See the passage in Frege’s letter to Husserl that I quoted in note 20 of Chapter
2, however. Frege does not explicitly introduce intersubstitutability preserving
truth value as a criterion of sameness of sense, but it follows from his views on
sense and denotation that it is a criterion. For, since in an intensional context
a sign denotes its ordinary sense, two signs are intersubstitutable in all contexts
preserving truth value if, and only if, they express the same sense (and have the
same denotation). I have used this criterion several times already (e.g. in Chapter
4) to argue that two sentences do not express the same sense.
17. Both “Function and Concept” and “On Concept and Object” contain references to “On Sense and Reference” and use the distinctions drawn there. The
distinction “between form and content, sign and thing signified” is clearly made
at the beginning of “Function and Concept” (pp. 1-3). In “On Concept and
Object”, p. 198, Frege comments on his characterization of concept and object in
The Foundations of Arithmetic:
When I wrote my Grundlagen der Arithmetik, I had not yet made
the distinction between sense and reference; and so, under the expression ‘a possible content of judgment’, I was combining what I
now designate by the distinctive words ‘thought’ and ‘truth-value’.
Consequently, I no longer entirely approve of the explanation I then
gave (op. cit., p. 77), as regards its wording; my view is, however, still essentially the same. We may say in brief, taking ‘subject’
and ‘predicate’ in the linguistic sense: A concept is the reference
of a predicate; an object is something that can never be the whole
reference of a predicate, but can be the reference of a subject.
This suggests that the ontological interpretation was at least part of the view in
The Foundations of Arithmetic.
18. This primitiveness is already emphasized in “Function and Concept” (p. 18):
When we have thus admitted objects without restriction as argu-
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ments and values of functions, the question arises what is it that we
are here calling an object. I regard a regular definition as impossible, since we have here something too simple to admit of logical
analysis. It is only possible to indicate what is meant. Here I can
only say briefly: An object is anything that is not a function, so that
an expression for it does not contain any empty places.
19. The whole passage is the following (p. 94):
In front of an expression for an assertible content, such as 2 + 3 = 5,
I put a horizontal stroke, the content stroke, distinguishable from
the minus sign by its greater length:
— 2+3=5
I take this stroke to mean that the content which follows it is unified,
so that other symbols can be related to it [as a whole]. In
— 2+3=5
absolutely no judgment is made. Thus, we can also write:
— 4+2=7
without being guilty of writing a falsehood.
If I wish to assert a content as correct, I put the judgment stroke on
the left end of the content stroke:
|— 2 + 3 = 5
How thoroughly one is misunderstood sometimes! Through this
mode of notation I meant to have a very clear distinction between
the act of judging and the formation of a mere assertible content;
and Rabus accuses me of mixing up the two!
20. The fragment begins with a table of contents and what we have is the introduction and the beginning (half a page) of the next section titled “Content of
possible judgment”. The points I mentioned are made very succinctly and very
clearly by Frege (pp. 2-3):
The goal of scientific endeavor is truth. Inwardly to recognize something as true is to make a judgment, and to give expression to this
judgment is to make an assertion.
What is true is true independently of our recognizing it as such. We
can make mistakes. The grounds on which we make a judgment
may justify our recognizing it as true; they may, however, merely
give raise to our making a judgment, or make up our minds for us,
without containing a justification for our judgment. . . .
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Now the grounds which justify the recognition of a truth often reside
in other truths which have already been recognized. But if there are
any truths recognized by us at all, this cannot be the only form that
justification takes. There must be judgments whose justification
rests on something else, if they stand in need of justification at all.
And this is where epistemology comes in. Logic is concerned only
with those grounds of judgment which are truths. To make a judgment because we are cognizant of other truths as providing a justification for it is known as inferring. There are laws governing this
kind of justification, and to set up these laws of valid inference is
the goal of logic.
The subject-matter of logic is therefore such as cannot be perceived
by the senses and in this respect it compares with that of psychology
and contrasts with that of the natural sciences. Instincts, ideas
etc. are also neither visible nor tangible. All the same there is a
sharp divide between these disciplines, and it is marked by the word
‘true’. Psychology is only concerned with truth in the way every
other science is, in that its goal is to extend the domain of truths;
but in the field it investigates it does not study the property ‘true’
as, in its field, physics focuses on the properties ‘heavy’, ‘warm’, etc.
This is what logic does. It would not perhaps be beside the mark
to say that the laws of logic are nothing other than the unfolding
of the content of the word ‘true’. Anyone who has failed to grasp
the meaning of this word – what marks it off from others – cannot
attain to any clear idea of what the task of logic is.
21. This account of the logical notions begins in p. 21 of “Function and Concept”.
When Frege introduces the judgment sign (p. 22) he emphasizes in a note the
difference in character with the other signs: “The assertion sign cannot be used
to construct a functional expression; for it does not serve, in conjunction with
other signs, to designate an object. ‘|— 2 + 3 = 5’ does not designate anything;
it asserts something.”
22. See Chapter 2, note 19. One of the people who was struck by this oddity of
Frege’s account was Russell, and the question of the denotation and sense of sentences is one of the substantial questions discussed in their correspondence. Frege
was quite adamant about it, as can be seen from his letter of December 28, 1902,
which begins: “You could not bring yourself to believe that the truth value is the
[denotation] of a proposition. I do not know whether you read my essay on sense
and [denotation] in. . .” Frege, Philosophical and Mathematical Correspondence, p.
152.
Frege argues very strongly for his views, and in a later letter of November
13, 1904, he actually offers a proof. He says (Ibid., p. 165):
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Indirect speech must here be disregarded; for we have seen that, in
it, the thought is designated, not expressed. Disregarding it, we can
therefore say that any true proposition can be replaced by any true
proposition without detriment from its truth, and likewise any false
proposition by any false proposition. And this is to say that all true
propositions [denote] or designate the same thing, and likewise all
false propositions; and this agrees with your definition:
x | ‘y = ϕx ⊃ϕ ϕy Df.
The formula is the definition of identity accepted by both Frege and Russell –
though Frege goes on to object to the equals sign ‘=’ used by Russell in the
formulation of definitions. Frege concludes that if all true sentences denote the
same thing, and all false sentences denote the same thing, then they must denote
truth values.
It is a neat argument, and I’m grateful to Luiz Henrique Lopes dos
Santos for calling my attention to it several years ago, but it is not convincing.
In the second half of “On Sense and Reference”, Frege tests his conclusion that
sentences denote truth values in several different contexts and does some very fine
analyses. One cannot say, however, that he establishes the main premise of the
present argument. The problem is that he doesn’t have an independent definition
of what it is for a context to be direct or indirect; so whenever substitutivity fails
one can always blame it on the context.
Over the years substitutivity has become the criterion for extensionality (directness) and intensionality (indirectness), which makes it even harder to
argue along those lines today. Quine offers an argument of this sort in “Identity,
Ostension, and Hypostasis” p. 71, but it is relative to a given context of discourse
– in this case the propositional calculus.
It is also somewhat ironic that after showing so much resistance to the
True and the False, Russell ended up postulating something very similar in “The
Philosophy of Logical Atomism”. For as I argued in Chapter 5, his pointings to
and away are a counterpart of Frege’s truth values conceptualized as connections
rather than as objects.
23. One can still hold that it is not necessary to distinguish notationally the
various readings, however, on the grounds that the sentence is true on one reading
if and only if it is true on any alternative reading. As I pointed out in earlier
chapters this is not convincing, but it is not unreasonable if one holds that all
terms in the sentence refer to something.
24. Given the importance of the notion of sense for Frege one would have expected
him to develop it more systematically. Although he had a very interesting analysis
of intensional contexts, he didn’t develop it, and in fact never developed a theory
of senses at all. He didn’t even succeed in making clear what sort of entity a
sense is supposed to be. I think that if Frege had tried to work out a theory of
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senses in the early 1890’s, he may have seen the general difficulty in adopting
the identity version of the context principle. And he may also have seen that his
view of senses was straining at the seams. For he didn’t draw a clear distinction
between the thought expressed by a sentence and what the sentence conveys in
communication – i.e., between thoughts and sentential meanings. This ambiguity
in the treatment of thoughts creates a good deal of tension in relation to his initial
account of senses of objects as manners of presentation, and makes his recovery
of conceptual contents as the senses of truth values seem rather artificial.
With time Frege’s notion of sense became a mixture of meaning with
meaning; ‘meaning’ as determinant of reference, and ‘meaning’ as what is grasped
in the communication of thought. The mixture is unstable, however, and a lot of
people got burnt.
25. But to reduce the horizontal function to the identity function in this way we
must make sense of the identity of ǫ to ǫ = ǫ, and to make sense of this we need
truth values as denotation of sentences. Actually, ǫ = ǫ could denote a state of
affairs, say, but then ǫ = (ǫ = ǫ) wouldn’t work as the horizontal function.
26. The passage is the following (p. 17):
Now the question whether one of the truth-values is a course-ofvalues cannot be decided from the fact that [(6)] is to have the same
denotation as [(7)]. It is possible to stipulate generally that
“η❡Φ(η) = η❡Ψ(α)”
shall denote the same thing as [(7)] without the identity of ǫ’ Φ(ǫ)
and η❡Φ(η) being derivable from this. We should then have a class
of objects with names of the form “η❡Φ(η)”, and for whose differentiation and recognition the same distinguishing mark held good as
for courses-of-values. We could now determine the function X(ξ) by
saying that its value shall be the True for η❡Λ(η) as argument, and
shall be η❡Λ(η) for the True as argument; further the value of the
function X(ξ) shall be the False for the argument η❡M (η), and shall
be η❡M (η) for the False as argument; for every other argument the
value of the function X(ξ) is to coincide with the argument itself.
If now the functions Λ(ξ) and M (ξ) do not always have the same
value for the same argument, then our function X(ξ) never has the
same value for different arguments, hence
“X(η❡Φ(η)) = X(α
❡Ψ(α)”
also always has the same denotation as [(7)]. The objects whose
names were of the form “X(η❡Φ(η))” would then be recognized by
the same means as the courses-of-values, and X(η❡Λ(η)) would be
the True and X(η❡M (η)) the False. Thus without contradicting our
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setting [(6)] equal to [(7)] it is always possible to stipulate that an
arbitrary course-of-values is to be the True and another the False.
And after stipulating that the True is (8) and the False is (9), Frege remarks (p.
18):
With this we have determined the courses-of-values so far as is here
possible. As soon as there is a further question of introducing a
function that is not completely reducible to functions known already,
we can stipulate what values it is to have for courses-of-values as
arguments; and this can then be regarded as much as a further
determination of the courses-of-values as of that function.
It is precisely for this reason that I referred to Frege’s system as a “constructive”
open-ended system.
27. P. 11:
We have been using the identity-sign as we went along, to form
examples; but it is necessary to stipulate something more precise
regarding it.
“Γ = ∆”
shall denote the True if Γ is the same as ∆; in all other cases it shall
denote the False.
28. Frege introduces the negation function as follows (p. 10):
We need no special sign to declare a truth-value to be the False, so
long as we possess a sign by which either truth-value is changed into
the other; it is also indispensable on other grounds. I now stipulate:
The value of the function
—|— ξ
shall be the False for every argument for which the value of the
function
—ξ
is the True; and shall be the True for all other arguments.
29. The argument is the following (p. 18, n. 17):
A natural suggestion is to generalize our stipulation [that the True
and the False are to be identified with (8) and (9)] so that every
object is regarded as a course-of-values, viz., as the extension of a
concept under which it and it alone falls. A concept under which
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the object ∆ and ∆ alone falls is ∆ = ξ. Suppose we attempt the
stipulation: let ǫ’(∆ = ǫ) be the same as ∆. Such a stipulation is
possible for every object that is given us independent of courses-ofvalues on the same basis as we have observed with the truth-values.
But before it may be generalized, the question arises whether it
may not contradict our notation for recognizing courses-of-values if
we take for ∆ an object that is already given us as a course-of-values.
In particular it is intolerable to allow it to hold only for such objects
as are not given us as courses-of-values; the way in which an object is
given must not be regarded as an immutable property of it, since the
same object can be given in a different way. Thus if we substitute
“α’Φ(α)” for “∆”, then we obtain
“ǫ’(α’ Φ(α) = ǫ) = α’ Φ(α)”,
and this would denote the same as
a −− (α’ Φ(α) = a) = Φ(a)”,
“−−∪
which however denotes the True only if Φ(ξ) is a concept under
which one and only one object falls, namely α’ Φ(α). Since this last
is not necessary, our stipulation cannot remain intact in its general
form.
30. That is, Frege’s definition amounts to:
1. If there exists an object ∆ such that α’Φ(α) = ǫ’(∆ = ǫ), then let \αΦ(α) be ∆;
2. otherwise let \αΦ(α) be α’Φ(α).
Since so far Frege has only introduced courses-of-values as objects, the function is
well-defined; but how can he ever introduce any other objects? And if he restricts
the logic to courses-of-values, then what becomes of the distinction between logical
objects and non-logical objects?
31. Frege defines (p. 35):
We now understand by
f —— f(Γ)”
“—∪
|a
–∪− f(a)
the truth-value of one’s always obtaining a name of the True, whatever function-name one may substitute in place of “f ” in
“——
| af (Γ)”
—∪—f (a)
32. The Basic Laws of Arithmetic, pp. 3-4.
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Chapter 9
Structuring Reality:
Properties, Sets, and States of Affairs
Frege’s ontology can be interpreted as a hierarchy of functions and
objects. Objects appear exclusively at the bottom level 0 and are characterized as everything that is not a function. Functions appear at all higher
levels and have as arguments and values entities of lower levels. Frege’s fundamental distinction between functions and objects is that the former are
incomplete and the latter complete. He also says that functions are unsaturated whereas objects are saturated. An object can be the reference of a
name and a concept can have a somewhat analogous relation to a predicate.
Concepts, including relations, are functions which applied to (or saturated
by) appropriate arguments give as value (or result in) one of the two truth
values1 .
The extensions of functions are logical objects for Frege, and therefore belong at level 0. With this move he “reflected” the whole hierarchy of
functions into level 0, which allowed him to formulate his logical account of
arithmetic as an account of logical objects but which was also instrumental
in the eventual collapse of his system. It is important to realize, however,
that Frege’s claim that arithmetical concepts are logical concepts is quite
independent of the move through extensions. His analysis does show that
arithmetical concepts are logical concepts. In fact, many of the concepts
which appear in Frege’s hierarchy are logical concepts; for instance, Existentiality, Universality, Identity, Subordination, Nullness, Oneness, Twoness,
Threeness, etc. Moreover, the usual combinations of logical notions, as expressed in the notation of predicate logic, refer to logical concepts in the
hierarchy. The problem, as he sees it, is that arithmetic appeals not only
to the arithmetical concepts and relations but also to arithmetical objects.
Numbers are treated as objects in mathematics, and it doesn’t seem to make
sense to treat them as concepts; therefore, a proper account of the ontology
of arithmetic must have the numbers as objects. Thus the move through
extensions. But Frege himself expresses the view that the introduction of
extensions is not essential to his logical analysis of arithmetic2 .
I will follow Frege in this structuring of reality but with some
modifications – and instead of talking of concepts and functions I will talk
of properties. These modifications are partly motivated by the account
of truth as denotation of states of affairs, instead of truth values, partly
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motivated by the paradoxes and other problems that derive from Frege’s
introduction of extensions at level 0, and partly motivated by additional
considerations which I will discuss as we go along.
As I conceive it the hierarchy goes on transfinitely, both vertically
and horizontally (at least at each level higher than 0). One can set lower
bounds but no upper bounds or limits; it is absolutely unlimited and cannot
be treated ontologically as a unit. There are entities that are like objects
in Frege’s sense at all levels of the hierarchy, but aside from the objects at
level 0, among which I would include physical and mental objects of various sorts3 , all these entities are states of affairs which can be conceived as
instantiations of a property in appropriate entities4 . Properties (can) exist independently of being instantiated and may relate finitely or infinitely
many entities – i.e., the arity of a property can be any cardinal κ. The level
of a property is an ordinal which is larger than the level of its arguments,
and the level of a state of affairs that is the instantiation of a property in
appropriate entities is the same as the level of the property. There are no
extensions at level 0; there are only objects that can be grouped into pluralities, but these pluralities are not level 0 ontological unities. Extensions as
objects may be conceived as certain states of affairs which appear at levels
higher than 0. Whether one may hold that there are mathematical objects
at the bottom of the hierarchy is a question that I will consider later.
Since a difference with Frege’s hierarchy is the introduction of
states of affairs, I will begin with a few remarks about them.
Consider the state of affairs that Quine is a philosopher. In my settheoretic formulation I treated this state of affairs as an ordered sequence –
and I also treated the first component as the set of philosophers, or as the
extension of the property ‘philosopher’; i.e.,
(1) <set of philosophers, Quine> .
The sequence
(2) <set of dentists, Quine>
is a perfectly good sequence, however, and there seems to be no good reason
to disqualify it just because Quine isn’t a dentist. In fact, this sort of
approach to states of affairs is open to the objection that what determines
whether a sequence is or is not a state of affairs is precisely the truth of the
corresponding sentence. I do not think that the explanation of truth in terms
of states of affairs is circular, but I agree that a set-theoretic approach is
rather artificial. The set-theoretic semantics of Chapter 6 is only interesting
in that it is quite close to the standard set-theoretic semantics and can be
used to illustrate the discussion of the notions and distinctions introduced
there.
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In Fregean terms states of affairs correspond to something like
saturation; i.e., in the example above, to the combination of a property
with an object. There is no such thing as the state of affairs that Quine
is a dentist because Quine does not saturate the property ‘dentist’. This
differs from Frege’s account in that he held that Quine does saturate the
property ‘dentist’ into the False. Since I think that Frege’s account is rather
artificial, as I have already argued at some length, I conceive the falsity of
the statement that Quine is a dentist as saturation by Quine of the property
‘non-dentist’ – which also involves his not saturating the property ‘dentist’.
As I see it this is Plato’s idea, and is also related to Aristotle’s formulation
in terms of combination and lack of combination. Frege’s the False drops
out, therefore, while the True is divided into states of affairs conceived as
roughly what Frege meant by “parts” of the True. Thus the True is divided
into its parts and these parts are spread out throughout the hierarchy5 .
In my view Frege was mislead by the imagery of saturation and
unsaturation, or completeness and incompleteness, that he seems to have
derived from language. He sees a predicate such as ‘is a philosopher’ as
something with a gap – as in ‘( ) is a philosopher’ – that can be filled in
by a name such as ‘Quine’. The name and the resulting sentence have no
gap. From this he may have concluded that the concept ‘philosopher’ has
a similar nature to the predicate – it has a gap as well, as it were, or is
incomplete, or requires further determination – whereas the object Quine
has a similar nature to the name – it is complete, saturated, requires no
further determination. Thus his fundamental distinction between concepts
and objects. This imagery leads Frege to several conclusions that seem to
me both unnatural and unjustified. One is the conclusion that one cannot
refer to a concept by means of the definite article. The expression ‘the
concept ‘horse’ ’ must refer to an object, not to a concept. Why? Because
if the expression ‘the concept ‘horse’ ’ is a subject that can be used to fill in
the gap of a predicate of objects – as, for example, in ‘the concept ‘horse’
is a horse’, which we would take to be false – then the entity referred to
must be able to saturate the concept to which the predicate refers; hence, it
must be an object. Another is the conclusion that truth values are objects
because they are the reference of sentences which are themselves complete.
Moreover, since the gap of the predicate ‘is a dentist’ can be filled in by the
name ‘Quine’ just as easily as the gap of the predicate ‘is a philosopher’,
Quine must be capable of saturating both the concept ‘philosopher’ and the
concept ‘dentist’. Thus saturation cannot be related to instantiation and
it is necessary to have the False as well as the True. The conclusion that
numbers must be objects rather than concepts is another related conclusion,
as is the conclusion that the senses of objects must themselves be objects.
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By partially accepting the imagery of saturation I do not mean to accept
any of these conclusions6 .
I shall refer to states of affairs by means of sentences, but since
sentences are often ambiguous with respect to logical structure I shall either
use the logical notation I introduced in chapters 1 and 6 or I shall use big
corner brackets listing a property and the arguments to which it applies;
i.e.,
(3) <philosopher, Quine>.
This notation can also be combined with the logical notation for referring
to properties and objects. Thus, we can refer to the state of affairs that
Quine is self-identical by
(4) <[x = x](x),Quine>.
We can use this notation to indicate properties of states of affairs as well.
The property of being a level 1 state of affairs that combines an object with
a level 1 unary property that applies to at least one more object may be
expressed as
(5) [∃y(Zx & Zy & x 6= y)](<Z, x>).
Similarly,
(6) [Zx & W x & ∀y(Zy ⇒ W y)](<Z, x>,<W, x>)
is a level 2 relation between level 1 states of affairs <Z, x> and <W, x>
which holds when Z is extensionally subordinate to W. If we introduce
boldface letters as variables for states of affairs, we can also use the following
alternative notations
(7) [∃Z∃x∃y(Zx & Zy & x 6= y & s =<Z, x>)](s),
(8) [∃Z∃W ∃x(Zx & W x & ∀y(Zy ⇒ W y) & s =<Z, x> & u =
<W, x>)](s, u) to indicate these properties.
The structuring of properties and states of affairs into levels gives
raise to some difficult questions concerning both the nature and identity
of these entities. The natural distinction of orders that Frege emphasizes
in The Foundations of Arithmetic is the distinction between a property of
properties, such as existential quantification, and a property of objects. Although this can lead to a further distinction between properties of properties
of properties and properties of properties of objects, and thus to a sequence
of levels, it doesn’t follow that it must lead to a stratification of properties
into levels such that a property of a certain level can only apply to entities
of lower levels. We could conceive of existential quantification as a property of properties that applies to a property if and only if this property is
instantiated. Although distinct from a property of objects, existential quantification thus characterized would be significant for all properties, including
itself. But it is precisely this very general conception of properties that can
lead to paradox and that seems to force a stricter distinction of levels. Thus,
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instead of a single property Existentiality, we have an indefinite number of
Existentiality properties of different levels7 .
But even if we divide properties strictly into levels, and restrict
the range of applicability of a property to entities of lower levels, there are
several ways in which we can delimit the range of applicability of a property.
The assumptions I made so far are that the initial level 0 is a level of objects
which are neither properties nor states of affairs, that the level of a property
is an ordinal which is larger than the level of its arguments, and that the
level of a state of affairs that results from a combination of a property with
appropriate arguments that instantiate this property is the same as the level
of the property. These assumptions are compatible with an assignment of
types to properties and states of affairs according to which the level of
a property is determined by the levels of its arguments and is the lowest
level greater than the levels of the arguments – which is essentially how I
classified predicates and sentences in Chapter 6 (note 4). That is, if to level
0 objects we assign type 0, then the type of a κ-ary level 1 property is a
sequence < 0, 0, . . . > of length κ, and the type of a level 1 state of affairs is
<< 0, 0, . . . >, 0, 0, . . . >. In general, the type of a κ-ary property of level λ
is a sequence of length κ of types τν such that each τν is the type of an entity
of level lower than λ, and λ is the smallest ordinal satisfying this condition,
and the type of a state of affairs of level λ is a sequence beginning with the
type of a λ level property followed by the types of the arguments.
It follows from this that a relation such as Frege’s relation Equinumerical for level 1 unary properties – i.e., the relation that holds between
two such properties if there is a 1-1 correlation between the objects to which
they apply – is a level 2 binary relation of type << 0 >, < 0 >>. Even
though Frege’s definition of this relation involves quantification over level
2 properties, and hence a level 3 property, the relation is a level 2 relation
because we are classifying properties in terms of what they apply to and not
in terms of how they may be defined by means of certain other properties.
But we could also say that Frege’s definition of the relation Equinumerical
for level 1 unary properties determines a level 3 property and leave it open
whether there is a level 2 property that in some sense “coincides” with it –
in the sense of being extensionally equivalent to it, say, or even necessarily
extensionally equivalent to it. In this case the assignment of types would
have to be made differently so that the level of a property is not determined
by the level of its arguments. For example, the type of a level λ κ-ary
property could be a sequence
<< λ, κ >, τ0 , τ1 , τ2 , . . . >
of length κ+1, where < λ, κ > gives the level and arity of the property and
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each τν is the type of an entity of level lower than λ. Level λ states of affairs
could be assigned type
<<< λ, κ >, τ0 , τ1 , τ2 , . . . >, τ0 , τ1 , τ2 , . . . >
and level 0 objects type 0. Then by Frege’s definition we would have an
Equinumerical property of type
<< 3, 2 >, << 1, 1 >, 0 >, << 1, 1 >, 0 >>,
and the question we may raise is whether this property, or a property extensionally equivalent to it, or a property necessarily extensionally equivalent
to it, appears at level 2. This is related to Russell’s axiom of reducibility,
which could be formulated as the principle that for any level λ property
there is an extensionally equivalent property of the lowest level higher than
the level of its arguments8 . But if the distinction between these two properties is merely a distinction of level, then why bother to distinguish them?
And if it isn’t merely a distinction of level, then what distinguishes them
as properties? My view is that it is better to separate the question of level
from the question of definability, and that’s why I prefer to think of the
level of a property as determined by the levels of its arguments. Without
getting into this question now, however, the more general question I want
to raise about the division of properties into levels concerns the relationship between a logical notion and the plurality of properties into which it
is divided.
What seems to characterize a logical notion is universality. Notions
such as instantiation, existential quantification, etc., are logical notions in
this sense because they are significant for all properties. Since it does not
seem to be possible to treat them ontologically as units, they are divided
into an indefinite number of “parts” which are treated as ontological units.
Each of these parts is a logical property with a limited range of significance
(or applicability) given by its type, but together they are significant all over
the hierarchy above a certain level. Existentiality first appears at level 2, as
a property of level 1 properties, and corresponding Existentiality properties
for higher level properties will appear at every level higher than 2. There
is no corresponding property at level 1 because existential quantification is
significant only for properties. The question is whether we can legitimately
conceive of this plurality of Existentiality properties as having the sort of
unity that is suggested by such words as ‘together’ or ‘corresponding’ that I
used above. One way to think of this unity is to conceive of types as being
cumulative in essentially the sense in which one thinks of the cumulativity
of sets.
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Set theorists generally work with a cumulative hierarchy of pure
sets organized in levels as follows. At the bottom is the empty set ∅, which is
of level 0. Level 1 consists of ∅ and {∅}; level 2 of ∅, {∅}, {{∅}}, and {∅, {∅}};
level 3 consists of the sets of level 2 plus the sets that can be “generated”
out of them by single applications of the operation set of; etc. The first
level at which a set appears is the rank of that set, and the hierarchy is
cumulative in that each level includes all sets of lower rank. If we think
in terms of properties, then we can think of the levels as a division of the
notion of set into an indefinite sequence of properties Set0 , Set1 , Set2 , etc.,
such that each property Setλ extends the property Setλ′ for λ′ < λ. We
may even think of the notion of set as a sort of limit of this sequence of
properties, but it is in the nature of sets that there is no absolute limit –
and, therefore, that there is no absolute property Set.
We can think of a logical notion such as existential quantification
in a similar way. That is, we may hold that at level 2 there is a property
Existentiality2 that applies to a κ-ary level 1 property if and only if this
property is instantiated by a κ-sequence of objects. This would encompass
the properties I have been expressing as [∃xZx](Z), [∃x∃yZxy](Z), etc., as
well as infinitary existential quantifications. At level 3 there is a property
Existentiality3 that applies to a κ-ary property of level 1 or level 2 if and
only if this property is instantiated by a κ-sequence of appropriate entities.
In general, at level λ there is a property Existentialityλ that applies to a κary property of level lower than λ if and only if this property is instantiated
by a κ-sequence of appropriate entities. We may again think of the notion
of existential quantification as a sort of limit of this sequence of properties,
but it is in the nature of logical properties that there is no absolute limit
– i.e., that there is no absolute property Existentiality. Still, there is a
reasonable sense in which we may talk of unity in connection with such
logical properties.
My idea, therefore, is that one should restrict the level of a property
to be the lowest level higher than the level of its arguments but that one can
otherwise be fairly flexible in assigning types to properties. Some properties
may be significant for all entities below their level, others may be significant
only for properties, or for properties of a certain arity, etc. And we could
agree on the following system of notation. If a κ-ary property is significant
for all entities of lower level, then its type is simply the pair < λ, κ >
giving its level and arity. A property may also have variable arity – or be
multigrade9 – in which case we can indicate this by placing some bounds
on the arity. Thus, a property of level 0 objects that can relate κ-uples of
objects for any κ ≥ 2 is a multigrade property of type < 1, κ >2≤κ . Let’s
also agree that the type of a level 0 object is 0 and that the type of a level
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λ state of affairs is a sequence consisting of the type of a level λ property
followed by the types of the arguments in which it is instantiated. If there
are restrictions on the arguments of a property, then we indicate that by
indicating the type of its arguments. Thus, the type of Existentiality2 could
be given as
<< 2, 1 >, < 1, κ >>0<κ ,
the type of the more restricted property [∃xZx](Z) could be given as
<< 2, 1 >, < 1, 1 >>,
and the type of Existentialityω could be given as
<< ω, 1 >, < λ, κ >>0<λ<ω,
0<κ .
Although I will not discuss the logical properties of level higher
than 1 in detail at this point, I will assume that there are logical properties
corresponding to the usual logical notations. I also assume that the combination of logical notations with terms that refer to properties also refer to
properties. In particular, I assume that for each property there is a complementary property of the same type, to which I normally refer by means
of negation, such that for any arguments of the appropriate types one or
the other and not both of these properties apply to those arguments. Also
the question as to which non-logical properties appear in the hierarchy –
i.e., whether there is in fact such a property as ‘philosopher’, or ‘human’, or
‘chair’, etc. – is a question that I will not consider now. For the time being
I will simply treat these things as properties.
The only level 1 properties that have a claim to being logical are
the properties Existence, Non-Existence, Identity and Diversity. Aside from
the usual Identity and Diversity relations, there are infinitely many Identity
and (pairwise) Diversity relations of all arities, as well as multigrade Identity and Diversity of type < 1, κ >1≤κ (which include Self-Identity and SelfDiversity), and mixed Identity-Diversity relations of various types. For the
moment I’ll restrict myself to the binary relations. Such relations supposedly
appear at all levels of the hierarchy and that’s why they are traditionally
considered to be logical properties. One could wonder, of course, whether
there may not be entities to which neither of these relations applies. Part
of the ontological content of Quine’s dictum “No entity without identity”10
may be expressed by the claim that Identity and Diversity are complementary properties and that there are Identity and Diversity relations at all
levels in the hierarchy. And, as Quine points out, for objects the thesis may
be traced back to Frege in The Foundations of Arithmetic11 .
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Quine and Frege could in principle agree with this ontological formulation, at least for objects, but the main point for them is an epistemological point concerning the introduction of objects of one kind or another
(by definition or by postulation). It may be taken to be a consequence of
the ontological view that identity is universal in the previous sense, that in
order to introduce (define, postulate) objects of a certain kind it is necessary
to provide a criterion of identity for such objects. For Frege the criterion
must be completely general, showing the determinateness of Identity and
Diversity for such objects in relation to each other and to any other objects.
For Quine the criterion of identity would appear to be specific for the objects (or entities) in question; for although he holds that there is a criterion
of identity for sets – namely, that two sets are identical if and only if they
have the same elements – he does not claim that this criterion settles the
question whether Julius Caesar is some set or other, or is a set at all.
Leibniz’ principles of identity of indiscernibles and of indiscernibility of identicals may be considered to be general criteria for Identity and
Diversity for all objects. These principles, in the form
(9) ∀x∀y(∀Z(Zx ⇔ Zy) ⇒ x = y),
and
(10) ∀x∀y(x = y ⇒ ∀Z(Zx ⇔ Zy)),
where the latter can also be formulated as
(11) ∀x∀y(∃Z¬(Zx ⇔ Zy) ⇒ x 6= y),
are independent of the thesis that Identity and Diversity are complementary
properties, for we could conceive (or introduce) entities that are different
but cannot be distinguished by their properties – as in the traditional geometrical conception of points. Thus, although one may define Identity by
(12) ∀x∀y(x = y ⇔ ∀Z(Zx ⇔ Zy)),
or hold that it is a general criterion of identity for objects, the adequacy of
the definition or criterion is open to question12 . Nevertheless, (12) seems
a good candidate for a general criterion of identity for objects relative to
properties. It is useful to establish Diversity, but it is not so useful to establish Identity. A useful (or more informative) criterion of identity for entities
of a certain kind would be something of the form
(13) ∀x∀y(C(x, y) ⇒ x = y),
which could derive from (9) and
(14) ∀x∀y(C(x, y) ⇒ ∀Z(Zx ⇔ Zy)),
where C(x, y) is a condition, or combination of conditions, referring to a
relation which can be verified or established in a more or less direct way for
specific cases. Combining (13) with Leibniz’ principle (10) of indiscernibility of identicals, which is reasonable on any view of identity and properties,
we get
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(13#) ∀x∀y(x = y ⇔ C(x, y)),
where the variables range over a specific kind of entities. This seems to be
essentially what Quine has in mind when he talks of a criterion of identity
for entities of a given kind. His epistemological and ontological thesis is
that the introduction of a certain kind of entities requires for its legitimacy
a criterion of identity of form (13#).
Now what is it to introduce, or define, or postulate, a certain
kind of entity? In general, this would seem to mean to introduce (define,
postulate) a property, or concept, and some existential assumptions about
it. Frege assumes that a concept of objects must be everywhere defined
for objects, and if all and only the objects that fall under the concept
can be specified by a description of a standard form, then the question
whether the concept is everywhere defined can be seen as being reducible
(or even equivalent) to the question whether there is a general criterion that
determines for any name of an object whether it denotes the same thing as
a description of that form. Thus, if all and only numbers can be specified
by a description of the form ‘the number which belongs to the concept F’,
and if there is a general criterion that determines the truth or falsity of
an identity of the form ‘Julius Caesar = the number which belongs to the
concept F’, then the question whether Julius Caesar is a number or not is
determined as well13 .
I do agree with Frege that every property of objects is defined for
all objects in the sense that for every object the property applies or does not
apply to that object. And I would also agree with Frege that a definition of
objects of a certain kind should make clear that Identity and Diversity are
universally applicable to such objects in relation to each other and to any
other objects.
In The Foundations of Arithmetic Frege is trying to show that
numbers can be defined in purely logical terms as logical objects. When he
complains in p. 68 that the proposed definition does not allow for a decision
as to whether or not Julius Caesar is a number, he may be interpreted as
pointing out that this definition does not succeed in defining numbers as
objects because Identity is universal and the definition gives no criterion
for Identity and Diversity in general. And, again, when in pp. 77-78 he
complains that the definition of direction does not allow for a decision as
to whether or not England is identical to the direction of the Earth’s axis,
he may be interpreted as pointing out that although he has a criterion for
deciding when two directions are the same or different, the definition does
not succeed in establishing that directions are objects.
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Thus there are both ontological and epistemological considerations
involved. If one defines directions and numbers as the extension of certain
concepts and assumes that extensions are objects, then this settles the question that directions and numbers are objects. Moreover, given the ontological assumption that Identity and Diversity are complementary for objects,
and that (12) is a general criterion of identity, the questions about Julius
Caesar and England are ontologically settled as well. If, on the other hand,
one takes a more absolute position and holds that there must be a general
criterion for extensions and for any objects that may be involved in the
formulation of this criterion, then one gets landed with the problems that
Frege faces in The Basic Laws of Arithmetic.
There are at least three different issues, therefore. The first is
whether Identity and Diversity are complementary for objects. The second is whether there is a general criterion of identity for objects; and in
particular whether (12) can be considered to be such a criterion relative to
properties. The third relates to a certain view concerning the “introduction”
of objects. This view assumes the universality of Identity and Diversity and
requires that a definition of a certain kind of object must ensure that the
alleged objects being defined bear one or another of these relations to all
objects. Although the definition may provide a specific criterion of identity
of something like form (13#) for the kind of objects being defined, this is
not enough for Frege. The criterion must be completely general, though
as Frege says it need not be such that we can actually decide in each case
whether objects of that kind are the same or different to each other or to
other objects.
Thus, if numbers are objects, then the concept of number must be
everywhere defined for objects. It follows that if one claims that the concept
of number is a concept of objects, then it must be determined whether
Julius Caesar is a number or not; and if one is defining the concept, then
this determinateness should be clear from the definition. Moreover, since
Identity and Diversity are complementary for objects, it follows also that
if we can specify an object by the description ‘the number which belongs
to the concept ‘moon of Mars’ ’, then it must be determined whether this
object is identical to or different from the object specified by the name
‘Julius Caesar’, assuming that this name also specifies an object, and this
determinateness should be clear from the definition of the concept Number.
With this much I agree – though not thereby agreeing that numbers are
objects. Does it follow that we must also have a criterion for determining
when a term specifies an object or for when two terms specify the same
object? Here is where a certain confusion arises.
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To postulate extensions as objects and to hold that reality determines Identity and Diversity for extensions in relation to each other and to
any other objects is one thing, even if this determination by reality is formulated more specifically by postulating (12) as a general criterion of identity
for objects. To take (12) as a general criterion for sameness of denotation
is quite another.
In The Foundations of Arithmetic Frege uses Leibniz’ law (in the
form “Things are the same as each other, of which one can be substituted
by the other without loss of truth”) as the definition of identity (p. 76).
This dictum can be interpreted in various ways; in particular, it can be
interpreted as a schema
(15) a = b ⇔ (P a ⇔ P b),
where ‘a’ and ‘b’ stand for names and ‘P’ stands for a predicate (or context),
which gives a criterion for sameness of denotation of names by varying the
predicate (or context) in all possible ways. (12) would be a reasonable
formulation of this if one interprets the quantifiers substitutionally, as Frege
seems to do in Begriffsschrift – let’s refer to this interpretation as (12S), and
similarly for (9S) and (10S). But as a general criterion for sameness (and
difference) of denotation (15) is open to the objection that it doesn’t work
for intensional contexts – and that there is no accepted way of separating
extensional from intensional contexts except in terms of whether or not (15)
holds for that context. And as a definition of Identity (12S) is open to the
objection that it confuses sameness of denotation for names with identity of
objects; i.e., that it confuses a relation of names with a relation of objects14 .
Now Frege’s main concern was not with a criterion of sameness of
denotation but with a criterion of denotation, or objecthood, or objectivity.
If one has a criterion that determines every identity involving a term of
the form ‘the number which belongs to the concept F’, in the sense that
every such identity is either true or false according to the criterion, and
if contexts of the form ‘the number which belongs to the concept F is a
number’ are always true in virtue of the definitions of the concepts involved,
then by (10S) every context of the form ‘a is a number’ has a determinate
truth value. If one assumes that every object can be named, then from
this one can conclude that the concept Number is everywhere defined for
objects and that expressions of the form ‘the number which belongs to the
concept F’ denote objects. In fact, this a somewhat misleading way of
stating the conclusion, for Frege’s idea in The Foundations of Arithmetic
is that satisfaction of the previous conditions is precisely what it means to
say that numbers are objects, or are objective. The problem is that there
is no general way of guaranteeing the determinateness of every identity
statement. Frege’s attempt to do so in The Basic Laws was ingenious, and
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almost inevitable given the conditions of the problem, but as I pointed out
in the last chapter it doesn’t work15 .
The only completely general criterion of identity that has ever
been formulated by anyone is (12). It can be used not only as a criterion
of identity for objects, but can be generalized to properties themselves and
to any other entities. When we define logical and mathematical entities we
either reduce the question of identity for these entities to the question of
identity for other entities – of which they are “made up” in some way, for
instance – or we give a criterion of identity of something like form (13) for
these entities in relation to each other, and let (12) – or reality – handle the
rest. The latter is what one normally does for sets. The usual criterion of
identity for sets is restricted to sets and does not decide the question whether
Julius Caesar is the empty set – and does not decide the question whether
Julius Caesar is any other set either, for that matter. One may claim
that Julius Caesar has no elements, and that therefore it cannot be a nonempty set, but there is nothing about the membership relation (as usually
conceived), or about Julius Caesar, that determines that Julius Caesar has
no elements. To postulate that only sets have elements does not settle the
question, for the question is precisely what distinguishes sets from non-sets.
If sets are a legitimate category of entities, then reality will settle this, one
way or the other. But as far as the principles of set theory are concerned,
every object could be a set. Of course, one can postulate explicitly that
Julius Caesar is not a set, but one cannot formulate a general criterion of
identity that will determine this question for all objects that we normally
conceive of as not being sets.
These considerations about questions of identity were intended to
make three conclusions initially plausible. The first is that one must distinguish criteria of identity from criteria of denotation, and that the question
of whether Identity and Diversity are everywhere defined (complementary)
for objects should be independent of having criteria for denotation or for
sameness of denotation for terms. The second is that the universality of
Identity and Diversity must be postulated in some way, directly or indirectly. A third conclusion related to this is that general criteria of identity
are always relative, either because the criterion of identity for a certain kind
of entities depends on the determinateness of identity for some other kinds
of entities, as is the case with (12) as a criterion of identity for objects, or
because the criterion involves a certain “regress” akin to the “regress” in
the general formulation of (12) as a criterion of identity for properties as
well.
Although I think that (12) is a reasonable criterion of identity for
objects, my view is that Identity and Diversity must be postulated at all
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levels as basic logical properties, and that to adopt (12) as a definition of
identity begs important logical and ontological issues. For as a definition
(12) will not be adequate if there are objects at level 0 that have no distinguishing unary properties as required by (12) and yet are “different”. This
seems to me at least conceivable, and I see no “logical” reason why it should
be ruled out ab initio by means of a definition16 .
I will hold, therefore, that Identity and Diversity are fundamental
logical properties that appear at every level (above level 0) in the hierarchy.
It follows that the properties Existence, which also appear at every level
higher than 0, coincide with the properties Self-Identity of the same level
for one or another category of entities of lower level. There are infinitely
many Identity and Diversity relations at all levels (higher than 0) for entities
of lower level of the same type or of different types. It should also be
noticed that although there is only one Existence property at level 1, there
are infinitely many Existence properties at all levels higher than 1. At
level 2, for example, there is an Existence property of type << 2, 1 >,
<< 1, 0 >, 0 >> for states of affairs of type << 1, 0 >, 0 >, which is in fact
the property of being a state of affairs of type << 1, 0 >, 0 >, and there are
Existence properties for level 1 properties of each arity, of types << 2, 1 >,
< 1, 1 >>, << 2, 1 >, < 1, 2 >>, etc., which are the properties of being
a level 1 unary property of objects, of being a level 1 binary property of
objects, etc.
I will also hold with Frege and Quine that in defining certain kinds
of entities, or properties, there should be criteria of identity that may be
related to criteria of sameness of denotation for certain terms that are themselves related to the property in question. The precision and specificity of
these criteria will depend on cases, and there may be reasons for demanding
more precision in some cases than in others. But as far as very broad categories are concerned, such as the category of level 0 objects, it seems to me
unreasonable to demand anything beyond a criterion like (12) as a general
criterion of identity. The generalizations of (12) can be considered to be
general criteria of identity for properties and states of affairs at all levels,
but this involves the “regress” I was talking about before. This seems to
me inevitable, however, because in my view the notion of identity is indeed
presupposed by the notion of property – as in the Platonic conception of
forms as “expressing” what is the same in each particular instance (see note
32). Moreover, all attempts that I know aiming at reducing the question of
identity for properties to other entities – possible objects, possible worlds,
space of possible applications, etc. – seem to me to raise more problems than
they solve and to presuppose an even more problematic notion of identity
for such entities. But, evidently, to claim this primitiveness for the notion
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of identity in relation to properties requires further argumentation. And
it requires some kind of answer to Quine’s objection that the postulation
of properties is both illegitimate and unnecessary – illegitimate precisely
because we lack a reasonable criterion of identity for them; unnecessary because we can make do with extensional entities such as sets instead. This
is something I’ll discuss in the next chapter, however.
Let me turn now to the question of extensions. Frege held that
every property has an extension, and that extensions are level 0 objects.
Nowadays one talks about sets of objects and says that the extension of
the property ‘human’ is the set of all humans. In particular, one talks
about such things as the set {Frege, Russell}, which is considered to be the
extension of the property [x = Frege ∨ x = Russell](x). But what is this set?
What are its essential features? There seem to be two basic features. One
is that the set is an object; something that needs no further determination,
in Frege’s image. The other is that the identity of the set is determined
by its elements considered as units. The latter means that although both
Frege and Russell can be seen as wholes composed of parts, it is the object
Frege and the object Russell that are elements of the set {Frege, Russell},
not their parts. In other words, although an object may be treated as a
plurality of parts, it is also treated as a unity of some sort17 .
My suggestion is to identify the set {Frege, Russell}, or the extension of the property [x = Frege ∨ x = Russell](x), with the state of
affairs
<Diversity, Frege, Russell>;
i.e., with the state of affairs that Frege is different from Russell. In general,
given any plurality of level 0 objects which can be arguments of a pairwise
Diversity relation of level 1, the set whose elements are the objects of the
plurality is the state of affairs consisting of the appropriate Diversity relation and all the objects18 . A singleton such as {Frege}, or the extension
of the property [x = Frege](x), can be identified with the state of affairs
<Existence, Frege>. Evidently, this idea can be generalized to characterize not only sets of objects but sets of properties and of mixed pluralities
at all levels of the hierarchy. The qualification ‘which can be arguments’ is
necessary because if level 0 is absolutely unlimited, then there is no pairwise
Diversity relation that can “collect” all level 0 objects into a set. This will
actually happen for level 1 properties; there is no set of all level 1 properties
because level 1 is absolutely unlimited.
Every property in the hierarchy will either have or not have an extension in this sense, depending on whether there is an appropriate pairwise
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Diversity relation that can “collect” all the entities to which the property
applies. Ordinary properties such as ‘human’, ‘chair’, etc., have extensions,
but many logical properties of level higher than 1 don’t have extensions.
Sets of level 0 objects are level 1 states of affairs, and therefore
the extensions of level 1 unary properties of objects are at the same level
as these properties. This conception of sets as extensions seems to me
to answer well to what we normally consider to be the basic features of
such sets; namely, that the identity of a set is in some sense determined
by its elements and that sets are somewhat like objects. Moreover, some
questions and objections concerning the usual conception of sets of objects
receive rather natural answers here.
It is quite obvious, for instance, that if Frege had not existed, then
the set {Frege, Russell} wouldn’t have existed either. And the set {Frege,
Russell} is just as temporal as Frege and Russell are, in the sense that
the set did not exist before both of them did, and if they don’t exist now
neither does the set. Naturally, our conception of the temporal features of
the set will be related to our conception of the temporal features of objects
generally.
Goodman’s objection that sets can be distinct objects without a
distinction of content is also met by this characterization. The set
{Frege, Russell, Gödel},
which is the state of affairs
<Diversity<1,3> , Frege, Russell, Gödel>,
has quite a different content from the set
{{Frege, Russell}, Gödel},
which is the state of affairs
<Diversity, {Frege, Russell}, Gödel>.
The latter set is a level 2 state of affairs and the Diversity relation in question
is of type
<< 2, 2 >, <<< 1, 2 >, 0, 0 >, 0, 0 >, 0 > .
Thus, although there is no difference in the objects of the two sets, their
logical content is indeed different19 .
Ordered pairs could be defined according to Kuratowski’s definition, as well as in other ways, and sets of them are states of affairs which
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appear at various levels. Hence relational properties also have extensions.
There are no empty extensions, but since to say that a property has an
empty extension is to say that it doesn’t apply to anything, there is a state
of affairs that a property has an empty extension. For the level 1 property
[x 6= x](x) – i.e., Self-Diversity, or Non-Existence – this state of affairs is
<[¬∃xZx](Z), [x 6= x](x)>,
appearing at level 2. One can’t identify “the” empty extension with such
states of affairs because the states of affairs are different for different properties, but one could arbitrarily identify the empty extension of level 1
properties with the state of affairs above – I won’t do this, however.
Although it would be artificial to do mathematics with these sets,
the artificiality does not derive from the conception of extensions that I
am suggesting, which seems to me quite natural. What is artificial is to
conceive mathematical objects as extensions of properties, or as sets in this
sense of ‘set’.
An alternative is to hold that there are mathematical objects at the
bottom of the hierarchy. In fact, following a certain tradition in logic and
the foundations of mathematics, mathematical objects could be conceived
as pure set structures of a certain sort. What does this mean? Well, in a
sense it is Plato’s idea about mathematics interpreted in a particular way.
One can simply postulate the entire cumulative hierarchy of abstract sets at level 0 of the general hierarchy; i.e., postulate a level 1 property
set and a level 1 binary relation membership and some axioms about them –
which need not be first order axioms. This may seem to have the advantages
of theft over honest toil, in Russell’s famous simile, but it is more natural
(and better) than postulating extensions at level 0.
This hierarchy of abstract sets is extremely rich and powerful but
limited in certain ways. For although we have infinitely many set structures
of cardinality 2, there isn’t one among them that can be said to be purely
a cardinality 2 set; each has some additional structure as well. A pure
cardinality 2 set would be something like {, }, but the first cardinality 2 set
we get of rank 2 has the structure {, {}}; and as we go on the additional
structure gets richer and richer. In order to get a denumerable set all whose
elements have cardinality 1, the simplest structure is something like
{{}, {{}}, {{{}}}, . . .}
of rank ω+1, whereas it would be much more natural to have something
like
{{}, {}, {}, . . .}
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of rank 2. In fact, a natural conception of pure set structures would be to
conceive of them as structures obtained by “iteration” from pure cardinality
structures, allowing mixing of lower levels. Thus, at set-level 1 we would
have all pure cardinality structures
∅, {}, { , }, { , , }, . . .
At set-level 2 we would have all cardinality structures any of whose positions
may be unstructured or may have level 1 structures. In general, at any setlevel α, where α is an ordinal, we would have all cardinality structures any
of whose positions may be unstructured or may have lower level structures.
Just as one can postulate the set structures that arise from the
empty set by transfinite iterations of the operation ‘set of’, one could postulate this hierarchy of pure set structures at level 020 . These pure set
structures seem to me to “reflect” (at least some) mathematical properties
in a rather natural sense. The initial set-level reflects the cardinality properties Nullness, Oneness, Twoness, etc., and subsequent set-levels reflect such
properties as a Twoness of a Oneness and a Threeness. It may seem misleading to treat these structures as objects, and it may be more reasonable
to hold that they are really abstract properties of some sort. I agree, and
one could actually conceive the pure cardinality structures as the pairwise
Diversity relations themselves. Still, it is a rather generalized feeling that
there is a difference between a property and a structure, and that a pure
structure is something like an object which doesn’t have any specific content
except for its structural features. The problem is how to conceive of such
structures without objectifying their content21 .
As I mentioned earlier, one of the difficulties for the foundations
of mathematics that Frege raised is that mathematical discourse seems to
be about objects and not about properties22 . So even if one shows that
there is a property Twoness, a logical property in fact, this may not be
enough for mathematics because one needs an object 2. Frege dealt with this
problem by appealing to abstraction and by objectifying these abstractions
as extensions, which in his conception are level 0 objects which can be
manipulated in the way required by mathematical discourse. But properties
cannot be reflected into level 0 via their extension without paradox. My
present speculations go in the direction of preserving Frege’s solution by
reflecting only the mathematical part of the hierarchy into level 0, but I am
not suggesting that one must do this.
Of course, for the purposes of doing mathematics one can perfectly well restrict oneself to the usual cumulative hierarchy of sets. But
as many people have pointed out23 , this is somewhat artificial and leads to
a rather formalistic conception of sets and of mathematical structures gen-
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erally. Frege showed that mathematics is a part of logic in the sense that
all arithmetical properties are logical properties, but he did not succeed in
showing that mathematics is a part of logic in the sense that arithmetical
objects are logical objects. One could hold that in this respect mathematics
has its own specificity and that these objects must be postulated.
That is, one could hold that what the mathematician uses are exemplars of the mathematical properties, and the problem is how to talk
about them and how to characterize them. The pure set structures I described above may not seem appropriate as mathematical objects because
it is not clear how to talk about them without objectifying the structure
positions in some way. A natural solution is to appeal to something that
Frege didn’t like; namely, units24 . These units are essentially what set theorists call ‘Urelemente’. Suppose we are thinking of the axioms of set theory
as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Instead of
starting with the empty set and building up from there, we start with a
stock of units and build up from there using the operation set of. We thus
obtain sets of units such as {u1 , u2 }, which is a mathematical structure
representing the property Twoness. Since there are many units, there will
be many such structures representing this property. This is again artificial,
but it is essential in order to have such set structures as {{u1 , u2 }, {u3 , u4 }},
which is a cardinality Two structure with two cardinality Two structures as
elements. In fact the difference between the standard hierarchy of abstract
sets and this hierarchy built out of units is that one conceives the standard
hierarchy as having just one unit, ∅, and hence a structure such as {u1 , u2 }
collapses to {∅} and must be represented by something like {∅, {∅}}. Having
many units solves this problem but makes for a proliferation of the “same”
structure.
Now usually the totality of units is supposed to be collectable
into a set structure, but this would spoil the idea that all cardinalities are
represented at the first set-level. So in addition to ZFC we can assume an
axiom of units (AU) which says that for any set structure there is a set of
units of the same cardinality; i.e., a set of units that can be put into 1-1
correspondence with the given set structure. This means that the hierarchy
of set structures is just as wide as it is high; the units cannot be collected
into a set structure and all set structures can be seen as obtainable from
units by finite or transfinite iterations of the operation ‘set of’.
But what are these units? Are they level 0 objects of the general
hierarchy? There are two ways one can conceive of them. One is as level
0 objects that have no properties except Existence, Identity, Diversity and
the membership relation to the set structures; they are objectified structure
positions. The other is that the units are an idealization that is useful to
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theorize about the pure set structures25 . The units are structure positions
that have no independent existence. This is the most natural alternative,
but it would have to be developed in more explicit terms. If we assume
that we are postulating units as level 0 objects, then we would have a
property unit as well as the properties set and membership about which we
can formulate the axioms of ZFC+AU26 . If we take units as an idealization,
then ‘is a unit’, ‘is a set’ and ‘is an element of’ would be predicates of the
mathematical theory that do not actually refer to properties. Given that
units are structure positions, objectified or not, the axiom of units is fully
justified, because if there is a set structure of some cardinality, then there
must be that many structure positions and a pure cardinality structure of
that cardinality.
Although these set structures seem to correspond rather closely
to Plato’s conception of mathematical objects27 , my main motivation for
thinking of sets as pure set structures, of which these sets built up from units
may be idealized representations, came from some remarks of Gödel28 . He
suggests that sets can be thought of as certain kinds of structures which can
be identical to proper parts of themselves, and refers to some early work
by Mirimanoff where he defines such structures as {, } and {{, }, {, }} as
isomorphism-types of structures made up of units (or indecomposables)29 .
The problem is how to construe these isomorphism-types. Instead of trying
to derive the pure structures from the sets of units by isomorphism, one
can postulate the pure structures and introduce the sets of units as an
idealized (or fictional) way of speaking about them. Evidently, the set
{u1 , u2 , u3 , . . .} contains infinitely many proper parts such as {u2 , u3 , u4 , . . .}
which are isomorphic to it though not equal from the point of view of the
mathematical theory. From the absolute point of view however, i.e., from
the point of view of the Identity relation conceived as indistinguishability,
these sets are identical because they have exactly the same properties –
all that is involved is a “renaming” of the structure positions. We could
understand the claim that the pure structure {, , , . . .} involves itself as a
proper part in infinitely many ways in this sense30 .
If one doesn’t have units, then one can work with the pure set
structures themselves and try to axiomatize them directly. One can’t do this
just by having membership as a primitive relation, but one can introduce
another primitive relation ≪ such that if A and B are sets built up from
units, then A ≪ B means that A can be mapped 1-1 onto a subset of B.
In particular, this gives us a “part-whole” relation at the bottom level of
pure cardinality structures ∅, {}, {, }, {, , }, . . ., which has the feature of well
ordering these structures. When one develops the hierarchy of these pure
set structures one has to distinguish notions defined in terms of ≪ from
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notions defined in terms of ∈. The analogue of the power set construction,
for instance, defined in terms of ≪, has the interesting property that for each
level 1 pure cardinality structure x, Πx is the corresponding well ordering
by ≪ of all the pure cardinalities up to that:
Π∅ = {∅}
Π{} = {∅, {}}
Π{, } = {∅, {}, {, }}
Π{, , } = {∅, {}, {, }, {, , }}
etc. Thus, for finite cardinality structures x, Πx gives us the successor of
the corresponding ordinal. It will skip the limit ordinal
{∅, {}, {, }, {, , }, . . .}
corresponding to ω, and for {, , , . . .} it will give us
{∅, {}, {, }, {, , }, . . . , {, , , . . .}}
corresponding to ω+1. As one goes on to higher levels, even the next higher
level, and the structures have elements which are level 1 structures, then Πx
gets more complicated; and if one takes as x the limit ordinal corresponding
to ω, then Πx is something like the usual power set defined in terms of ∈.
Although we may thus postulate set structures at level 0, in one
way or another, we may wonder about the point of doing so, for the usual
cumulative hierarchy of (pure) sets will appear as a hierarchy of extensions
in the sense I characterized earlier. The empty set ∅ can be identified with
the level 2 state of affairs
<Existence, Self-Diversity>;
{∅} with the level 3 state of affairs
<Existence,∅ >;
{{∅}} and {∅, {∅}} with the level 4 states of affairs
<Existence,{∅}>
and
< Diversity, ∅, {∅}>;
and so on31 . Thus, it appears that there is really no need to postulate sets
at level 0.
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One axiom that is often postulated for hierarchies, in some form,
is the comprehension axiom. This axiom really goes back to Plato when
he argues that if a plurality of particulars have some feature in common,
then they exemplify a form32 . In its modern version it is usually stated as
asserting that for any well-defined condition about particulars, say, there
is a property that applies precisely to those particulars. In a more general
version it would assert that for any well-defined condition for entities below
a certain level, there is a property of that level that applies precisely to
those entities. Whether this axiom is reasonable or not depends on our
conception of properties. If one thinks of properties as “intrinsic” in some
sense (for instance, as necessarily existing independently of the existence
of the entities to which they apply), then it may be an implausible axiom;
what reason can one give for thinking that whatever condition one can cook
up, however well-defined, picks up a property in this sense? If, on the
other hand, one thinks of properties as including not only “intrinsic” but
also “extrinsic” properties (whose existence may depend on the existence of
some or all entities to which they apply), then the axiom would seem to be
justified as long as one respects the distinctions of level33 .
Axioms of comprehension are usually postulated to obtain sets,
or extensions. The “naive” intuition is supposed to be that one should be
able to do set theory based on an axiom of comprehension and on an axiom
of extensionality, and many people hold that once this naive approach is
lost, then any intuitive conception of sets is also lost34 . Given the way I
characterized extensions, it does follow that for any plurality that is not too
big, i.e., for which there is a pairwise Diversity relation of the appropriate
type, there is an extension which is an object at the same level as the
Diversity relation. So if we have a condition that determines such a plurality,
then there is an extension. And this holds also for properties, although many
properties do not have extensions.
Thus, although an axiom of extensionality does not hold for the
hierarchy, which is to be expected because it is a hierarchy involving properties, we may hold that the hierarchy has a certain “extensional completeness” characterized by the three principles I mentioned – extensional cumulativity, reducibility, and comprehension for properties and extensions.
Although the hierarchy is an ontological hierarchy, nothing prevents us from organizing parts of it in terms of knowability or definability.
Hierarchies such as Russell’s predicative hierarchy (without the axiom of
reducibility) can be viewed as structurings of a part of the ontological hierarchy. One can also try to “reflect” parts of the hierarchy into level 0,
or other levels, by means of inscriptions, or open sentences, or mental constructions, or whatever – as I illustrated in the case of sets as abstract
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structures. And, evidently, one may also try to structure various levels, or
parts of them, in terms of certain specific properties and relations. Something like Goodman’s ontological structuring of appearances (which may
appear at level 0) in mereological terms, or like Brouwer’s structuring of
consciousness (or understanding) in terms of experience and abstraction35 .
In fact, acceptance of the ontological structure does not deny us the use of
any of the interesting work that has been done nominalistically, mentalistically, linguistically, etc. On the contrary, it may help to illuminate such
work in various ways without rejecting its specificity.
One feature that our hierarchy has in common with other similar
hierarchies is that there aren’t absolute properties that are significant at all
levels. Although there are Twoness properties for all levels higher than 1,
there is no absolute Twoness. This is an ontological feature of the hierarchy. We do have notions that are absolute in this sense, however, since we
certainly do know what it is to be Twoness. What is most general is not
necessarily what is most abstract from an ontological point of view. That’s
how we may comprehend and describe various aspects of the hierarchy.
What I mean by ‘notions’ are intersubjective abstractions closely
connected to language and thought. Such a conception is present in Dedekind
when he refers to numbers as a “free creation of the human mind”36 . The
creation is not quite free though, because it depends on the aprehension of
ontological structures which Dedekind has characterized up to isomorphism.
If one introduces the pure set structures thorough sets of units, isomorphism,
and abstraction, one would be following essentially Dedekind’s procedure.
But even if the pure set structures are objects in their own right, and we
have also the cardinality properties higher up in the hierarchy, none of these
correspond to our notion of number; which seems to me indeed to be a notion and not a property, although there are many properties closely related
to it. And the same goes for the notions of one, two, three, etc.; they are
simply too general to be properties. At the same time, these notions do not
seem to me to be really good candidates for an ontological interpretation of
mathematical discourse. I see Plato’s dilemma concerning the proper treatment of the objects of mathematics as arising from this complex mixture of
universality and particularity.
A similar problem came up in intuitionistic mathematics. Brouwer’s
conception of a purely alinguistic mental mathematical activity may seem
rather problematic – although perhaps a mathematical structuring and
aprehension of the understanding may not – and it gave way to an intersubjective conception as in Heyting. Unfortunately, this intersubjective
conception has been formulated in ways that do not seem to do justice to
Brouwer’s original ideas37 .
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The real problem comes when we try to objectify the content of
these notions as bona fide independently given entities. Why can’t we abstract from Conan Doyle’s construct Sherlock Holmes to an intersubjective
Sherlock Holmes and to an objective Sherlock Holmes? We definitely do
the first step, but we generally refuse the second. Yet, why not? I think
that there is an important distinction between our intersubjective notion
of Napoleon and our intersubjective notion of Sherlock Holmes in that the
former is ultimately rooted in reality whereas the latter is not – at least
not in the same way. Which doesn’t mean that it is the “content” of the
intersubjective notion that is real and objective in that sense. Similarly for
mathematical and logical notions38 .
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Notes
1. Although the distinction of levels is one of Frege’s most important insights, and
is central both to his analysis of quantification and to his analysis of number, he
never developed this classification in a systematic way. It is somewhat misleading
to talk literally of a hierarchy, therefore, because Frege did not develop the hierarchy as such – in The Basic Laws of Arithmetic he basically restricts himself to
the first three levels. The development was begun by Russell in “Mathematical
Logic as Based on the Theory of Types” and in Principia Mathematica, building
on his earlier analysis in The Principles of Mathematics (Chapter 10 and appendixes A and B). Still, the fundamental idea is expressed by Frege quite clearly
already in The Foundations of Arithmetic and is developed further in “Function
and Concept”. An interesting passage in the former is in p. 65:
It would also be wrong to deny that existence and oneness can ever
themselves be component characteristics of a concept. What is true
is only that they are not components of those particular concepts to
which language may tempt us to ascribe them. If, for example, we
collect under a single concept all concepts under which there falls
only one object, then oneness is a component characteristic of this
new concept. Under it would fall, for example, the concept “moon
of the Earth”, though not the actual heavenly body called by this
name. In this way we can make one concept fall under another higher
or, so to say, second order concept. This relationship, however,
should not be confused with the subordination of species to genus.
The distinction between concepts and objects is also a main theme in
The Foundations of Arithmetic which finds a more definite expression in “Function
and Concept” – see Chapter 8 note 17. As to the distinction between functions
and objects, Frege says (“Function and Concept”, pp. 6-7):
I am concerned to show that the argument does not belong with the
function, but goes together with the function to make up a complete
whole; for the function by itself must be called incomplete, in need
of supplementation, or ‘unsaturated’. And in this respect functions
differ fundamentally from numbers. Since such is the essence of the
function, we can explain why, on the one hand, we recognize the same
function in ‘2.13 + 1’ and ‘2.23 + 2’, even though these expressions
stand for different numbers, whereas, on the other hand, we do not
find one and the same function in ‘2.13 + 1’ and ‘4-1’ in spite of their
equal numerical values. Moreover, we now see how people are easily
led to regard the form of the expression as what is essential to the
function. We recognize the function in the expression by imagining
the latter as split up, and the possibility of thus splitting it up is
suggested by its structure.
In The Basic Laws he says (p. 6):
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The function is completed by the argument; what it becomes on
completion I call the value of the function for the argument. . . .
Thus the argument is not to be counted as part of the function, but
serves to complete the function, which in itself is unsaturated. In the
sequel, where use is made of an expression like “the function Φ(ξ)”,
it is always to be observed that “ξ” contributes to the designation
of the function only so far as it renders recognizable the argumentplaces, but not in such a way that the essence of the function is
altered if some sign is substituted for “ξ”.
And a little later (p. 7) he characterizes objects by contrast with functions:
Objects stand opposed to functions. Accordingly I count as objects everything that is not a function, for example, numbers, truthvalues, and the courses-of-values to be introduced below. The names
of objects – the proper names – therefore carry no argument places;
they are saturated, like the objects themselves.
2. The Foundations of Arithmetic, p. 80 note and p. 117. The passage in p. 117
is the following:
One doubt, however, still remained, which was this. A recognition
statement must always have a sense. . . . We were thus led to give
the definition:
The Number which belongs to the concept F is the extension of the
concept “concept equal to the concept F”, where a concept F is
called equal to a concept G if there exists the possibility of one-one
correlation referred to above.
In this definition the sense of the expression “extension of a concept”
is assumed to be known. This way of getting over the difficulty
cannot be expected to meet with universal approval, and many will
prefer other methods of removing the doubt in question. I attach no
decisive importance even to bringing in the extensions of concepts
at all.
It should be noticed, though, that this somewhat detached attitude toward extensions gives way to a rather different attitude in The Basic Laws (p. 14):
a−− Φ(a) = Ψ(a), is the True, then by our earlier stipulation
If −−∪
(§3) we can also say that the function Φ(ξ) has the same courseof-values as the function Ψ(ξ); i.e., we can transform the generality
of an identity into an identity of courses-of-values and vice versa.
This possibility must be regarded as a law of logic, a law that is
invariably employed, even if tacitly, whenever discourse is carried on
about the extensions of concepts. The whole Leibniz-Boole calculus
of logic rests upon it. One might perhaps regard this transformation
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as unimportant or even as dispensable. As against this, I recall the
fact that in my Grundlagen der Arithmetik I defined a Number as the
extension of a concept, and indicated then that negative, irrational,
in short all numbers were to be defined as extensions of concepts.
And a little later (pp. 15-16):
The introduction of a notation for courses-of-values seems to me to
be one of the most important supplementations that I have made
of my Begriffsschrift since my first publication on this subject. By
introducing it we also extend the domain of arguments of any function.
3. In The Foundations of Arithmetic (p. xxii) Frege distinguishes ideas (or representations) from concepts and from objects, although it appears (p. 72) that
the distinction he wants to draw may be between objective objects [objektives
Gegenstand] and subjective objects.
4. I shall often refer to states of affairs as higher level objects in what follows.
Since Frege characterizes objects by contrast with functions, states of affairs would
literally be objects in his sense, but placing them at higher levels in the hierarchy
makes a significant difference. In other words, states of affairs, level 0 objects and
properties are three different categories.
5. The image in terms of saturation or combination is only helpful as an image, not
as a definition. It is somewhat like the image of sets as collections or aggregates; it
may help to visualize sets but it does not characterize them in any way. Moreover,
if we are not careful, this sort of imagery may lead us astray, as can be the case
with the image of the empty set as an empty collection, or as an empty bag.
6. It is debatable whether Frege derived the conception of saturation and unsaturation from language, and he certainly did not consider language a good guide to
ontological and logical distinctions, but in any case he does draw a close connection between an ontological distinction and a linguistic distinction. The last three
quotations in note 1 illustrate this, and the following passage from “Comments
on Sense and Meaning” summarizes well his view on various subjects I mentioned
in the text (pp. 119-120):
The name of a function is accompanied by empty places (at least
one) where the argument is to go; this is usually indicated by the
letter ‘x’ which fills the empty places in question. But the argument
is not to be counted as belonging to the function, and so the letter
‘x’ is not to be counted as belonging to the name of the function
either. Consequently one can always speak of the name of a function as having empty places, since what fills them does not, strictly
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speaking, belong to them. Accordingly I call the function itself unsaturated, or in need of supplementation, because its name has first
to be completed with the sign of an argument if we are to obtain
a meaning that is complete in itself. I call such a meaning an object and, in this case, the value of the function for the argument
that effects the supplementing or saturating. In the cases we first
encounter the argument is itself an object, and it is to these that
we shall mainly confine ourselves here. Now with a concept we have
the special case that the value is always a truth-value. That is to
say, if we complete the name of a concept with a proper name, we
obtain a sentence whose sense is a thought; and this sentence has a
truth value as its meaning. To acknowledge this meaning as that of
the True (as the True) is to judge that the object which is taken as
the argument falls under the concept. What in the case of a function is called unsaturatedness, we may, in the case of a concept, call
its predicative nature.* This comes out even in the cases in which
we speak of a subject-concept (‘All equilateral triangles are equiangular’ means ‘If anything is an equilateral triangle, then it is an
equiangular triangle’.)
Such being the essence of a concept, there is now a great obstacle
in the way of expressing ourselves correctly and making ourselves
understood. If I want to speak of a concept, language, with an
almost irresistible force, compels me to use an inappropriate expression which obscures – I might almost say falsifies – the thought.
One would assume, on the basis of its analogy with other expressions, that if I say ‘the concept equilateral triangle’ I am designating
a concept, just as I am of course naming a planet if I say ‘the planet
Neptune’. But this is not the case; for we do not have anything
with a predicative nature. Hence the meaning of the expression ‘the
concept equilateral triangle’ (if there is one in this case) is an object.
We cannot avoid words like ‘the concept’, but where we use them
we must always bear their inappropriateness in mind.
[Note *] The words ‘unsaturated’ and ‘predicative’ seem more suited
to the sense than the meaning; still there must be something on the
part of the meaning that corresponds to this, and I know of no better
words. Cf. Wundt’s Logik.
(In this passage ‘meaning’ is a translation of ‘Bedeutung’.)
7. Russell’s initial distinction of logical types in The Principles of Mathematics
combines the idea that each propositional function should have a range of significance which consists of certain “types” of entities with the idea of a hierarchic
organization of such ranges. Russell does not discuss types specifically in relation
to properties but in relation to propositional functions and to classes. When he
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introduces the notion of type in connection with his paradox in Chapter 10 of The
Principles of Mathematics he sums up the idea as follows (p. 107):
Finally, in the present chapter, we examined the contradiction resulting from the apparent fact that, if w be the class of all classes
which as single terms are not members of themselves as many, then
w as one can be proved both to be and not to be a member of itself as many. The solution suggested was that it is necessary to
distinguish various types of objects, namely terms, classes of terms,
classes of classes, classes of couples of terms, and so on; and that
a propositional function φx in general requires, if it is to have any
meaning, that x should belong to some one type. Thus xǫx was held
to be meaningless, because ǫ requires that the relatum should be a
class composed of objects which are of the type of the referent.
Russell comes back to this question in connection with several alternative views
of classes in Appendix A (pp. 515-518) and his hierarchization is essentially a
hierarchization of classes as many (p. 518):
According to the view here advocated, it will be necessary, with
every variable, to indicate whether its field of significance is terms,
classes, classes of classes, and so on. . . . We shall have to distinguish
also among relations according to the types to which their domains
and converse domains belong; also variables whose fields include
relations, these being understood as classes of couples, will not as
a rule include anything else, and relations between relations will be
different in type from relations between terms. This seems to give
the truth – though in a thoroughly extensional form – underlying
Frege’s distinction between terms and the various kinds of functions.
In Appendix B he says (p. 523):
Every propositional function φ(x) – so it is contended – has, in
addition to its range of truth, a range of significance, i.e. a range
within which x must lie if φ(x) is to be a proposition at all, whether
true or false. This is the first point in the theory of types; the second
point is that ranges of significance form types, i.e. if x belongs to
the range of significance of φ(x), then there is a class of objects, the
type of x, all of which must also belong to the range of significance
of φ(x), however φ may be varied; and the range of significance is
always either a single type or a sum of several whole types.
And he goes on to characterize individuals as follows:
A term or individual is any object which is not a range. This is
the lowest type of object. If such an object . . . occurs in a proposition, any other individual may always be substituted without loss
of significance. What we called . . . the class as one, is an individual,
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provided its members are individuals: the objects of daily life, persons, tables, chairs, apples, etc., are classes as one. (A person is a
class of psychical existents, the others are classes of material points,
with perhaps some reference to secondary qualities.) These objects,
therefore are of the same type as simple individuals. It would seem
that all objects designated by single words, whether things or concepts, are of this type. Thus e.g. the relations that occur in actual
relational propositions are of the same type as things, though relations in extension, which are what Symbolic Logic employs, are of a
different type. (The intensional relations which occur in ordinary relational propositions are not determinate when their extensions are
given, but the extensional relations of Symbolic Logic are classes of
couples.)
8. See Russell “Mathematical Logic as Based on the Theory of Types” §V.
9. This terminology derives from Quine – see “Intensions Revisited”, p. 113.
10. “On the Individuation of Attributes”, p. 102. On the universality of Identity
Quine says (Philosophy of Logic, p. 62):
Another respect in which identity theory seems more like logic than
mathematics is universality: it treats of all objects impartially. Any
theory can indeed likewise be formulated with general variables,
ranging over everything, but still the only values of the variables
that matter to number theory, for instance, or set theory, are the
numbers and the sets; whereas identity theory knows no preferences.
I think that the remark on number theory and set theory is misleading, though,
because as incompleteness and analytic number theory show one can’t really do
number theory just quantifying over numbers.
11. In “Identity, Ostension, and Hypostasis” Quine comments (pp. 74-75):
The accessibility of a term to identity contexts was urged by Frege
. . . as the standard by which to judge whether that term is being
used as a name. Whether or not a term is being used as naming
an entity is to be decided, in any given context, by whether or not
the term is viewed as subject in that context to the algorithm of
identity: the law of putting equals for equals.
It is not to be supposed that this doctrine of Frege’s is connected
with a repudiation of abstract entities. On the contrary, we remain
free to admit names of abstract entities; and, according to Frege’s
criterion, such admission will consist precisely in admitting abstract
terms to identify contexts subject to the regular laws of identity.
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Frege himself, incidentally, was rather a Platonist in his own philosophy.
Frege held that Identity is specifically a relation of objects and not
literally of concepts. Thus following the first passage I quoted in note 6 he says
(Op. Cit., p. 120):
. . . the relation of equality, by which I understand complete coincidence, identity, can only be thought of as holding for objects, not
concepts. If we say ‘The meaning of the word “conic section” is the
same as that of the concept-word “curve of the second degree”’ or
‘The concept conic section coincides with the concept curve of the
second degree’, the words ‘meaning of the concept-word “conic section”’ are the name of an object, not of a concept; for their nature
is not predicative, they are not unsaturated, they cannot be used
with the indefinite article. The same goes for the words ‘the concept conic section’. But although the relation of equality can only
be thought of as holding for objects, there is an analogous relation
for concepts. Since this is a relation between concepts I call it a
second level relation, whereas the former relation I call a first level
relation. We say that an object a is equal to an object b (in the sense
of completely coinciding with it) if a falls under every concept under
which b falls, and conversely. We obtain something corresponding
to this for concepts if we switch the roles of concept and object. We
could then say that the relation we had in mind above holds between
the concept Φ and the concept X, if every object that falls under Φ
also falls under X, and conversely. Of course in saying this we have
again been unable to avoid using the expressions ‘the concept Φ’,
‘the concept X’, which again obscures the real sense.
12. The interpretation of (9), (10) and (12) also depends on the interpretation of
the range of the variable ‘Z’ – which may be restricted only to properties that are
“intrinsic” in some sense. For a discussion of this and other questions in connection
with Leibniz’ formulations see Mates The Philosophy of Leibniz, Chapter 7.
13. This presupposes that every object can be named or described in some way
– or that every object is the referent of some name or description. One could
formulate the same idea more generally for senses as manners of presentation, in
which case the assumption is that for every object there is at least one sense that
presents (or individuates) that object. Whether these are reasonable assumptions
depends partly on our conception of objects. One may hold that geometric points
are legitimate objects which can neither be named nor presented – except in a
relative sense.
14. Quine remarks (Word and Object, pp. 116-117):
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Similar confusion of sign and object is evident in Leibniz where he
explains identity as a relation between the named object and itself:
“Eadem sunt quorum unum potest substitui alteri, salva veritate.”
Frege at some point took a similar line. . . . Identity evidently invites
confusion between sign and object in men who would not make the
confusion in other contexts.
I am not suggesting that (12S) involves a straightforward confusion between sign and object – as when it is said that 3 is not identical to 2+1 on the
grounds that the signs ‘3’ and ‘2+1’ are different – but between a relation that
signs have when they have the same denotation and the relation of identity for
objects. This is the problem with section 8 of Begriffsschrift when Frege characterizes identity as a relation between names that have the same content. This
characterization was a deliberate choice motivated by similar considerations to
those Frege used to justify the notion of sense at the beginning of “On Sense
and Reference” – as he remarks himself in p. 25 of that paper. Nevertheless
I agree with Quine that there may have been a confusion of sign and object in
Begriffsschrift and that there may be a similar confusion in The Foundations of
Arithmetic.
In his later work Frege seems to use (12) as the definition of identity, as
when he says in the passage I quoted in note 10:
We say that an object a is equal to an object b . . . if a falls under
every concept under which b falls, and conversely.
But the proper interpretation of this formulation is not completely unambiguous
either, especially if one thinks of a and b as standing for names – see notes 15 and
16.
15. One could argue that there is a sense in which a criterion of identity such as
(12) yields a criterion of sameness of denotation along the lines of (15), namely:
(15′ ) a = b ⇔ ∀Z(Za ⇔ Zb),
where ‘Z’ ranges over properties of objects, not contexts. But if Number is such
a property, then the truth of the right hand side of (15′ ) presupposes that both
names denote and that either both denote a number or both denote objects that
aren’t numbers. Thus (15′ ) does not help much in determining whether Number
is everywhere defined for objects or whether terms of the form ‘the number which
belongs to the concept F’ denote objects.
Quine’s suggestion in the first paragraph of the passage I quoted in
note 11 also doesn’t seem to be very helpful in connection with Frege’s problem.
Take the term ‘the number which belongs to the concept ‘is a moon of Mars’ ’. I
presume that to say that this term is subject to the algorithm of identity in a given
context is to say that if, for example, Julius Caesar = the number which belongs
to the concept ‘is a moon of Mars’, then ‘Julius Caesar’ and ‘the number which
belongs to the concept ‘is a moon of Mars’ ’ are intersubstitutable in that context
preserving truth value. But Quine’s formulation gives us no way of determining
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whether the identity is defined – i.e., whether the term in question is indeed the
name of an object. If it isn’t, then the identity context itself is truth-valueless
and such substitutions of alleged equals may result in a truth-valueless context.
Of course, if a person does perform such substitutions in a given context, then
it may be reasonable to conclude that the term is meant to refer in that context
by that person. Even if we agree to this, however, it doesn’t follow that the term
actually has a reference.
Similar problems may arise from Frege’s formulation at the end of the
previous note. (12) is universally quantified and to obtain particular instances of
it using names, as in (15′ ), one must presuppose that the names denote. Moreover,
since in The Basic Laws Frege characterizes second order quantification substitutionally (see Chapter 8 note 31), it is not really clear how one should interpret his
formulation. That Frege’s conception in The Basic Laws may have been somewhat confused can also be seen from the following passage when he introduces
basic law III (pp. 35-36):
If Γ = ∆ is the True, then
f —— f(Γ)
—∪
|
— f(∆)
is also the True; i.e., if Γ is the same as ∆, then Γ falls under every
concept under which ∆ falls; or, as we may also say: then every
statement that holds for ∆ holds also for Γ. But also conversely; if
Γ = ∆ is the False, then not every statement that holds for ∆ also
holds for Γ, i.e., then
f —— f(Γ)
—∪
|
— f(∆)
is the false. For example, Γ does not fall under the concept ξ = ∆,
under which ∆ does fall. Thus, Γ = ∆ is always the same truth-value
as
f —— f(Γ)
—∪
|
— f(∆)
Consequently, [the truth-value]
f —— f(Γ)
—∪
|
— f(∆)
falls under every concept under which [the truth- value] Γ = ∆ falls.
Thus,
|−−
− f(a)
| −g −∪−
−|− f(b)
|
|
−−−g(a = b)
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III.
16. In my view the proper way to express (12) is as a relation between two
properties, namely Identity and [∀Z(Zx ⇔ Zy)](x, y). One can formulate this in
my notation as
(12#) [[∀x∀y(Vxy ⇔ Wxy)](V, W )]([x = y](x, y), [∀Z(Zx ⇔ Zy)](x, y)),
where [∀x∀y(Vxy⇔ Wxy)](V,W) is the relation Mutual Subordination. To accept
(12) as a definition in the strict sense is to hold that Identity for objects is the
specific property [∀Z(Zx ⇔ Zy)](x, y).
To postulate this property as Identity for objects seems to presuppose
identity, however, for what we are postulating is that objects are the same when
they have the same level 1 properties, or have all level 1 properties in common.
Perhaps one can argue directly as follows. Suppose that F and G are properties
of objects, that a and b are objects which have all other properties in common
and that a has F and b has G. It would seem that whether a and b are the same
object might depend on whether F and G are the same property. Of course, for
the definition it would be enough that a has G and b has F, but this might depend
in turn on whether a and b are the same object.
In fact, I think that Frege’s argument for the undefinability of truth
applies even with more force to the undefinability of identity, and that his own
attempt (in the passage I quoted in note 11) to characterize the identity of objects
in terms of concepts and the identity of concepts in terms of objects is rather problematic if the definition of identity for objects is interpreted as (12). It amounts
to defining identity for objects by means of the condition
(a) ∀Z(Zx ⇔ Zy),
where the variable ‘Z’ ranges over concepts of objects, and identity for such concepts by means of the condition
(b) ∀x(V x ⇔ W x).
If we formulate this linguistically, there seems to be an obvious circularity: objects
are the same when they fall under the same concepts, and concepts are the same
when they apply to the same objects. If we distinguish levels (and properties)
more carefully, then the circularity is not so obvious, because we may say that
Identity for objects is the level 1 property
(c) [∀Z(Zx ⇔ Zy)](x, y),
and Identity for level 1 properties of objects is the level 2 property
(d) [∀x(V x ⇔ W x)](V, W ).
But if we follow Frege’s procedure, then we should define identity for level 2 properties in terms of the level 1 properties to which they apply, and similarly for
higher levels and for properties of more than one argument. Consider now the
level 2 property
(e) [∀Y(YV ⇔ YW )](V, W ).
Whether (d) and (e) are the same property depends on whether they apply to the
same level 1 properties, which depends in turn on whether two level 1 unary properties apply to the same objects if and only if they have the same level 2 unary
properties in common. Thus the procedure again raises questions of circularity.
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If we consider sets of objects instead of level 1 properties, then these
circularities are quite apparent. Suppose that we define identity for objects as
(f) u = v ⇔ ∀x(u ∈ x ⇔ v ∈ x),
and define identity for sets of objects as
(g) x = y ⇔ ∀u(u ∈ x ⇔ u ∈ y).
Then, if u and v are objects, whether u = v depends on whether u ∈ {v} – i.e.,
on whether {u} = {v} – which depends in turn on whether u = v.
In my view (12) is at best a definition of identity for objects relative
to properties, for which the definiteness of identity and diversity must either be
assumed, or characterized in terms of generalizations of (12) for properties of each
level, or characterized in some other way.
I think that identity and diversity are fundamental logical (and metaphysical) notions that cannot be explained or defined in terms of anything more
ultimate. Although there are notions such as uniqueness that may be more primitive in some respects, such notions are inseparably connected to identity and
diversity. This is not to say that I consider (12) and its generalizations an inadequate characterization of Identity and Diversity. On the contrary, I think that it
is a very good characterization and that the relation between level 1 Identity and
the property (c) is not merely coextensionality but necessary coextensionality. I
don’t want to get into the question of necessity right now, however, and I also
want to have a certain latitude in my discussion of some general issues concerning
objects, properties, logical properties, identity, etc.
17. Gödel remarks that this already involves an abstraction of some sort (“What
is Cantor’s Continuum Problem?”, pp. 271-272):
That something besides the sensations actually is immediately given
follows (independently of mathematics) from the fact that even our
ideas referring to physical objects contain constituents qualitatively
different from sensations or mere combinations of sensations, e.g.,
the idea of object itself, whereas, on the other hand, by our thinking
we cannot create any qualitatively new elements, but only reproduce
and combine those that are already given. Evidently the “given”
underlying mathematics is closely related to the abstract elements
contained in our empirical ideas.
18. This characterization is related to some remarks of Frege in The Foundations
of Arithmetic (p. 46):
We cannot succeed in making different things identical by dint of
operations with concepts. But even if we did, we should then no
longer have things in the plural, but only one thing; for, as Descartes
says, the number (or better, the plurality) in things arises from
their diversity. . . .W. S. Jevons makes this point with unusual force:
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‘Number is but another name for diversity. Exact identity is unity,
and with difference arises plurality’.
My characterization of sets ties very closely the set that “collects” a plurality of
objects with the number that “belongs” to that plurality. It doesn’t follow from
this that I am disagreeing with Frege that what gives the “unit” to the plurality
is a concept, which is why I prefer to think of this conception as a conception of
extensions rather than a conception of sets in a more abstract sense.
Another approach to sets as states of affairs is developed by Armstrong
in A Combinatorial Theory of Possibility (Chapter 9 §IV) and “Classes are States
of Affairs”.
19. Goodman may not approve of my Diversity relations either, but my characterization is compatible with his idea that there must be a difference of content.
See “A World of Individuals”, pp. 158-162, and The Structure of Appearance, pp.
34-37. In the latter Goodman remarks (p. 36):
. . . it is clear that two classes, however defined, are indistinguishable
if they have the same members; classes are in a sense distinguished
only by what is comprised within them. But the nominalist goes still
a step further. If no two distinct entities whatever have the same
content, then a class (e.g., that of the counties of Utah) is different
neither from the single individual (the whole state of Utah) that
exactly contains its members nor from any other class (e.g., that
of acres of Utah) whose members exactly exhaust this same whole.
The platonist may distinguish these entities by venturing into a new
dimension of Pure Form, but the nominalist recognizes no distinction
of entities without a distinction of content.
Although the Diversity relations may be characterized as pure forms in a rather
literal sense, Goodman’s claims about the sets in question only follow if by ‘content’ he means ‘material content’. But this is not the point of Goodman’s principle
about content, and his notion of individual is completely independent of considerations of materiality. Thus, in “A World of Individuals” he says (pp. 159-160):
Nominalism describes the world as composed of individuals. To
explain nominalism we need to explain not what individuals are but
rather what constitutes describing the world as composed of them.
So to describe the world is to describe it as made up of entities no
two of which break down into exactly the same entities.
This is clearly the case for sets as I characterized them, for no two different sets
break down into exactly the same entities, which is the point I am making in the
text.
20. Formulating axioms for these pure set structures involves selecting appropriate
primitives besides membership. I mention two approaches below. I shall not
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develop the mathematical part of the general hierarchy in this book. This will
be done elsewhere and will include both the mathematical properties and various
notions of set.
21. We may be intellectually handicapped in some way that makes it difficult
for us to treat abstract structures as we properly should. In Republic 510b Plato
says:
Then consider next how the intelligible part of the line is to be divided. In one section the mind uses the originals of the visible world
in their turn as images, and has to base its inquiries on assumptions
and proceed from them to its conclusions instead of going back to
first principles: in the other it proceeds from assumption back to
self-sufficient first principle, making no use of the images employed
in the other section, but pursuing its inquiry solely by means of
Forms.
22. The Foundations of Arithmetic, sections 29-33.
23. That it is artificial follows from the fact, mentioned earlier, that it forces us
to conceive of each pure cardinality structure as involving additional structures
which get more and more complex as the cardinalities go into the transfinite.
Moreover, as Benacerraf argues in “What Numbers Could Not Be”, the multiplicity of structurings for each pure cardinality forces us to select a particular
sequence of them as a representative of the pure cardinalities. Undoubtedly, von
Neumann’s definition of ordinals was a very clever choice which represents beautifully the structure of ordinals and which allows the representation of cardinals
as special kinds of ordinals. I think, in fact, that it is an excellent representation
of ordinal numbers and cardinal numbers as sets, though it is not so good as a
representation of well-orderings or of pure cardinalities generally. By adding the
other structures we lose nothing that we had before, but we get a more complete
and accurate picture of set structures generally.
24. The Foundations of Arithmetic, sections 34-39. In Republic 525d Plato already
discusses some of the objections that Frege raises against using units in arithmetic:
“As we have just said, it draws the mind upwards and forces it to
argue about pure numbers, and will not be put off by attempts to
confine the argument to collections of visible or tangible objects.
You must know how the experts in the subject, if one tries to argue
that the unit is divisible, won’t have it, but make you look absurd
by multiplying it if you try to divide it, to make sure that their unit
is never shown to contain a multiplicity of parts.”
“Yes, that’s quite true.”
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“What do you think they would say, Glaucon, if one were to say to
them, ‘This is very extraordinary – what are these numbers you are
arguing about, in which you claim that every unit is exactly equal
to every other, and at the same time not divisible into parts?’ What
do you think their answer would be to that?”
“I suppose they would say that the numbers they mean can be apprehended by thought, but that there is no other way of grasping
them.”
25. The problem is that there is a very strong pull toward objectification, but
to satisfy this by appealing to real objects seems problematic. Plato has some
observations about this strong pull in Republic 511a:
“This sort of reality I described as intelligible, but said that the
mind was forced to use assumptions in investigating it, and because
it was unable to ascend from these assumptions to a first principle,
used as illustrations objects which in turn have their images in a
lower plane, in comparison with which they are themselves thought
to have superior clarity and value.”
“I understand”, he said. “You are referring to what happens in
geometry and kindred sciences.”
“Then when I speak of the other sections of the intelligible part of the
line you will understand that I mean that which reason understands
directly by the power of pure thought; it treats assumptions not as
principles, but as assumptions in the true sense, that is, as starting
points and steps in the ascent to the universal, self-sufficient first
principle; when it has reached that principle it can again descend, by
keeping to the consequences that follow from it, to a final conclusion.
The whole procedure involves nothing in the sensible world, but
deals throughout with Forms and finishes with Forms.”
I am not excluding the possibility of dealing directly with the forms, but the usual
conception in mathematics is indeed in terms of units. If we start with the natural
numbers as given and build our sets from there, then the natural numbers are our
units. And as I mentioned before, even in pure set theory one conceives of ∅ as a
(and only) unit.
26. The axiom of units can be formulated as
(AU) For every set x there is a set of units y and a 1-1 function f whose domain
is x and whose range is y.
More formally,
(AU) ∀x(Sx ⇒ ∃y(Sy & ∀z(zǫy ⇒ U z) & ∃f (f : x → y & f is 1-1 & onto))).
Also in the formulation of the axioms of ZFC one must add a qualification to sets
in various places. Thus, the axiom of extensionality must be formulated as
(E) ∀x∀y((Sx & Sy & ∀z(z ∈ x ⇔ z ∈ y)) ⇒ x = y),
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and the empty set ∅ must be introduced as a set characterized by the condition
that nothing is an element of it – for although all units satisfy this condition, the
empty set is now a set rather than a unit.
27. An interesting book on Plato’s views on mathematics is Wedberg Plato’s
Philosophy of Mathematics. He summarizes his interpretation of Plato’s view of
numbers as follows (p. 65):
The Mathematical Numbers are characterized by the following properties:
(1) They are made up of certain ideal “units” or “1’s”. The Mathematical Number N is a set of N such units: 2 is a set of two, 3 a set
of three, and so on.
(2) Of such ideal units, or 1’s, there exists an infinite supply.
(3) There is no difference between the ideal units: two such units
are completely indistinguishable.
(4) An ideal unit does not contain any plurality of parts, or constituents, or characteristics: from whatever point of view we consider
such a unit, it is One and One only.
(5) Of each Mathematical Number there are infinitely many copies.
From the infinite supply of ideal units we may choose N units in
infinitely many ways, and every choice gives us a representation of
the Mathematical Number N.
(6) The elementary arithmetical notions are simple set-theoretical
notions.
(7) Mathematical Numbers are the numbers studied by arithmetic.
It is for them, and only for them, that the concepts of arithmetic
are defined.
Besides these Mathematical Numbers there are also Ideal Numbers –
what I have been calling ‘mathematical properties’ – and in pp. 65-66 Wedberg
gives a sketch of their features:
The Ideal Numbers are characterized by the following properties:
(1) They are Ideas, viz. the Ideas of Twoness, Threeness, and so on.
(2) As Ideas the Ideal Numbers are simple entities.
(3) In particular, they are not sets of units like the Mathematical
Numbers.
(4) The notions of arithmetic, which – as already mentioned – are of
a set-theoretical kind, are not defined for the Ideal Numbers. The
equation, 2+3 = 5, e.g., says only that the addition of the Mathematical Numbers 2 and 3 gives raise to the Mathematical Number 5;
it says nothing of the Ideal Numbers, for which arithmetical addition
is not defined. Likewise, the arithmetical statement, 2 < 5, holds
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only for the Mathematical Numbers 2 and 5. For Ideal Numbers the
relation < is not defined.
(5) However, there is a relation of “priority” among the Ideal Numbers, by which they are ordered in a series that runs parallel to the
series of Mathematical Numbers, ordered according to size: 2, 3, . . .
(6) The study of the Ideal Numbers belongs to the general theory
of Ideas, Dialectic.
28. “Russell’s Mathematical Logic”, p. 222. I discuss Gödel’s ideas, and their
motivation, in the next chapter.
29. See Mirimanoff “Les Antinomies de Russell et de Burali-Forti et le Probléme
Fondamental de la Thèorie des Ensembles”, pp. 40-41.
30. Gödel says (Op. Cit., pp. 221-222):
Nor is it self-contradictory that a proper part should be identical (not
merely equal) to the whole, as is seen in the case of structures in the
abstract sense. The structure of the series of integers, e.g., contains
itself as a proper part and it is easily seen that there exist also
structures containing infinitely many different parts, each containing
the whole structure as a part.
It is very difficult to literally make sense of such claims because the very notion
of proper part seems to involve the notion of difference, though as I mentioned
before the difficulty may lie in our conception of structures.
The notation I use in the text is misleading, because units cannot be
named and one cannot distinguish two “isomorphic” set structures notationally –
as, e.g., with the notation ‘{u1 , u2 }’ and ‘{u3 , u4 }’. What one can show is that
there are different sets of units of the same cardinality. Thus the claim that the
structure {, , , . . .} involves itself as a proper part in infinitely many ways amounts
to the claim that for any denumerable set of units – i.e., a set of units that can be
put into 1-1 correspondence with such a set as {∅, {∅}, {∅, {∅}}, . . .} – there are
infinitely many denumerable sets of units that are proper subsets of it. If units
are postulated as bona fide level 0 objects that cannot be distinguished by means
of their properties, then such sets also cannot be distinguished by means of their
properties. This means that the level 3 “Identity” relation characterized by (12)
does not coincide with level 1 Identity. If units are a fictional device, then the
mathematical theory itself is a fictional discourse and its predicates do not refer to
properties. But it doesn’t follow that this fictional discourse is entirely fictional,
because although the units may be fictional the set structures themselves may
be real. The fictional character of the discourse is just a way to get around a
difficulty in talking about real entities.
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31. A similar idea, in terms of properties and intensional abstraction, is developed
by Bealer in Quality and Concept pp. 115-118.
32. In Plato’s Parmenides 132a, Parmenides says to Socrates:
I imagine your ground for believing in a single form in each case is
this. When it seems to you that a number of things are large, there
seems, I suppose, to be a certain single character which is the same
when you look at them all; hence you think that largeness is a single
thing.
33. I shall use ‘property’ in this broader sense, so that properties are not extensional but have many of the characteristics usually attributed to sets. In particular, I hold that analogues of the usual axioms of ZFC, except for extensionality,
hold for properties in this sense. The analogue of the power-set axiom, for instance, says that for any property X of level α there is a property Y of level α + 1
such that Y applies to a property Z of the same type as X if, and only if, Z is
extensionally subordinate to X – this “power-property” Y need not be unique,
however.
34. Quine, for example, sees various (re-)formulations of set theory as a sort of ad
hoc patching up for the loss of the naive conception, and tends to blame Platonism
for it. In “Logic and the Reification of Universals” he says (p. 127):
The platonist can stomach anything short of contradiction; and
when contradiction does appear, he is content to remove it with
an ad hoc restriction.
This is actually quite far from the truth, as can be seen from looking at the work
of such famous Platonists as Plato and Gödel, for instance. Quine, on the other
hand, formulated some systems of set theory like New Foundations (NF) and
Mathematical Logic (ML) which are still today intuitively unintelligible. Why?
Because they are based on the purely formal device of stratification which has
one purpose and one purpose only; to preserve as much as possible of the general
comprehension principle while preventing the known contradictions. This is not
to say that these systems are uninteresting, but they are certainly motivated by
formal ad hoc considerations.
35. See Goodman The Structure of Appearance.
36. He says (Essays in the Theory of Numbers, p. 68):
If in the consideration of a simply infinite system N set in order by
a transformation ϕ we entirely neglect the special character of the
elements, simply retaining their distinguishability and taking into
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account only the relations to one another in which they are placed
by the order-setting transformation ϕ, then these elements are called
natural numbers or ordinal numbers, or simply numbers, and the
base-element 1 is called the base-number of the number-series N.
With reference to this freeing the elements from every other content
(abstraction) we are justified in calling numbers a free creation of
the human mind.
37. Heyting tries hard, in the figure of INT., to defend Brouwer’s views, but
he makes major concessions to formalism. To a question as to the meaning of
equality for natural numbers, he replies (Intuitionism: An Introduction, p. 15):
Indeed this point needs some clarification; it forces me even to revise
somewhat our notion of a natural number. If a natural number
were nothing but the result of a mental construction, it would not
subsist after the act of its construction and it would be impossible
to compare it with another natural number, constructed at another
time and place. It is clear that we cannot solve this problem if we
cling to the idea that mathematics is purely mental. In reality we
fix a natural number, x say, by means of a material representation;
to every entity in the construction of x we associate, e.g., a dot on
paper. This enables us to compare by simple inspection natural
numbers that were constructed at different times.
I would like to see intuitionists of this persuasion pulling out little pieces of paper
from their pockets, with date and time, to make the comparisons. If identity
is not well-defined for the mental constructions as such, then it won’t become
well-defined by this procedure. It is a confusion between identity and sameness of
representation somewhat along the same lines as the confusion between identity
of signs and identity of things. Moreover, the very possibility of such a correlation
depends on the notion of identity.
The intuitionist must also postulate identity and diversity as fundamental logical properties. In fact, the very intuitionist conception of number as arising
by abstraction from a process of division presupposes diversity (and identity) as
primitives. Thus, in “Consciousness, Philosophy, and Mathematics” Brouwer says
(p. 1237):
Mathematics comes into being, when the two-ity created by a move
of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as
basic intuition of mathematics, is left to an unlimited unfolding, . . .
And (p. 1235):
By a move of time a present sensation gives way to another present
sensation in such a way that consciousness retains the former one
as a past sensation, and moreover, through this distinction between
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present and past, recedes from both and from stillness and becomes
mind.
As mind it takes the function of a subject experiencing the present
as well as the past sensation as object. And by reiteration of this
two-ity-phenomenon, the object can extend to a world of sensations
of motley plurality.
It seems quite clear that this presupposes identity and diversity, and that
it may not even be beside the mark to think of the “empty form of the common
substratum of all two-ities” as the (abstract) Diversity relation. I think that one
of the merits of Griss’ approach to intuitionistic mathematics is the recognition of
this primitiveness of identity and diversity – in “Logic of Negationless Intuitionistic
Mathematics” he says (p. 44): “In negationless intuitionistic mathematics the
notion of distinguishability is equally fundamental as the notion of identity.” In
my view this must be so for any version of intuitionistic mathematics, even if one
allows negation in the usual intuitionistic sense.
38. These remarks about notions, or concepts in a traditional interpretation
different from Frege’s, are admittedly rather vague. Russell introduced the idea
of typical ambiguity in a somewhat related sense. I will come back to this issue
later.
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Chapter 10
Identity and Extensionality
Questions of extensionality are intimately tied up to questions of
identity, both when one says that entities of one sort or another are extensional or intensional, and when one says that certain kinds of contexts are
extensional or intensional. The basic criterion for extensionality of contexts
is in terms of substitutivity; for a context is normally said to be extensional
if substitutivity of identicals holds for it. One way to formulate this criterion
is in terms of denotation. Let C(v1 , . . . , vn ) be a context (complex expression) involving variables v1 , . . . , vn that are not quantified in the context,
let e1 , . . . , en and e′1 , . . . , e′n be expressions that are appropriate for substitution in place of v1 , . . . , vn , let C(e1 , . . . , en ) be the result of substituting ei
for one or more occurrences of vi in C(v1 , . . . , vn ), and let C(e′1 , . . . , e′n ) be
obtained by replacing one or more of these occurrences of ei in C(e1 , . . . , en )
by occurrences of e′i (for 1 ≤ i ≤ n). Then
(1) C(v1 , . . . , vn ) is extensional if, and only if, for any e1 , . . . , en and e′1 , . . . ,
e′n , if ei has the same denotation as e′i , then C(e1 , . . . , en ) has the same
denotation as C(e′1 , . . . , e′n ).
If we assume that the notion of denotation is defined not only for
singular terms but also for general terms and sentences, and that having the
same denotation means that either both expressions denote and denote the
same thing or neither denotes, then this is a fairly general formulation of
extensionality for contexts. If we don’t assume that denotation is defined for
general terms and sentences, then we may have to give different formulations
for different categories of expressions. Thus, for n = 1, if e, e′ , C(e) and
C(e′ ) are sentences, then we may substitute ‘has the same truth value as’
for ‘has the same denotation as’; and if e and e′ are singular terms and
C(e) and C(e′ ) are sentences, then we can use a mixed formulation. For
predicates we can use ‘has the same extension as’ or ‘applies to the same
entities as’.
When we say that a certain kind of entities – sets, meanings, properties, etc. – are extensional or intensional, which is quite often, it is not
generally clear how to translate this in terms of substitutivity. Are material objects extensional? Persons? Elementary particles? Since extensional
entities are considered to be better off in terms of identity criteria than
intensional entities, and this has been an important issue in ontological
discussions, I shall consider this question in this chapter especially in connection with sets and properties.
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The paradigm of extensional entities are sets and there are several
senses in which sets are said to be extensional. In one sense, sets are extensional because they obey the axiom of extensionality
(2) ∀x∀y(∀z(z ∈ x ⇔ z ∈ y) ⇒ x = y).
This is often stated informally by saying that if two sets have the same
elements, then they are the same set. One also expresses the same idea
by saying that a set is completely determined by its elements; though the
exact sense of this formulation depends on how we understand ‘completely
determined by its elements’. If by this one means (2), then naturally it’s
the same sense as before; but it need not be. Finally, sets are said to be
extensional in the sense that the manner in which the elements of a set are
specified does not affect the identity of the set. This can be understood
in the sense that if C(x) and C ′ (x) are conditions, including listings, that
specify sets {x : C(x)} and {x : C ′ (x)}, then
(3) ∀x(C(x) ⇔ C ′ (x)) ⇒ {x : C(x)} = {x : C ′ (x)};
which is a special case of (1).
The first sense of extensionality is the standard set-theoretic formulation. The second sense, aside from being a reformulation of the first,
can also be understood as saying that the identity (or nature) of a set is
completely determined by having the elements it has. We can formulate
this idea more specifically as follows.
Suppose that for a certain kind of entities there is a relation S such
that
(4) ∀x∀y(Sxy ⇒ x = y).
We can reasonably say in this case that the relation S determines the identity
of the entities in question, and also that (4) is a criterion of identity for those
entities. One way in which we can generalize (4) is
(5) ∀x∀y(∀z(Rzx ⇔ Rzy) ⇒ x = y),
where the variable ‘z’ may have a different range than the variables ‘x’ and
‘y’. Thus, if we are only considering sets of objects of some sort, and R is
the membership relation, (5) gives us a criterion of identity for such sets in
terms of the objects that do or do not bear the membership relation to them.
And if we go on to consider sets whose elements may be objects of that sort
or sets of such objects, then by allowing ‘z’ to vary over all of those, (5) is a
criterion of identity for sets having them as elements. Therefore, even for a
specific relation R, (5) can be thought of as a schema which depends on the
specification of the range of the variable ‘z’. Evidently, we can generalize
(5) by allowing more complex relations involving many variables which can
have different ranges and which can be quantified in many different ways.
Suppose now that we are considering a cumulative hierarchy of sets
starting with the natural numbers as a basis. We can then formulate (5) as
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a sequence of criteria (5-1), (5-2), . . . , where R is the membership relation,
such that in (5-1) ‘z’ ranges over the natural numbers, in (5-2) ‘z’ ranges
over natural numbers and sets of natural numbers, and so on. Assuming
that identity and diversity are well-defined for the natural numbers, if we
have two conditions on natural numbers which are satisfied by the same
numbers, then they determine the same set of numbers. We can think
of this as following from (3), but if the set abstraction operator is also
introduced gradually subject to the axiom schema
(6) ∀z(z ∈ {x : C(x)} ⇔ C(z)),
then (3) follows from (5-1) and (6-1) via
(7) ∀z(C(z) ⇔ C ′ (z)) ⇒ ∀z(z ∈ {x : C(x)} ⇔ z ∈ {x : C ′ (x)}).
It may seem that this is a very roundabout way of getting to the
axiom of extensionality (2), but this is not so. In (2), as formulated in
abstract set theory, the variable ‘z’ ranges over all sets, and therefore the
determination of identity for any set depends on all sets, no matter how
high up in the hierarchy. Moreover, if there are sets x such that x ∈ x, or
sets x and y such that x ∈ y ∈ x, or other things of the sort, then (2) may
hold for such sets without there being any way in which we can determine
the identity of sets in the sense that I was trying to capture above. For
if by determining the identity of a set through its elements we mean that
the identity of those elements is determined independently of the set, then
clearly we won’t be able to do it if the set can be an element of itself. And
even if we don’t have those odd sets, as is the case in the usual hierarchy
of sets considered in abstract set theory, the ontological determination of
the identity of each set depends on the entire universe of sets. At least
that’s what (2) tells us, and although it is an axiom of identity it may be
misleading to call it an axiom of extensionality1 .
Moreover, an alternative criterion of identity for sets is
(8) ∀x∀y(∀z(x ∈ z ⇔ y ∈ z) ⇒ x = y),
which can also be taken as the basic principle of identity for sets together
with other axioms. Yet (8) by itself does not guarantee (2), just as (2) by
itself does not guarantee (8). But if two sets can have the same elements
and not be elements of the same sets, then it is not clear in what sense
the identity of a set is completely determined by its elements. I.e., if the
identity of sets is characterized by (8), then (2) by itself is not necessarily
a criterion of identity for sets. It seems, therefore, that the question of
extensionality of sets must be considered in broader terms, and that it is
reasonable to look for a more general characterization of extensionality for
entities of a given kind. To motivate such a characterization let’s look now
at some kinds of entities that are not considered to be extensional.
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Consider properties, for instance. Why are properties not extensional? Because different properties may apply to the same things; i.e.,
because the things to which a property applies do not determine its identity. No doubt there is an important similarity between the relation of
things to properties and the relation of things to sets – both deriving from
Plato’s relation of participation between particulars and forms – and the
intensionality of properties is inferred from the contrast mentioned above.
But why should the comparison be made in those terms? If the identity
of a set may depend on all sets, then why can’t the identity of a property
depend on all properties? It is a general criterion of identity for properties
(of a given level) that they have all properties (of higher levels) in common;
which is analogous to (8) as a criterion of identity for sets. The difference is
that we can get (8) as a consequence of (2) plus some other axioms, whereas
there seems to be no natural way to get a criterion of identity for properties
along similar lines.
But what if instead of considering the things to which a property
applies we consider the things to which a property possibly applies? Then
properties may be extensional in somewhat the same sense in which sets are
extensional. In fact, many accounts of properties treat them as extensional
entities in the image of sets, either by characterizing them in terms of their
extensions in possible worlds – which is a way of understanding the ‘possibly
applies’ – or by introducing possible objects, or by introducing a “space” of
possible applications, etc. So: are properties extensional or not?
Take meanings now. Why are meanings not extensional? Here
the comparison with sets is not so obvious because we don’t seem to have
anything like the membership relation relating meanings to things. Most of
the time we talk of meanings in relation to sentences and to other linguistic
entities. Of course, many philosophers conceive the meanings of predicates
as something like properties or concepts, the meanings of names and definite
descriptions as something like Frege’s senses, or as individual concepts, and
the meanings of sentences as something like propositions. If we do this,
then the analogy is clear again, even in the case of sentential meanings.
If the connection between a sentence and reality is the circumstance of its
being true or false, and if this is determined by its meaning, then even if we
don’t view these circumstances as the objects the True and the False, we
can say that the connection to reality does not determine the meaning of
a sentence. And if we actually take Frege’s view, then the analogy is quite
clear because the relation we want is denotation. Two sentential meanings
may denote the same thing without thereby being the same thing. And
similarly for predicate meanings and for name meanings. Moreover, even if
sentences were to denote states of affairs, there are many sentential meanings
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denoting each state of affairs. So, again, we apply the set-theoretic model
of extensionality.
This conception of the intensionality of meanings is related to the
usual classification of contexts as extensional or intensional. Thus, belief
contexts are held to be intensional on the grounds that what’s relevant to
the truth or falsity of such sentential contexts is not merely the denotation of
the terms but (generally) their meaning. Since the denotations of predicates
are conceived as sets, and sets are extensional, the denotation of names
and descriptions are conceived as objects, and objects are extensional, and
the denotation of sentences are conceived as truth values, and truth values
are extensional, then a context whose denotation depends exclusively on
the denotations of the parts must be extensional. Intensional contexts are
intensional by contrast, and the mark of their intensionality is failure of
substitutivity of terms or sentences with the same denotation.
This view seems to me to leave a lot unexplained. Why are objects
extensional, for example? Well, not all objects are extensional because there
are also intensional objects, such as meanings. But tables and chairs are
extensional. How come? What is it about a chair that makes it extensional
in the appropriate sense? Is a chair a set of molecules, for example? This
would at best reduce the question of extensionality of chairs to the question
of extensionality of molecules. But perhaps chairs are extensional by being
mereological sums. The counterpart of membership is now the part-whole
relation, and this is claimed to be an extensional relation. The extensionality
of this relation depends on the notion of atom, in the sense of Goodman’s
calculus of individuals, and the extensionality of a chair depends on its
identity being ultimately determined by its atoms. So the question seems to
reduce again to the question of extensionality for atoms (in the appropriate
sense).
Moreover, we are now talking about extensionality in connection
with such relations as the part-whole relation. Is set membership extensional? Why? Because sets are extensional? Shouldn’t we then have some
kind of independent account of the extensionality of sets? Our account of
the extensionality of sets depends on membership, so we cannot very well
say that membership is extensional because sets are extensional. And what
about denotation? What makes denotation extensional?
Although there are answers to these questions, at least on the
surface it appears that the notions of extensionality and intensionality are
somewhat obscure as they stand, and that it is not altogether clear how to
interpret some claims that are made concerning these notions; in particular,
claims to the effect that a certain kind of entity is or is not extensional. To
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formulate what I consider to be an appropriate sense of extensionality for
entities of a given kind, deriving from some remarks by Gödel in “Russell’s
Mathematical Logic”, let me go back to the case of sets.
The basic idea is that sets are extensional because they can be
organized hierarchically in levels – and we can attribute ranks to them
according to the level in which they first appear in this organization – so
that the identity of a set of a given rank depends only on the sets of lower
ranks and on objects that stand at the basis of the hierarchy (for which
identity is assumed to be well-defined). This means that sets are predicative
in a certain sense. My earlier discussion of extensionality in connection with
schema (5) was designed to bring out this idea. The natural numbers at the
basis, which are of rank 0, are sufficient to determine the identity of the sets
of natural numbers, which are of rank 1; and together these are sufficient
to determine the identity of the sets of rank 2; and so on. At any level α,
the identity of the new sets appearing at that level, i.e., the sets of rank α,
is completely determined by the sets and objects of rank less than α.
It is important to realize that this is not a construction of sets by
means of definitions, or whatever, but that it is a structuring of sets. This
structuring derives fundamentally from our conception of their nature, but
to say that sets are predicative in this sense is not to say that they are predicative in a definitional sense emphasized by Russell. The reason I do not
consider the usual axiom of extensionality (2) an axiom of extensionality in
the sense I’m trying to bring out, is due to the fact that it is highly impredicative and could hold independently of such a hierarchical organization.
What makes it seem like an axiom of extensionality are the other axioms of
standard set theory, especially the axiom of foundation. This can be seen
very clearly in such set theories as Quine’s NF and ML which contain an
axiom like (2) but which cannot be organized hierarchically in the sense I
am discussing; i.e., in that conception sets are not extensional in this sense2 .
The notion of predicativity was first emphasized by Poincaré in
connection with Richard’s paradox of definability3 . Poincaré’s idea was to
classify classifications, which was the way he thought of sets, into good and
bad. The good ones were predicative, the bad ones impredicative. The
paradoxes came about because people made bad classifications.
Suppose that I’m classifying some pieces of paper with numerals
on them that I am pulling out of a hat – I’ll talk about the corresponding
numbers, though. So far I have taken 5, 7, 12, 18, and 29. I have two containers, A and B, and I say: I’ll put the smallest number and every other
one by order of magnitude into A, and the others into B. That’s my rule
for classifying the numbers I pull out of the hat. So 5, 12, and 29 go into
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A, and 7 and 18 go into B. If I now pull out 33, that’s fine; I just put it
into B. Suppose, however, that the next number I pull out is 4. This forces
me to exchange the contents of the two containers and to place 4 into A;
which means that the earlier classification has been disturbed and does not
coincide with the new classification – i.e., the new classification is not an
extension of the earlier one. As I pull more and more numbers out of the
hat this process may repeat itself indefinitely, so that at no point in time
I will know how my classification is going to end up. Of course, if there
are only finitely many pieces of paper in my hat, once I take them all out
we’ll have a definite classification into A and B. But if there is no end to
the pieces of paper I take out of the hat, then it may never be determined
what the content of A and B will be. I may claim that the actual content
of the hat does determine what the classification will ultimately be at the
limit, but one may very well deny that there is an actual content of the hat
independently of the pieces of paper I pull out of it4 .
This is Poincaré’s idea about mathematics and about sets. My
hat is my head; the pieces of paper are definitions; sets are classifications
determined by rules. The rule I used is a bad rule because it does not
necessarily determine what the classification will be5 . Rules need not be
given a priori, before I start taking things out the hat, but if they classify
things that I have already taken out of the hat at time t, they must satisfy
the condition that this classification will not be disturbed by things I take
out of the hat after time t. A rule that satisfies this condition is predicative,
and so is the classification; other rules are impredicative. It follows from
this that at any time t the classification up to that time is completely
determined by the rule and by what I have taken out of the hat up to that
time. This gives us a criterion of identity for sets even if we conceive of them
as idealized limiting processes. For any predicative classification brings with
it a structuring of the idealized limit into levels which build up to it.
This is a very interesting idea, which is in fact completely independent of whether the stuff we are classifying is created by us in some way or
exists apart from any creative or classificatory acts on our part. We could
say that a kind of entities is predicative if there is a well-founded structuring
into levels and a criterion of identity for each level α which appeals only
to entities of level lower than α. If a kind of entities is predicative in this
sense, it means that we can conceive of them as if they were built up in
a step by step process of idealized construction. That’s in fact how most
people conceive of the standard hierarchy of sets. This process of idealized
construction need not be definitional, of course. I can conceive of the sets
of rank α as “built up” from materials below α not because I have a rule
for building them up, or because I can characterize in some “constructive”
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way how they are built up, but because that’s how they are – as in the
case of sets, for which we may say that the structuring into levels reflects a
certain conception of their nature. Obviously this was not Poincaré’s view,
but the additional aspects of his view do not seem to me to follow from his
characterization of predicativity.
Poincaré’s notion of predicativity was taken up by Russell and
made into one of the central features of Principia – it was Russell’s paradox
buster. All the known paradoxes were blasted out of existence as involving
impredicativity in some way6 . As with the theory of descriptions, out went
the baby together with the bath water, and Russell had to rescue him with
the axiom of reducibility. It was a good cause, but if anything was theft over
honest toil, that was it. Be that as it may, however, Russell’s conception of
predicativity was formulated rather differently than Poincaré’s – though in
effect I think that it is essentially the same conception.
Russell’s idea was that if a totality is well-defined, then its members must be well-defined independently of that totality. This much can
be interpreted exactly as above, in the sense of there being a hierarchical
well-founded structuring of the totality which allows us to conceive it ontologically in a certain way. But Russell formulated the idea in terms of
definability, declaring that it is illegitimate to suppose “that a collection
may contain members which can only be defined by means of the collection
as a whole”, which is an entirely different matter7 .
The notion of definability is primarily an epistemological notion
which shares many of the features of the notion of proof. If to say that an
entity is definable in terms of some entities is to say that it can be identified
through those entities in a specific way, then it doesn’t follow that the
entity in question depends ontologically on the entities that are used in
the definition. If I identify a word as the longest word occurring in this
paragraph, I am appealing to a totality of words to which that very word
belongs. Evidently, in this particular case that’s not the only way in which I
can identify that word – but at this point it is, because the longest word up
to now need not be the longest word (if any) by the time I’m finished with
the paragraph. Is it reasonable to suppose that all members of a collection
must be independent of that collection in the sense that they can be defined
without appealing to the collection in some way? That’s Russell’s claim.
This is false for mathematics generally, and by formulating the
claim more precisely it can be proved to be false in many specific cases.
In particular, although the standard hierarchy of sets can be structured
predicatively in the sense I was talking about earlier, it definitely does not
satisfy this epistemological requirement. But, naturally, this supposes that
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the objects of mathematics, including sets, do not in some sense “arise”
from definitions. If each mathematical entity, or each such entity aside
from certain given entities, were “created” by definitions, then definitions
would have a fundamental ontological role in mathematics – to be would
largely be to be defined.
Poincaré took this view of mathematics. For him the given were
the natural numbers, and all other mathematical entities were introduced by
definition. He felt, therefore, with good reason, that if from the definition
of a classification new entities were to arise to be classified by this very
same classification, one would never be sure that the classification would be
stable. Suppose, for instance, that my hat contained finitely many pieces of
paper. In spite of the odd rule I used, by the time I pulled them all out we
would get a definite classification – this is like the case of the longest word in
the last but one paragraph. But if, by some magic, from the “completion” of
the classification new pieces of paper started appearing in the hat, then we
would be back to the earlier case, and it would seem to be nonsensical to talk
about a “completion”. Russell felt the same way, and followed Poincaré in
ruling out such classifications. Another formulation he gives of the viciouscircle principle above is:
(9) If, provided a certain collection had a total, it would have members only
definable in terms of that total, then the said collection has no total.
Gödel pointed out that if we look at the various formulations of the
vicious-circle principle offered by Russell, we can distinguish several different vicious-circle principles. He contrasts (9) with the following two other
formulations:
(10) Whatever involves all of a collection must not be one of the collection,
and
(11) [G]iven any set of objects such that, if we suppose the set to have a
total, it will contain members which presuppose this total, then such a set
cannot have a total8 .
Moreover, Gödel distinguishes two readings of (11) – say, (11E) and (11K)
– according as to whether we read ‘presuppose’ as ‘presuppose for its existence’ or as ‘presuppose for its knowability’. I shall refer to (9) and (11K)
as the strong and weak epistemological versions, and to (10) and (11E) as
the strong and weak ontological versions9 .
What I pointed out earlier is that the standard hierarchy of sets is
impredicative in the strong epistemological sense but is predicative in both
ontological senses. Whether or not the standard sets are impredicative in the
weak epistemological sense is hard to say unless one analyses more precisely
the notion of knowability. What I would like to suggest as a criterion of
(relative) extensionality for a kind of entities is that they be predicative in
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the strong ontological sense. My understanding of this is that there is a
well-founded hierarchical structuring of the entities in question based on a
bottom level of entities such that the identity of entities of any higher level
α depends only on entities of lower levels – i.e., that there is a criterion of
identity for entities of level α that appeals only to entities of level lower
than α. Naturally, any such structuring will be relative to the entities at
the bottom level, for which the determinateness of identity must either be
presupposed or characterized in some other way.
As in the last chapter, we could say that a criterion of identity for
a kind of entities is something like
(12) ∀x∀y(C(x, y) ⇒ x = y),
where C(x, y) is a condition of some sort. If all quantifiers in C(x, y) range
over the entities in question, and all terms and predicates are defined for
those entities, then we can think of (12) as an absolute (or intrinsic) criterion
of identity. If, on the other hand, we appeal to other kinds of entities, or
distinguish some entities of the same kind for which the determinateness of
identity is presupposed, then (12) is relative to those entities.
The usual criterion of identity for objects in terms of indiscernibility
(13) ∀x∀y(∀Z(Zx ⇔ Zy) ⇒ x = y),
is a relative criterion which appeals to all properties of objects, or to all
properties of a certain kind.
Frege’s criterion of identity for line directions
(14) ∀x∀y(∃z∃w(Dir(x,z) & Dir(y, w) & z k w) ⇒ x = y),
is a relative criterion which appeals to lines and to a relation between lines
and line directions, as well as to a relation between lines.
A criterion of identity for material objects in terms of overlapping
in the sense of Goodman’s calculus of individuals
(15) ∀x∀y(∀z(zox ⇔ zoy) ⇒ x = y),
is an absolute criterion of identity for material objects – as long as the
variable ‘z’ ranges only over material objects and not over individuals in
some other sense10 .
The condition that we use in formulating a criterion of identity
may also be predicative or impredicative depending on the ranges of values
of the quantified variables. This is, in fact, the way in which predicativity
is usually treated in connection with definitions. The idea is that if we
are defining an entity, then the quantified variables in the definiens should
not range over a totality which includes the entity being defined, or over
totalities which in some way “presuppose” that totality11 . Since in a way a
criterion of identity is “characterizing” the identity of entities of a certain
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kind, we may say that a criterion of identity is impredicative if the condition
C(x, y) involves quantification over the totality of such entities, or over
totalities which in some way presuppose that totality. The absolute criteria
of identity for abstract sets (2) and (8) and the absolute criterion of identity
for material objects (15) are impredicative in this sense.
What makes an impredicative definition seem inadequate from the
epistemological point of view is that it does not give a direct path to the
thing being defined. If we think of quantification in terms of possibly infinite
conjunctions and disjunctions, and if we are quantifying over a totality to
which the entity being defined belongs, then if we “unpack” the definiens
we will obtain an expression which involves reference to that very entity.
That’s why people talk about a vicious circle. A definition, like a proof,
should not depend on the thing that is being defined, or proved. But this is
not quite right unless we think of definitions as in some sense “introducing”
the thing defined, or think of them as a justificatory process for the very
existence of that thing. These two ideas are connected, though they are not
exactly the same idea.
If we want to establish the existence of something or other – a
property, say, or an object – we may do it through a definition which appeals
to other entities whose existence is not in question. As long as we think
of this as a singling out process, there is no reason why the entities whose
existence is not in question should not include totalities to which the entity
to be singled out, and hence shown to exist, belongs. That’s the typical use
of impredicative definitions in mathematics. If, on the other hand, there is
a question as to whether the entity in question can exist, and be a member
of any totalities at all, perhaps because of some peculiar features it has that
would make the very notion of totality to which it belongs a problematic
notion, then the situation is quite different. This was precisely the feeling in
connection with Richard’s paradox and the sort of totality (of definitions)
that was in question there. It is quite natural also that this should have
been extended to other kinds of entities such as sets and propositions which
were also involved in paradoxical situations. Evidently, if the very notion
of totality for a certain kind of entities is in question, then a justificatory
definition of an entity of that kind must be predicative – the singling out
must be above board.
In the case of sets the situation was compounded by the fact that
sets were supposed to be totalities which were often thought of as classification processes given by rules, conditions, concepts, etc. It is quite clear that
that’s how Poincaré thought of them; and not only him. So if the notion of
set was in question, then the notion of totality was in question as well, for
any kind of entities. The answer to this questioning was precisely to show
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that sets could be conceived predicatively in a certain way; namely, in the
strong ontological sense. This was done by Zermelo, Fraenkel, and others,
and it was also Cantor’s conception of set, as Gödel has emphasized. That
this was so was only realized later, however, primarily by Gödel himself,
and what it shows is that the conception of set in terms of classifications
determined by rules is an altogether different conception; which though useful and significant in itself, is not particularly appropriate for the purposes
for which sets were introduced in mathematics. Once we realize this, we
realize that there is absolutely no problem about impredicative definitions
of an entity in terms of totalities to which it belongs, or other more complex
totalities that “presuppose” such totalities, as long as the notion of totality
for that kind of entities has been legitimized. This is Gödel’s claim about
impredicative definitions in mathematics and about sets.
In any case, it doesn’t follow that by a predicative or impredicative definition we are creating the thing being defined, though we may be
introducing it in the sense of legitimizing our claim that it exists. And even
if by ‘introducing’ we mean ‘creating’, it still doesn’t follow that our “creation” must be predicative. All that follows is that the entity, or kind of
entities, we have created isn’t predicative. This may seem paradoxical but
it is actually quite obvious.
Suppose we agree that I create my own thoughts; i.e., that they
are not independently existing entities which I simply single out in some
way. I can certainly create the thought that all my thoughts are created
by me. I can in fact characterize this thought as the thought that every
thought of mine is created by me. This is clearly an impredicative thought,
yet there doesn’t seem to be anything wrong with it. The thought I created
is impredicative in the sense that through quantification it involves all my
thoughts, including itself. It is true that I can identify that thought in other
ways as well – for example, as the thought I created today at exactly 2:55 pm
– but this doesn’t make it any the less impredicative12 . It is impredicative
in the strong ontological sense of involving reference to a totality to which
it itself belongs. Do we have the right to talk about such a totality as
the totality of all my thoughts? I should think so; though, naturally, since
thinking is a temporal process I cannot expect this totality to be a set in
the mathematical sense, or to be well-defined now – at each moment of time
it is only defined up to that time13 .
It seems to me, therefore, that even if mathematical entities are
created by means of definitions it is not at all clear that impredicative definitions must be ruled out. And this doesn’t seem to follow even if the entities
are identified with the definitions, in some sense. What does follow is that
my notion of classification of such entities is not the classical mathematical
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notion of set. But this is obvious anyway because these entities are temporal
entities, and classifications of them will have to be potential classifications.
If I want to use these classifications as bona fide mathematical objects, then
it may be reasonable to require that they be predicative in Poincaré’s sense.
Nothing prevents me, however, from introducing (classical) mathematical
sets by definitions as long as I don’t confuse them with these temporal
classifications. Since Poincaré grants me the natural numbers, I may introduce the hierarchy of sets built up from the natural numbers as a kind of
mathematical entities which satisfy the usual axioms of set theory. If I can
introduce a number by means of a definition, then why can’t I introduce
sets by means of a definition?
It is clear, in any case, that besides introducing something or justifying the existence of something, definitions have other roles as well. As I
mentioned before, a definition may serve to identify something in the sense
of singling it out, and may also serve to characterize the essence of something. In either of these cases there is no question of either creating the
something in question or of justifying its existence. The existence is presupposed, and an impredicative definition is just as good as a predicative
definition for identifying something. There is no difficulty in figuring out
that the impredicative definition several paragraphs back identifies the word
‘epistemological’. A predicative definition may give us a more direct identification, or, in many cases, may give us a more informative identification;
i.e., it may be epistemologically richer in some ways. One may want to
restrict oneself to identifications that are epistemologically richer in some
such sense, but this has nothing to do with the issue of whether or not the
alleged identification succeeds in singling something out in an ontological
sense. This seems to me to hold even for definitions of essences, because
part of the essence of a entity, or of a kind of entities, may be to be impredicative in one sense or another, and it may not be possible to characterize
that through a predicative definition.
The preceding remarks were not meant to pervert the philosophical position of predicativists, and it is not my purpose here to examine
predicativism as a philosophy of mathematics – though I do think that as
a philosophical position predicativism must be formulated in more definite
ontological and epistemological terms. Due to the “nominalistic” interpretation of Principia Mathematica there has been a tendency to think of predicativism as closely associated to some form of nominalism, but this doesn’t
seem to me to follow at all because one can raise the question of predicativity for all the kinds of objects that the nominalist takes for granted; material
objects, thoughts, appearances, etc. Moreover, one can raise the question
for the idealized syntax that the predicativist uses to formulate his systems
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of mathematics14 . The interesting issue is an epistemological one concerning our categorization and structuring of reality. If the predicativist were to
apply his strict epistemological criterion of classification and identification
to everything, he would end up with nothing. So he adopts a relativist position according to which certain things are legitimate, including the totality
of natural numbers (or something essentially equivalent), but other things
are not legitimate; and there he becomes a nominalist of sorts.
The main issue of predicativity seems to me to be an issue about
infinity, not definitions. One may hold that in the case of the impredicative
definition of the word ‘epistemological’ I gave above it is the finiteness of
the paragraph that allows us to “figure out” that the definition does indeed
identify this word. If the paragraph went on forever, then we wouldn’t
be able to do it. And it wouldn’t even be “objectively determined” which
word (if any) is identified by the definition unless we were to assume that the
infinite paragraph exists (or is “determined” in some way) as a completed
totality. I agree; and it is silly to assume this for paragraphs. The question is
whether the predicativist can really draw a legitimate distinction between
paragraphs and totalities of mathematical entities such as the totality of
natural numbers15 .
Let’s go back now to the question of extensionality for properties
in the light of our previous discussion. Whether properties are extensional
or not, relative to a certain kind of entities for which the determinateness
of identity is presupposed, depends on the possibility of structuring them
(and these entities) in a well-founded hierarchy so that the identity of each
property is determined by whatever comes below it in the hierarchy. The
hierarchy of the last chapter would show that properties are extensional
relative to level 0 objects only if we consider each property as determined
by the entities to which it applies. This is not the case, however, and the
general criterion of identity for properties of level 1 is their indiscernibility
with respect to properties of level 2. There certainly are properties of level 1
which apply to exactly the same level 0 objects and yet are different. This is
the usual conception of properties and is the intuitive reason for saying that
properties are intensional entities. It doesn’t follow, of course, that there
aren’t other ways of structuring properties hierarchically which would show
that they are extensional. For the traditional conception of properties this
seems out of the question, but we may be able to do it either by restricting
the conception of property in various ways, or by enlarging the levels in
various ways.
If we throw in enough possible entities, then we may be able to do
it. This is essentially what one does in one interpretation of the possible
worlds approach. One calls level 0 of the hierarchy ‘the actual world’ and
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introduces a plurality of similar hierarchies with different level 0’s which
one calls ‘possible worlds’ – or possible worlds “accessible” from the actual
world. A level 1 property is then determined by what it applies to in the
actual world and in all (accessible) possible worlds. This normally assumes
that the level 1 properties are fixed for all possible worlds, but one could
try to handle the entire hierarchy in the same way. That is, just as two
possible worlds may differ in the objects they contain, they may differ in
the properties they contain. One may say that two level λ properties are
the same in the actual world if for every (accessible) possible world in which
they exist they apply to the same entities.
To this sort of extensionalization of properties one might object
that it appeals to a notion of identity for objects and properties across possible worlds which seems no less problematic than the notion of identity for
properties, and that it involves treating the entire hierarchy as an entity
which is part of a plurality, or totality, of such entities – which goes against
the characterization of the original hierarchy as being absolute and uncollectable. If possible worlds have properties, and if the identity relation for
them is a real relation, this will lead to an additional ontology which is on
top of the ontology of possible worlds and which will give raise to an entire
new hierarchy for which the question of extensionality will come up again.
This would certainly be the case if one takes a “realist” approach to possible
worlds.
One may argue that if one can treat the pure set structures of
mathematics in terms of the fictional device of units, then one can treat
the extensionality of properties in a similar way. Possible worlds are, like
units, a helpful device to characterize properties – and other things as well.
I grant this, at least insofar as I grant that sets may be treated by similar
means in terms of fictional units, and it is one way to think of possible
worlds. The question is whether this can really be done in a satisfactory way
that goes beyond a purely mathematical treatment in set-theoretic terms,
and whether extensionalizing properties in this fictional way is helpful in
understanding the notion of property. In terms of criteria of identity, the
problem is that one needs a criterion of identity for determining when an
object of the actual world exists in a possible world, or is the same as an
object in that possible world. Similarly, one needs a criterion of identity for
determining when a property of the actual world exists in a possible world,
or is the same as a property in that possible world, and one obviously cannot
do it through the entities to which the property applies. It seems to me that
the attempt to formulate such criteria will very likely beg the question.
To deny that this approach to properties throws light on their nature is not to deny the technical virtues of the notion of possible world.
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But, as a matter of fact, the technical work with possible worlds does not
depend on possible worlds, since it is carried out entirely within set theory. Moreover, as Kripke argues in Naming and Necessity, the philosophical
discussion can proceed within the real world through such notions as the
notion of counterfactual situation – or possible state of the world, or possible history of the world16 . This approach of Kripke’s does not yield an
extensional explanation of the notion of property, however, for it amounts
to adopting as a criterion of identity for properties that two properties are
identical when they necessarily apply to the same entities. This criterion of
identity is completely independent of the notion of possible world, although
one of the alleged virtues of the notion of possible world is that it provides
an “extensional” explanation of the notions of possibility and necessity. But
on Kripke’s approach this cannot be so, because the notion of possibility
is presupposed by the account of possible worlds. To say that it is possible
that Quine is a dentist is to say that there is a possible state of the world,
or a possible history of the world, or a counterfactual situation, in which
Quine is a dentist. And to say that two properties necessarily apply to the
same entities is to say that for every possible state of the world, or possible
history of the world, or counterfactual situation, these properties apply to
the same entities.
My conclusion is that properties (as usually conceived) are not
extensional because they are impredicative in the strong ontological sense.
There doesn’t seem to be a non-fictional well-founded hierarchical organization which will make the identity of properties of level α depend only on
entities of level lower than α in the organization. Although the appeal to necessity does give a criterion of identity for properties, it does not show that
properties are extensional in the sense I am considering. Not because the
notion of necessity is intensional in the usual contextual sense, but because
the necessary mutual subordination of two level 1 properties, say, is presumably not determined by the objects at level 0 – unless one includes possible
objects. Besides, one can also raise some questions as to the adequacy of
this criterion of identity for properties.
It is not clear, for example, that it is adequate for mathematical
properties and logical properties. It is reasonable to suppose that if two
mathematical properties are coextensive, then they are necessarily coextensive. Therefore, according to the criterion, coextensive mathematical
properties are the same property – which seems rather unintuitive in general. This is also a problem for the approach to the extensionalization of
properties in terms of possible worlds, for if mathematical entities exist
necessarily, then one cannot distinguish mathematical properties in terms
of the entities to which they apply in various possible worlds.
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These conclusions are related to conclusions drawn by Quine in
many occasions to cast doubt on the legitimacy of postulating properties.
He grudgingly accepts sets, but that’s as far as his Platonism goes17 . I was
neither objecting to properties nor to the notion of necessity as such, and
I think that it is questionable whether Quine’s preference for sets is really
justified on those grounds. Of course, as I have already pointed out to some
extent, the notion of set is rather ambiguous – including the notion of set
as extension of a property, the notion of set as a classification of entities,
and the notion of set as a structure, among others – and in order to discuss
this issue one must make some distinctions.
It seems quite clear to me that the notion of set basically derives
from the notion of property. If we restrict the levels of the hierarchy by
“identifying” two properties that apply to the same entities, then we get a
hierarchy of properties that are essentially like sets. This “identification”
may also be conceived as a process of abstraction in which a set is what all
properties that apply to the same entities have in common. Thus we may
conceive of a set as a property, or form, expressing what is the same in a
certain plurality of properties. By treating sets (or “extensions”) as objects,
and drawing a fundamental distinction between objects and properties (or
concepts), Frege obscured this connection; so much so, that we still tend to
think of sets as more closely related to objects than to properties. But this
is quite misleading, because the role which sets often play is fundamentally
the role of properties – as, for example, in higher order logic. And this is
quite natural, especially when sets are used in lieu of properties in mathematics. For, as I mentioned earlier, although we may distinguish coextensive
properties in mathematics, if two mathematical properties are coextensive,
then they are supposed to be necessarily coextensive. It is quite natural,
therefore, to (in a sense) simplify the notion of property in this way.
But from the point of view of the abstraction involved this simplification is not really a simplification, for it has the same character as the
original motivation for postulating properties, or forms. Frege’s abstraction
in the characterization of the cardinality properties was based on the intuition that a cardinality property reflects what is common (or the same)
in all properties that apply to the same number of entities. Thus, the intuition is that the cardinality property Twoness, for example, is what is
common to all properties that apply to two entities. Defining ‘F applies to
the same number of entities as G’ in terms of 1-1 correlation, and choosing
some specific (logical) properties that apply to no entities, or one entity, or
two entities, etc., we could use essentially Frege’s procedure to define the
cardinality properties Nullness, Oneness, Twoness, etc. as specific properties that apply to a property if and only if this property applies to the
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same number of entities as one or another of those logical properties. We
can do something similar for sets as properties. For any property F, the
set-property corresponding to F is the property which applies to a property
G if and only if F and G are mutually subordinate (or coextensive). Since a
set-property reflects what all properties that apply to the same entities have
in common, we may conceive of the set-property as the set of those entities,
just as Frege initially conceives of the direction of a line as the property
that all lines parallel to that line have in common. In this sense of ‘set’,
sets do not replace properties but derive from properties – or are properties
introduced by a process of abstraction characteristic of the introduction of
properties in general18 .
It is true, however, that Frege ended up conceiving sets as objects
rather than properties, but he could never quite make up his mind as to
which objects they were. His appeal to the criterion of coextensiveness
as expressing the fundamental character of extensions (or courses-of-values)
seems to me to reflect a certain ambiguity in his conception. And, as we have
seen, he argues himself that this criterion cannot fully determine the nature
of extensions as objects19 . Moreover, his idea that concepts themselves
are extensional seems to suggest that the distinction between concepts and
extensions is mainly categorical and somewhat like the distinction between
the concept ‘horse’ and the object denoted by the expression ‘the concept
‘horse’ ’. In other words, what has a concept got (in Frege’s extensional
conception) that an extension hasn’t got?
What Frege objected to was the idea that a concept “arises” from
the entities to which it applies. And he objected to a conception of sets as
ontological units which “collected” pluralities. This conception of ‘set’ is
indeed a simplification and generalization of Plato’s notion of form. Two
fundamental questions for Plato’s theory of forms were questions as to the
nature of forms and the nature of participation. If one conceives of a set
as nothing but the collection of a plurality of entities into a unit and of the
relation between an entity thus collected and the collecting unit as “membership”, then those questions are to some extent “defused”. This is the
simplification, which in fact also involves a process of abstraction typical of
mathematics and other highly theoretical sciences closely related to mathematics. Just as the Greek question as to the nature of motion, and change
in general, was replaced by an abstract theory of motion, the question as to
the nature of forms and participation was replaced by an abstract theory of
sets. (I am not criticizing this development; on the contrary, I think that it
was a very important development.) The generalization was related to this
and to another central question concerning Plato’s theory of forms; namely,
the question as to which forms there are. Or, alternatively, for which plural-
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ities of entities is there a form reflecting what is common to those entities?
The initial set-theoretic idea was, by and large, that for any plurality of
entities there is a set collecting these entities.
All of this eventually led to the paradoxes of set theory, which have
points in common with the paradoxes that came up in Plato’s own conception of forms, but in the meantime these ideas also led to a very powerful and
fascinating theory. When the paradoxes struck, however, people were forced
to take a better look at their conceptions of sets. One obvious shortcoming, in ontological and epistemological terms, of the mathematical notion
of set is the extensional idea that a set “is completely determined by its
elements”, or is nothing but “a plurality treated as a unit”. In what sense
is a set completely determined by its elements? A set cannot be just the
elements, since it is supposed to be an entity in its own right; and to treat
it as a unit is to treat it as one thing. So what has it got in addition to
the elements? In other words, what’s the difference between a plurality of
entities and the set whose elements are those entities? Or, in still other
words, closer to Plato’s: what is membership and what is this one over
many? The old questions came back, including the question as to which
pluralities can be collected by a set, and Whitehead and Russell actually
appeal to the Platonic problems in Principia Mathematica to partly justify
treating classes as incomplete symbols20 .
There were a number of solutions to (some of) these problems,
eventually leading to the cumulative hierarchy of sets as the main mathematical notion of set. But the problem with which I am concerned here is
the problem of the nature of sets, about which very little is said in explicit
terms. If we take the conception of set I have just been discussing, as a
unit consisting of a plurality of entities, which can also be sets and which
are themselves treated as units that determine the identity of the set, then
it seems to me that the best way to conceive of these sets is as expressing
(or “codifying”) the difference of the entities that are elements of the set.
That’s the notion of set I suggested in the last chapter as the state of affairs
that such and such entities are different entities. The entities are different,
but the set as an entity, and as a unity, is an ontological “expression” of that
difference. As Frege says (see Chapter 9 note 18) “the plurality in things
arises from their diversity”, and a set as a collection of a plurality of things
into a unity is basically an ontological code for their diversity.
This conception of sets of entities, and I know no better for sets
of entities as something like objects, also presupposes properties, at least
in the sense that it presupposes the pairwise Diversity relations of all arities (including infinite arities), and it also presupposes that properties can
combine with various entities into (something like) states of affairs. More-
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over, as Quine emphasizes, the identity and diversity of sets of entities in
some such sense is only well-determined if the identity and diversity of the
entities in question is well-determined. Or, in more explicit terms, the very
existence of these sets presupposes the determinateness of Identity and Diversity for the entities which they “collect”. In fact, I entirely agree with
Quine that if the determinateness of Identity and Diversity for properties
is in question, then to suggest as a criterion of identity for properties that
two properties are identical if they belong to the same sets of properties is
ridiculous21 . And similarly for any other entities, of course.
Although Quine emphasizes this relativity of sets, and could even
agree with the basic intuition behind my characterization of sets of entities,
in “On the Individuation of Attributes” and other earlier works he did not
see this as a problem for his own acceptance of sets, because the only sets
he accepted were sets of material objects, sets of those sets, etc. – i.e.,
he accepted a cumulative hierarchy of sets based on material objects as
units. Since he held that there is no real problem of identity for material
objects, he also held that there is no real problem of identity for such sets.
Of course, there is a question as to the actual nature of sets, for which
Quine has no good answer except that they seem necessary for science and
mathematics – that’s why the “grudging” acceptance. Thus the argument
of sets vs. properties seems to boil down to two questions. One is whether
material objects are indeed better off than properties in terms of a criterion
of identity. The other is whether there is a reasonable conception of the
nature of sets which does not involve properties in some way and for which
similar problems of identity do not arise.
In “On the Individuation of Attributes” (p. 101) Quine says that
the criterion of identity for material objects is coextensiveness. Now what
is a material object supposed to be, according to Quine, and what is the
appropriate sense of “coextensiveness”? If a material object is not a set,
but is made up of molecules, then we must interpret the object as being
some kind of structure. Perhaps the most general sense would be to think
of material objects as space-time sums (in Goodman’s sense) of whatever
they are made up. Should we say that two material objects are the same
when every elementary particle that is part of one is part of the other. I.e.,
using Goodman’s notation:
(16) ∀x∀y(∀z(z < x ⇔ z < y) ⇒ x = y),
where ‘x’ and ‘y’ range over material objects in general, ‘z’ ranges over
elementary particles and < is the part-whole relation. This would reduce
the problem of identity for material objects to the problem of identity for
elementary particles, and raises the question of the criterion of identity
for elementary particles. This is actually quite problematic, and in some
later papers Quine himself has examined the matter and argued against his
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earlier solution22 . Alternatively, we may say that two material objects are
the same when they occupy the same space-time region? This would at
best reduce the problem of identity for material objects to the problem of
identity for space-time regions, for which we would need again a criterion
of identity. By identifying material objects with the space-time region they
“occupy” and identifying space-time regions with sets of quadruples of real
numbers, Quine reaches the rather remarkable conclusion that – since real
numbers can themselves be defined set-theoretically – the entire ontology
can be restricted to pure sets23 :
We are left with just the ontology of pure set theory, since the numbers and the quadruples can be modeled within it. There are no
longer any physical objects to serve as individuals at the base of
the hierarchy of classes, but there is no harm in that. It is common practice in set theory nowadays to start merely with the null
class, form its unit class, and so on, thus generating an infinite lot
of classes, from which all the usual luxuriance of further infinities
can be generated.
Thus, almost a century later, Quine is led by the identity version of the
context principle to a conclusion reminiscent of Frege’s in The Basic Laws
of Arithmetic – as I discussed at the end of Chapter 8.
But there are two main problems with this conclusion. One is that,
as I argued in Chapter 5, sets of quadruples of real numbers do not really
give us a reasonable criterion of identity for material objects. The proxy
function that Quine needs to effect the reduction of material objects to sets
of quadruples of real numbers is simply not well-defined. The other is that,
in terms of a criterion of identity, the ontology of pure sets is not really
better off than an ontology of properties. Although Quine visualizes his
ontology as starting from the empty set, there is nothing that intrinsically
distinguishes the empty set from any other set. As with any set, which
set is the empty set depends on all sets; sets are individually inscrutable,
in this sense. In fact, by Frege’s transformation argument24 , the universe
of sets could be mapped onto itself so that any set can be the empty set
and the criterion of identity (2) would continue to hold. Even if we agree
to leave aside consideration of such transformations, the point is that the
identification of pure sets is just as impredicative as the identification of
properties.
To avoid this one must either treat the empty set as a unit whose
identity is not determined by having no members, or, if one wants to have
the empty set as a set, introduce a unit to stand at the basis of the hierarchy
– any non-set can serve as such a unit25 . But, contrary to the usual conception, one cannot do both; i.e., one cannot have the empty set both as set
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and as unit. And the reason is very simple; namely, that if the empty set is
characterized as the set such that no set belongs to it, its identity depends
(or is determined by) all sets. The other axioms of set theory, including the
axiom of foundation, do not change this one bit. There is no way to identify
the empty set without appealing to all sets. Of course, if one has a unit,
then it will have the property that no set belongs to it, but that’s not what
identifies it. Its identity must be presupposed26 .
It may seem that this is a small price to pay, and Quine could
simply postulate the hierarchy of pure sets with the empty set as the basic
unit. But from an ontological point of view this is completely unnatural.
What is the empty set? If all sets derive from the empty set by iteration
of the operation “set of”, then one cannot hold that the empty set is a
fiction and that other sets are not27 . And if one wants to reduce everything
to sets, as Quine does, then one cannot identify the empty set with an
arbitrary object. It would certainly be odd to have an ontology of sets
based on one material object, for example28 .
It seems to me, therefore, that Quine’s idea that he can solve the
problem of identity by appealing to a pure ontology of sets is an illusion.
Moreover, given the various alternative conceptions of sets that I have briefly
examined, I would argue also that the notion of set is not as primitive as it
is generally taken to be, and that the notion of property is a better primitive
for logic and ontology29 .
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Notes
1. When Fraenkel, Bar-Hillel and Levy introduce the axiom of extensionality (2)
in Foundations of Set Theory, they remark (pp. 27-28):
This axiom affirms the extensional nature of sets, i.e., that each
set is completely determined by its members. The antonym of the
present meaning of ‘extensional’ is ‘intensional’. Sets would be of
an intensional character if identity of sets would depend not only on
their extension (i.e., on their members) but also on the way they are
presented (“defined”). Thus, from an intensional point of view the
set of all non-negative real numbers and the set of all squares of real
numbers are not necessarily identical, even though they have the
same extension. The purely extensional notion of set is chosen to be
the basic notion of set theory, rather than any intensional notion, for
the following reasons. First, the extensional notion of set is simpler
and clearer than any possible intensional notion of set. Second,
whereas there is just one extensional notion of set, there may be
many intensional notions of set, depending on the purpose for which
those sets are needed; so if we wanted to base set theory on some
intensional notion of set, we would have to choose among the various
intensional notions of set in a way which is bound to be at least
somewhat arbitrary. Third, as we shall see, starting with the simple
notion of extensional set we shall obtain, by means of the axioms, a
system of set theory in which much more complicated notions can be
constructed. In particular, we shall be able to construct intensional
notions of set within our system.
We normally read such remarks without giving them a second thought, but if we
do give them a second thought we realize that they are quite problematic and at
best unclear. Moreover, they don’t explain what extensionality is supposed to be,
and do not give any inkling of what justifies the strong claims that are made as if
they were direct consequences of the extensionality of sets, which is supposed to
be completely characterized by axiom (2).
What does it mean, for instance, to claim that “there is just one extensional notion of set”? In any reasonable interpretation this is plainly false, because
non-well-founded sets are quite different from well-founded sets. If it means compliance with (2), on the other hand, so that the one notion of extensional set is
completely captured by (2), then the same is true for any axiom whatever. But to
claim that in this sense “the extensional notion of set is simpler and clearer than
any possible intensional notion of set” seems rather absurd. This is shown by the
many different systems of set theory which adopt (2), and by the fact that if the
axioms of any such system are independent, then we can fool around with their
negations in a large number of ways, while keeping (2), to obtain notions of set
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which are extensional in that sense but which may be neither clearer nor simpler
than intensional notions of set.
What I am pointing out is that even the extensionality of sets is not
as simple as we are led to believe by such remarks as I quoted above. I should
emphasize, however, that the surrounding context (Op. Cit., pp. 22-30) does
clarify some aspects of the questions I raised above – as well as the question of
the relation between (2) and (8) coming up in the text. I’ll come back to these
questions at the end of the chapter.
2. See Quine “New Foundations for Mathematical Logic” and Mathematical Logic.
Quine makes these points in “On the Individuation of Attributes” pp. 102-103:
These thoughts on attributes remind us that classes themselves are
satisfactorily individuated only in a relative sense. They are as satisfactorily individuated as their members. Classes of physical objects
are well individuated; so also, therefore, are classes of classes of physical objects; and so on up. Classes of attributes, on the other hand,
are as badly off as attributes. The notion of a class of things makes
no better sense than the notion of these things.
We may do well, with this relativism in mind, to cast an eye on the
credentials of set theory. It turns out that the usual systems of set
theory still stand up to the demands of individuation very well, if
we assume that the ground elements or individuals of the system
are physical objects or other well-individuated things rather than
ill-individuated things such as attributes. For, we saw how classes
of well-individuated things are well individuated; therefore so are
classes of such classes; and so on up.
This takes care of the usual systems of set theory, which exclude
ungrounded classes. A class is ungrounded if it has some member
which has some member which . . . and so on downward ad infinitum,
never reaching bottom. The system of my “New Foundations” does
have ungrounded classes, and so does the system of my Mathematical Logic; and it could be argued that for such classes there is no
satisfactory individuation. They are identical if their members are
identical, and these are identical if their members are identical, and
there is no stopping. This, then, is a point in favor of the systems
that bar ungrounded classes.
3. Richard’s paradox was formulated as follows by Richard. Let E be the totality
of finite sequences of letters in the English alphabet. This totality can be enumerated – alphabetically, for instance. Each definition of a real number is a member
of the totality E, so cross out from E the sequences that are not definitions of
real numbers, and let u1 , u2 , etc., be the numbers which are defined by the definitions in the order of appearance in E. This is a denumerably infinite set of
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numbers. “Now,” says Richard, “here comes the contradiction” (“The Principles
of Mathematics and the Problem of Sets”, p. 143):
We can form a number not belonging to this set. “Let p be the digit
in the nth decimal place of the nth number of the set E; let us form
a number having 0 for its integral part and, in its nth decimal place,
p+1 if p is not 8 or 9, and 1 otherwise.” This number N does not
belong to the set E. If it were the nth number of the set E, the digit
in its nth decimal place would be the same as the one in the nth
decimal place of that number, which is not the case.
I denote by G the collection [sequence] of letters between quotation
marks.
The number N is defined by the words of the collection G, that is,
by finitely many words; hence it should belong to the set E. But we
have seen that it does not.
Richard’s argument is Cantor’s diagonal argument for showing the non-enumerability of the set of real numbers transposed to the case of definition. Richard
concludes (pp. 143-144):
Let us show that this contradiction is only apparent. We come back
to our permutations [sequences]. The collection G of letters is one
of these permutations; it will appear in my table. But, at the place
it occupies, it has no meaning. It mentions the set E, which has not
yet been defined. Hence I have to cross it out. The collection G
has meaning only if the set E is totally defined, and this is not done
except by infinitely many words. Therefore there is no contradiction.
We can make a further remark. The set containing [the elements of]
the set E and the number N represents a new set. This new set is
denumerably infinite. The number N can be inserted into the set E
at a certain rank k if we increase by 1 the rank of each number of
rank [equal to or] greater than k. Let us still denote by E the thus
modified set. Then the collection of words G will define a number
N ′ distinct from N, since the number N now occupies rank k and
the digit in the kth decimal place of N ′ is not equal to the digit in
the kth decimal place of the kth number of the set E.
(The bracketed additions in the last paragraph are the translator’s.)
4. Poincaré says (“The Logic of Infinity”, p. 47):
From this we draw a distinction between two types of classifications
applicable to the elements of infinite collections: the predicative classifications, which cannot be disordered by the introduction of new
elements; the non-predicative classifications in which the introduction of new elements necessitates constant modification.
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For a discussion of Poincaré’s views see Chihara Ontology and the Vicious-Circle
Principle, Chapter 4.
5. In his résumé Poincaré says (Op. Cit., p. 63):
As for me, I would propose that we be guided by the following rules:
1. Never consider any objects but those capable of being defined in
a finite number of words;
2. Never lose sight of the fact that every proposition concerning infinity must be the translation, the precise statement of propositions
concerning the finite;
3. Avoid non-predicative classifications and definitions.
6. Principia Mathematica, pp. 60-65. For a discussion of Russell, see Chihara
Op. Cit., Chapter 1.
7. Principia Mathematica, p. 37.
8. Russell formulates (9), (10), and (11) in p. 37 of Principia. Evidently he did
not mean to distinguish three different principles, nor is Gödel claiming this, but
to bring out the same principle in different ways. Nevertheless Gödel’s distinction
seems to me extremely interesting and important; as I shall try to show below.
9. The passage from Gödel is the following (“Russell’s Mathematical Logic”, pp.
218-220):
Now as to the vicious circle principle proper . . . it is first to be remarked that, corresponding to the phrases “definable only in terms
of”, “involving”, and “presupposing”, we have really three different
principles, the second and third being much more plausible than
the first. It is the first form which is of particular interest, because only this one makes impredicative definitions impossible and
thereby destroys the derivation of mathematics from logic, effected
by Dedekind and Frege, and a good deal of modern mathematics
itself. It is demonstrable that the formalism of classical mathematics does not satisfy the vicious circle principle in its first form,
since the axioms imply the existence of real numbers definable in
this formalism only by reference to all real numbers. Since classical
mathematics can be built up on the basis of Principia (including
the axiom of reducibility), it follows that even Principia (in the first
edition) does not satisfy the vicious circle principle in the first form,
if “definable” means “definable within the system” and no methods
of defining outside the system (or outside other systems of classical
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mathematics) are known except such as involve still more comprehensive totalities than those occurring in the systems.
I would consider this rather as a proof that the vicious circle principle
is false than that classical mathematics is false, and this is indeed
plausible also on its own account. For, first of all one may, on
good grounds, deny that reference to a totality necessarily implies
reference to all single elements of it or, in other words, that “all”
means the same as an infinite logical conjunction. . . .
Secondly, however, even in “all” means an infinite conjunction, it
seems that the vicious circle principle in its first form applies only
if the entities involved are constructed by ourselves. In this case
there must clearly exist a definition (namely the description of the
construction) which does not refer to a totality to which the object
defined belongs, because the construction of a thing can certainly not
be based on a totality of things to which the thing to be constructed
itself belongs. If, however, it is a question of objects that exist
independently of our constructions, there is nothing in the least
absurd in the existence of totalities containing members, which can
be described (i.e., uniquely characterized) only by reference to this
totality. Such a state of affairs would not even contradict the second
form of the vicious circle principle, since one cannot say that an
object described by reference to a totality “involves” this totality,
although the description itself does; nor would it contradict the third
form, if presuppose means “presuppose for the existence” not “for
the knowability”.
So it seems that the vicious circle principle in its first form applies
only if one takes the constructivistic (or nominalistic) standpoint
toward the objects of logic and mathematics, in particular toward
propositions, classes, and notions, e.g., if one understands by a notion a symbol together with a rule for translating sentences containing the symbol into such sentences as do not contain it, so that a
separate object denoted by the symbol appears as a mere fiction.
10. See The Structure of Appearance, pp. 46-58.
11. For technical presentations and discussions of predicativity see Kreisel “La
Prédicativité”, Feferman “Systems of Predicative Analysis”, and Wang A Survey
of Mathematical Logic.
12. Moreover, this sort of identification seems quite problematic in itself. No
matter how much we try to “objectify” it, it will involve a reference to me, and
my identity as a person is closely tied up to my conscious activity, including my
thoughts. But we cannot characterize that thought by appealing to me up to any
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time prior to its creation. So it seems to me that any characterization of that
thought will have to appeal to me, and to my thoughts, after the creation of the
thought; for example, the thought I just created, or the thought I created at 2:55
p.m. today, or the thought I created five minutes ago, etc.
This is related to the following remarks by Gödel (“Russell’s Mathematical Logic”, pp. 221-222):
Speaking of concepts, the aspect of the question is changed completely. Since concepts are supposed to exist objectively, there seems
to be objection neither to speaking of all of them nor to describing
some of them by reference to all (or at least all of a given type).
But, one may ask, isn’t this view refutable also for concepts because
it leads to the “absurdity” that there will exist properties ϕ such
that ϕ(a) consists in a certain state of affairs involving all properties (including ϕ itself and properties defined in terms of ϕ), which
would mean that the vicious circle principle does not hold even in
its second form for concepts or propositions? There is no doubt that
the totality of all properties (or of all those of a given type) does
lead to situations of this kind, but I don’t think they contain any absurdity. It is true that such properties ϕ[or such propositions ϕ(a)]
will have to contain themselves as constituents of their content [or of
their meaning], and in fact in many ways, because of the properties
defined in terms of ϕ; but this only makes it impossible to construct
their meaning (i.e., explain it as an assertion about sense perceptions
or any other non-conceptual entities), which is no objection for one
who takes the realistic standpoint. . . . In addition there exist, even
within the domain of constructivistic logic, certain approximations
to this self-reflexivity of impredicative properties, namely propositions which contain as parts of their meaning not themselves but
their own formal demonstrability. Now formal demonstrability of a
proposition (in case the axioms and rules of inference are correct)
implies this proposition and in many cases is equivalent to it. Furthermore, there doubtlessly exist sentences referring to a totality
of sentences to which they themselves belong as, e.g., the sentence:
“Every sentence (of a given language) contains at least one relation
word.”
(Bracketed insertions by Gödel.)
13. In “The Nature and Meaning of Numbers”, p. 64 (theorem 66), Dedekind uses
the totality of his thoughts to prove the existence of an infinite system (set), and
this is indeed problematic precisely for the reasons pointed out by predicativists.
14. The nominalistic tendency can be seen very clearly in Chihara’s Ontology and
the Vicious-Circle Principle. His project is to construct a nominalist alternative
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to Quine’s Platonism, and he does this by constructing a syntactic system for
mathematics which is interpreted syntactically (Chapter 5). As an alternative to
Quine’s Platonism this may be legitimate, because it is essentially in the same
syntactic formal spirit of Quine’s Platonism – the only version of Platonism which
Chihara regards as reasonably intelligible. As a version of nominalism, on the
other hand, it doesn’t seem to me to be convincing, because the assumptions
are much too strong – that kind of syntax isn’t justifiable on strict nominalistic
grounds. Chihara may claim that that’s where the predicativism comes in, and he
refers to objections along these lines as extreme finitism or extreme nominalism
(p. 181), but he gives no justification for predicativism aside from nominalistic
and pragmatic considerations.
15. The claim that the infinite totality of natural numbers is legitimate but that
other infinite totalities, such as the totality of real numbers, are not legitimate,
seems to me completely unconvincing. The totality of natural numbers is supposed
to be “given”. Given how? As a completed totality? How come? And, if not,
then what reason do we have to suppose that it is well-defined? Because induction
is given? But induction is an infinitary principle of proof which presupposes that
the totality of natural numbers (or the successor function) is well-defined. To
say, as Poincaré does, that “[m]athematical induction – i.e., proof by recurrence
– is . . . necessarily imposed on us, because it is only the affirmation of a property
of the mind itself” (Science and Hypothesis, p. 13), doesn’t seem to me to cut
much ice, because it is exactly on a par with Gödel’s claim that “despite their
remoteness from sense experience, we do have something like a perception of the
objects of set theory, as is seen from the fact that the axioms force themselves
upon us as being true.” (“What is Cantor’s Continuum Problem?”, p. 271). The
structuring of mind and reality assumed by Poincaré is just as hypothetical as the
structuring of mind and reality assumed by Gödel – and no less problematic from
an epistemological point of view.
For an early attempt to develop mathematics predicatively along quite
different philosophical lines than Russell’s and Poincaré’s see Weyl The Continuum.
16. Op. Cit., pp. 18-20, 43-45.
17. See “On the Individuation of Attributes”.
18. Thus I disagree with Quine when he says (Op. Cit., p. 100) that “[a]ttributes
are classes with a difference.” What I am suggesting is that it goes the other way
round; i.e., that classes are attributes with a sameness. But Quine himself makes
this point in Quiddities (p. 23): “A class in the useful sense of the word is simply
a property in the everyday sense of the word, minus any discrimination between
coextensive ones.”
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19. See the quotation in Chapter 8 note 26. I will come back to this argument of
Frege’s later.
20. They say (p. 72):
In the case of descriptions, it was possible to prove that they are
incomplete symbols. In the case of classes, we do not know of any
equally definite proof, though arguments of more or less cogency can
be elicited from the ancient problem of the One and the Many.
[Note] Briefly, these arguments reduce to the following: If there is
such an object as a class, it must be in some sense one object. Yet
it is only of classes that many can be predicated. Hence, if we admit
classes as objects, we must suppose that the same object can be
both one and many, which seems impossible.
21. Op. Cit., pp. 101ff.
22. The main paper is “Whither Physical Objects?” but its conclusions were later
incorporated into “Things and Their Place in Theories”.
23. “Things and Their Place in Theories”, pp. 17-18. In “Whither Physical
Objects?” Quine says (pp. 501-502):
There remains the question of ground elements. Take the members
of my sets; then take the members of those members, if such there
be, and so on down, until you get to rock bottom: to non-sets, to
individuals in some sense. These are the ground-elements; and what
are they to be? Not physical objects; they gave way to space-time
regions. But space-time regions gave way in turn to sets of quadruples of numbers; so nothing offers. However, this is all right. Since
Fraenkel and von Neumann, a set theory without ground elements
has even been pretty much in vogue. There is the empty set, there
is the unit set of the empty set, there is the set of these two sets, and
so on. We get infinitely many finite sets in this way. Then we take
all the finite and infinite sets having these as members. Continuing
thus, we suffer no shortages. This is known as pure set theory, and
I seem to have ended up with this as my ontology: pure sets.
24. See Chapter 8, p. 275.
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25. So we have set theories with many units and set theories with only one such.
The latter conform to Frege’s conclusion in The Foundations of Arithmetic (§39)
that if there are units, then they must all be identical to each other.
26. In Foundations of Set Theory, pp. 23-24, Fraenkel, Bar-Hillel and Levy
remark:
Let us refer to those elements which have members as sets, and to
those elements which have no members as individuals. When we
develop a system of set theory we have to make up our mind as to
how many individuals we want to have. The same question arises
also with respect to the sets, i.e., we have to make up our mind
as to “how many” sets we want to have. The latter question is
indeed the central question of set theory, and in answering it we
are guided mostly by the idea of preserving the intuitive rules for
“construction” of sets available in Cantor’s naive set theory, to the
extent that they do not lead to contradictions. In contrast, the
former question is of much less significance. In our answer to the
question we cannot rely much on intuitive arguments since there are
no ways of “constructing” individuals and, as a consequence, there
is nothing to tell us how many individuals to admit. Therefore we
shall be guided here by arguments of simplicity and elegance rather
than by deep insights into the nature of the mathematical universe.
The existence of at least one individual is called for by both philosophical and practical reasons. An individual is needed to serve as
the foundation of the universe. Once we have an individual a, we
can construct a set b whose only member is a, a set c whose only
member is b, a set d whose only members are a and b, and so on.
The way in which the universe of set theory is constructed, starting with a single individual, will be discussed at length in §5. The
practical reasons that call for the existence of an individual are as
follows. When we define the intersection of two sets r and s to be
the set t which consists of those elements which belong to both r
and s, we want the intersection to be defined even in the case when
r and s have no members in common. In this case the intersection t
has to be a memberless element, i.e., an individual. There are also
many other examples where the existence of a memberless element
makes things simpler. The same practical reasons which call for the
existence of such an element also call for using always the same element for the intersection of any two sets r and s with no members
in common, and for referring to this element as a set. Therefore we
shall call this element the null-set and our sets are, from now on,
the elements which have members as well as the null- set. Let us,
however, stress at this point that whereas the existence of at least
one individual is required for serious philosophical reasons, referring
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to one of the individuals as the null-set is done only for reasons of
convenience and simplicity, and can be regarded as a mere notational
convention.
Having decided that we need an individual we now face the question
of whether we need more than one individual. It turns out that for
mathematical purposes there seems to be no real need for individuals
other than the null-set. Therefore, we shall indeed not admit any
such individuals in ZF. Thus all our elements are either sets which
have members or the null-set.
When I read this I thought that their intention was to postulate the null-set as
an individual, i.e., as a memberless element, but not as an element that would be
individuated by the condition of being the memberless element. This is what they
do, however, by defining a null-set as a memberless set, and proving that there is
such a set, and that it is unique by extensionality (p. 39). But this means that
the identity of the null-set is determined by all sets.
27. Russell drew the conclusion that the empty set is a fiction in The Principles
of Mathematics, p. 81:
We agreed that the null-class, which has no terms, is a fiction, though
there are null class-concepts. It appeared throughout that, although
any symbolic treatment must work largely with class-concepts and
intension, classes and extension are more fundamental for the principles of Mathematics.
Later, in Principia, Whitehead and Russell held that all classes (as extensions)
are fictions. And Gödel comments (“Russell’s Mathematical Logic”, p. 223):
Russell adduces two reasons against the extensional view of classes,
namely the existence of (1) the null class, which cannot very well be a
collection, and (2) the unit classes, which would have to be identical
with their single elements. But it seems to me that these arguments
could, if anything, at most prove that the null class and the unit
classes (as distinct from their only element) are fictions (introduced
to simplify the calculus like the points at infinity in geometry), not
that all classes are fictions.
I agree with Gödel, but this is a different discussion. Here one is talking about
classes of entities of one sort or another, not just classes deriving from the empty
class.
28. The same issue that arises for the empty set arises for zero. To identify zero
as the number which is not the successor of any number makes the identification
of zero depend on all numbers and is just as impredicative as the identification of
the empty set as the set which has no set as element. If one wants to “derive” all
numbers from zero by means of successor, then one must take zero as a unit whose
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identity is presupposed. In “The Nature and Meaning of Numbers” (Theorem 66),
Dedekind takes his own ego as the basic unit. This is a natural choice, and the
same choice could be used for sets – though it seems unlikely that it would be a
satisfactory choice for Quine.
29. This is argued by Bealer in Quality and Concept, Chapter 5.
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Chapter 11
Senses
Frege never developed a theory of senses in his published works.
In particular, he didn’t address the question as to the status of senses as
entities. Are senses objects or concepts? There is not a word about this
in “On Sense and Reference” and other related articles. When he contrasts
objects with functions in §3 of The Basic Laws of Arithmetic, he lists numbers, truth values and courses-of-values, but not senses, as objects1 . Since
senses are often referred to by means of descriptive phrases of the form ‘the
sense of . . .’ or ‘the sense expressed by . . .’, one might expect that senses
are objects – which would be in agreement with Frege’s doctrine that the
use of the article ‘the’ always indicates an object. In one of his unpublished
manuscripts, however, Frege draws a distinction for senses that parallels his
distinction between objects and functions. Senses of objects, or senses that
purport to be senses of objects, are objects; whereas senses of functions, or
senses that purport to be senses of functions, are functions2 .
Frege’s doctrine that the use of the article ‘the’ always indicates
that we are talking about an object seems to me to be based on a confusion;
the confusion between the two uses of ‘the’ which I distinguished in Chapter
3. If one holds that ‘the’ always expresses a logical operator which applied
to a level 1 property yields the unique object to which this property applies,
if any, then obviously we either get an object or we get nothing. Since Frege
adhered consistently to this interpretation of ‘the’, his doctrine was clearly
justified for applications of ‘the’ to level 1 properties. If we apply ‘the’ to
a level 2 property of level 1 properties, however, it would seem that what
we should get is the unique level 1 property to which the level 2 property
applies, if any, and it is not at all clear that this should be an object, in any
sense of object. But, for Frege, the fact that we could talk about it as the soand-so indicated a certain kind of completeness, or saturatedness, which he
attributed to objects, and which was precisely what distinguished an object
from a property; a property could only be denoted by a predicate, which is
an expression in need of complementation. Therefore, he saw all phrases of
the form ‘the so-and-so’ as denoting objects, if anything, no matter what
the so-and-so was.
I think that this stands in the way of getting a theory of senses
which reflects accurately Frege’s main idea about them; namely, that a sense
is (or contains) a manner of presentation. In my view, a sense of a level
0 object is any level 1 unary property of that object having unicity as a
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logical characteristic. I.e., a sense (of a level 0 object) is a level 1 property
which can be expressed as
(1) [F x & ∀y(F y ⇒ y = x)](x).
In Chapter 3 (46′ ) I introduced the notation
(2) [!xFx](x)
as short for (1). This was intended to capture the fact that the property is
not merely the property of being an F but that it is the property of being
the F; the expression is still a predicate though, not a name. If we think
of (2) as characterizing a property for any property F, whether F applies
to one, none, or many things, then if F is a unary level 1 property of level
0 objects, is the F is a unary level 1 property of level 0 objects which has
the logical feature that it either applies to nothing, if F applies to nothing
or to more than one thing, or it applies to exactly one thing, if F applies
to exactly one thing. If F is a unary level 2 property of level 1 properties,
then is the F is a unary level 2 property of level 1 properties with the same
logical feature; and so on for all levels.
A (singular) sense as characterized in (1)-(2) is not the property
F, even if it applies to exactly one thing, but is the property [!xFx](x). The
important point is that it is a logical feature of a sense that it purports
to apply to a thing uniquely. But if a sense is the sense of an object,
i.e. it identifies that object, then it does coincide extensionally with the
property F. This coinciding is not a necessary coinciding, however, because
the unicity of the object in question may be a contingent matter. Thus, the
property of being a natural satellite of the Earth applies to only one thing
but it is not a sense; whereas the property of being the natural satellite of
the Earth, as characterized above, is a sense.
There are properties, on the other hand, that although not of the
form is the F are necessarily equivalent to such a property. The property ‘is
an even prime’ is necessarily equivalent to the property ‘is the even prime’,
and could thus be taken to be a sense. We may generalize, therefore, and
characterize a sense as either a property is the F, which has unicity as a
logical feature of it, or a property that is necessarily equivalent to such a
property – for the time being, however, I’ll continue to speak of senses as
properties of the form is the F.
The reason for characterizing a sense as involving unicity as a logical or necessary feature is pretty obvious; namely, that a sense may not
identify anything and still be a sense. One cannot just say that a sense is
a property that applies to at most one thing because it is equally obvious
that the property of being a flying giraffe, which does not apply to anything,
is not a sense. And if we restricted senses to properties that uniquely present
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something, then we wouldn’t be able to talk about senses which do not
present anything.
That’s why senses are generally identified with definite descriptions. One takes something like ‘the pupil of Plato and teacher of Alexander
the Great’ and thinks of it as giving the (or a) sense of the name ‘Aristotle’;
and if there is a unique pupil of Plato who was a teacher of Alexander the
Great, then one says that the definite description determines the denotation
of ‘Aristotle’. But what is the sense here? Is it just the actual definite description? The phrase? What makes that a sense? Well, it’s the properties
of being a pupil of Plato and of being a teacher of Alexander the Great. But
then why not say that it is the conjunction of these two properties which
is the (or a) sense of ‘Aristotle’ ? Because if there had been several pupils
of Plato who were teachers of Alexander, then the conjunction of the two
properties wouldn’t have been a sense of ‘Aristotle’. If, however, the condition is that in order for a property, or conjunction of properties, to be a
sense it must uniquely present an object, then one wouldn’t be able to talk
about the sense expressed by the name ‘Pegasus’. This is the main reason
why Frege’s theory of senses was transformed by the description theorists
into a theory of references and their determination, rather than a theory of
senses3 .
Frege’s ‘the’ in his presentation of senses has some of the features
of the notion of sense which I suggested above. But since Frege did not
recognize the two distinct uses of ‘the’, he did not formulate his theory of
senses as a theory of properties with certain logical features; so, he formulated it as a theory of objects and functions whose nature remains rather
unclear. But if we look at the way he used the notion of sense, particularly
in connection with names, we can see that the interpretation of senses as
properties is quite natural. It seems clear to me, therefore, that Frege was
not a description theorist in the usual sense4 .
For a level 0 object or a property in the hierarchy there may be
many senses of that object or property, though there are also many senses
which are not senses of anything. To be a sense of a level 0 object or of a
property is to be a property that uniquely presents that object or property
and such that the unicity of the presentation is a logical feature of the
sense, which agrees completely with Frege’s idea that senses are manners of
presentation.
Let me turn now to states of affairs and their senses. A sense of
the state of affairs
(3) <Philosopher, Quine>
presumably combines in some way a sense of the property of being a philoso-
377
pher with a sense of the object Quine. How does this combination take
place? In the case of (3) we may have a level 2 unary property [!ZGZ](Z) of
level 1 unary properties which is a sense of the property of being a philosopher, i.e. such that
(4) [[!ZGZ](Z)]([x is a philosopher](x))
is true, and a level 1 unary property [!xFx](x) of level 0 objects which is a
sense of Quine, i.e. such that
(5) [[!xFx](x)](Quine)
is true. What we want as a sense of (3) “resulting” from the senses [!xFx](x)
and [!ZGZ](Z), is a property of level 1 states of affairs which applies uniquely
to (3) by combining those senses in some way. But what is a sense of the
state of affairs (3) as we understand it in terms of instantiation? It is that
the property of being a philosopher applies to Quine. So, using the notation
introduced in (7) of Chapter 9, it would seem that a sense of the state of
affairs (3) is given by
(6) [[!ZGZ](Z) & [!xFx](x) & Zx](<Z, x>).
This is a completely general analysis of the senses of states of
affairs, because any state of affairs consists of a property of a certain type
and of arguments of appropriate types to which the property applies. By
using different senses of the property or arguments we get different senses
of the state of affairs, but it will be the same state of affairs.
Properties of the form (6) exist independently of their being senses
of a given state of affairs. If [!ZKZ](Z) is a sense of the property of being
a dentist, and [!xFx](x) is a sense of Quine, then
(7) [[!ZKZ](Z) & [!xFx](x) & Zx](<Z, x>)
is also a property of states of affairs that is a sense, except that it does not
identify a state of affairs.
These senses, which may or may not identify a state of affairs, are
my suggestion for an ontological interpretation of the notion of proposition
– and are also an interpretation of Frege’s notion of thought. Given the
property plus arguments analysis of the logical form of sentences, any sentence thus analyzed would express a proposition in this sense; and the true
sentences will be precisely the ones whose sense identifies a state of affairs
as their denotation.
It follows from the preceding account that senses, including propositions, are completely independent of language. Hence, the basic ontological features of Frege’s account of senses are clearly preserved in this
interpretation of their nature. Senses are ontological determinants of reference, existing independently of language and of our knowledge of them5 .
Nevertheless, senses can be coded in linguistic signs.
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Frege held that a linguistic sign expresses a sense which may or
may not determine an entity; if it does, then the sign denotes that entity.
With reference to names he says6 :
A proper name (word, sign, sign combination, expression) expresses
its sense, stands for or designates its reference. By means of a sign
we express its sense and designate its reference.
There seem to be three different relations involved here; the relation between
a sign and its sense, the relation between a sense and the entity presented
by it, if any, and the relation between a sign and its reference, if any. For
many years I was puzzled by the fact that Frege says almost nothing about
the relation between a sign and the sense expressed by that sign. What sort
of relation is that? Shouldn’t one characterize it in some way? How does
a sign express a sense? In virtue of what? Since the usual view in logic
and philosophy is that the semantics is “attributed” to an independently
given syntax, I felt that it was a failure on Frege’s part not to tell us how a
syntactic expression expresses a sense. This puzzlement was compounded by
the fact that for Frege a sign is not a purely syntactic thing, but something
that already contains a sense; so it doesn’t seem to make much sense to
separate the sign and the sense contained in the sign and to say that the
sign expresses that sense. If the sense exists independently of the sign, then
what is the sign7 ?
The problem is not Frege’s, though, but derives from a conception
of syntax and semantics which puts the cart before the horse. We don’t
start from an independently given syntax and then go on to attribute senses
and denotations to our syntactic strings – at least not in any real language,
ordinary or scientific. Syntactic strings are a way of capturing independently
existing senses and entities; we start with the semantics, so to speak, and
then go on to codify it syntactically. A syntactic string is a code for a
sense, and by thus expressing a sense it is a sign. We grasp senses, as Frege
says, and codify them syntactically8 . This means that the relation between
a syntactic string and a sense does not go from syntactic string to sense,
but from sense to syntactic string. And it makes perfectly good sense to
say that a sign, which is a code for a sense, expresses that sense, which
exists independently of being coded by that sign. Moreover, since what
determines the reference of the sign, if any, is the sense, the relation of
denotation between sign and entity goes via the sense.
Actually, the situation is somewhat more complex, because we
don’t really start either from the syntax or from the semantics but from
both. It is important to distinguish senses of entities in general, as I characterized earlier, from senses of signs. A sign has an additional dimension
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which has to do with its syntax and semantics. In order to make this distinction perspicuous, I will discuss the case of definite descriptions.
The primary sense of a definite description ‘ιxF x’ involves the
property [!xFx](x) but is not merely the property [!xFx](x); for it involves
also the syntactic and semantic character of ‘ιxF x’ as a sign. That is, it
involves that ‘ιxF x’ is a singular term that denotes an entity if, and only
if, the property F applies uniquely to that entity. Thus, the primary sense
of ‘ιxF x’ is the complex property
(8) [!z(z is a singular term & ∀x(z denotes x ⇔ (F x & ∀y(F y ⇒ y =
x))))](z).
If we consider it fixed by the syntactic and semantic rules of the language
that
(9) ‘ιxF x’ is a singular term,
and
(10) ∀x( ‘ιxFx’ denotes x ⇔ (F x & ∀y(F y ⇒ y = x))),
then we may say that, relative to the rules of the language, the sense expressed by ‘ιxF x’ is the property [!xFx](x) – which is the manner of presentation of the denotation of ‘ιxF x’, if any. But to say that the property
[!xFx](x) is the (or a) sense of ‘ιxF x’ would lead to misunderstanding, because there is nothing in the character of this property that determines the
character of ‘ιxF x’ as sign9 .
Transposing Frege’s ideas to the present context, we may say that
every sign in a language expresses a sense, though many signs may express
different senses for different people – or may have different senses associated
to them. An ordinary proper name expresses a sense which may or may
not determine a reference for that name; if it does, the reference will be a
level 0 object. A (simple) predicate expresses a sense which, again, may or
may not determine a reference for that predicate; if it does, the reference
will be a property of some level. Complex designators such as descriptions
or sentences, and complex predicates of various sorts express senses which
are determined by the senses of their parts and by their logical structure.
These senses may or may not determine entities as referents of the sign in
question. The reference of a definite description may be any entity in the
hierarchy; the reference of a declarative sentence is a state of affairs; the
reference of a complex predicate is a property.
This raises some questions, however. In particular, it raises the
much discussed question as to what is an adequate account of the senses
expressed by proper names. In order to suggest such an account, and also
to clarify some other points raised in earlier chapters, I shall examine briefly
Kripke’s views in Naming and Necessity concerning the reference of proper
names and natural kind terms.
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Suppose you got a dog. You may decide to give him a name, and
for various reasons you have deliberately chosen ‘Freddie’. By your decision
to name your dog ‘Freddie’, you have fixed the reference of ‘Freddie’; you
intend that name to be the name of your dog and act upon this intention by
calling him ‘Freddie’. You may, however, get the dog, and because of some
circumstance or other start calling him ‘Freddie’; he reminds you of your
friend Fred, for instance, or you’ve had in the past another Freddie, and
he reminds you of him, or he has some characteristics which suggest that
name to you. Thus the many Blackies in the world, and Happys, and Ladys,
etc10 . In these cases you can fix the reference without making a deliberate
choice; you start calling him ‘Freddie’, maybe as a joke, and then it sticks.
You fix the reference of ‘Freddie’ by your acts of calling him ‘Freddie’ with
the intention that you are calling him ‘Freddie’.
This idea of fixing the reference of a proper name is one of the
central ideas in Kripke’s view of naming. There are other ways in which we
can fix the reference of a name, and one important case is when we do it by
description. I wake up during the night by what feels like a mosquito bite.
Being a philosopher, I decide to fix the reference of ‘Bloody’ to the mosquito
that’s just been biting me, and set out on a hunt for Bloody; maybe I find
him, maybe not. If there is a unique mosquito that was biting me, then my
deliberate decision to refer to that mosquito as ‘Bloody’ fixes the reference
of the name. It is essential for this that the description picks out a referent,
however; if what I have is a funny kind of allergy that feels and looks like
mosquito bites, or if there were several mosquitoes feasting on me, then I
haven’t fixed the reference of ‘Bloody’11 .
Kripke extends the notion of fixing the reference to other terms
as well, such as natural kind terms – i.e., terms like ‘gold’, ‘water’, ‘tiger’,
etc., that refer to natural kinds. This depends on the idea of using samples
and paradigmatic cases to fix the reference of such terms. The reference of
‘water’, for instance, gets fixed by using the word to refer to certain aspects
of reality that are encountered from time to time, in various forms and
shapes, and by intending to refer to that kind of “thing” that is common
to, or underlies, or that is the nature of, the various samples to which the
word is applied. Similarly, the reference of ‘tiger’ gets fixed to a certain
kind of thing, depending on a sample or even on a single paradigmatic case,
and the reference of ‘gold’ gets fixed to a certain kind of thing as well. As
before, this may be deliberate or not, depending on cases, and it can also
be done by description rather than by acquaintance12 .
As in the case of proper names, there are several reasons why the
reference of a word may fail to get fixed. If the sample to which ‘gold’
starts being applied consists not of one substance but of two or more, then
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there may be nothing (no kind) common to that sample to which the word
‘gold’ applies; and, in that case, a reference has not been fixed. If we used a
description such as ‘the substance that has such and such properties’, and
there is no such substance, then again the reference doesn’t get fixed. If
we intend to refer to a certain kind as ‘tiger’, and our sample (or paradigmatic case) consists of clever robots manufactured by the Japanese, then
we didn’t fix the reference of ‘tiger’. Similarly, if what you got is a Japanese
manufactured robot that you think is a puppy, then you fail to fix the reference of ‘Freddie’ to the robot. The uniqueness, uniformity, existence, and
intentional fit are essential for reference fixing to be successful13 .
Kripke talks about reference fixings as ‘baptisms’, but it is important to understand this in a sufficiently broad sense that goes far beyond
actual ceremonial baptisms – of people, animals, boats, cities, substances,
etc. It is also important to realize that in practice reference fixings can be
very messy affairs and that this is an oversimplified view of them. This
is one of the reasons why Kripke calls his views a picture rather than a
theory14 .
Once reference has been fixed, however, names and natural kind
terms can be transmitted and acquired with their reference. If you tell me
that the reason your pants are torn up is that your little dog Freddie had a
go at them, you are transmitting to me, and I am acquiring from you, the
name ‘Freddie’ with its reference to Freddie. If I go on to tell other people
that they shouldn’t think you eccentric for going around wearing torn up
pants, but that Freddie pulled at them as you came into the house, and I
intend to use ‘Freddie’ to refer to whatever you were referring to, then there
has been a successful transmission of the name from you to me.
Kripke goes on to make many interesting and innovative points
about naming, necessity, essence, etc., based on this picture of naming. In
particular, he makes the important point that as a name gets transmitted
many false beliefs can arise, so that by the time ‘Freddie’ gets to someone
else, he may think that Freddie is a cat, or a baby, or whatever; yet, if
the transmission is successful, he is also referring to Freddie when he uses
the name with the intention to refer to the same thing as the person from
whom he acquired it. Kripke appeals to our intuitions about various cases
of this sort to refute Russell’s view that proper names abbreviate definite
descriptions. In my view he is quite convincing about this, even if there
may be occasions in which we may properly say that we are using a name
to abbreviate some definite description15 .
In the addenda to his lectures, Kripke argues that the commonly
held view that although Sherlock Holmes did not exist in our world, Sherlock
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Holmes could have existed in other possible worlds, is wrong. Given that
the name ‘Sherlock Holmes’ is a fictional name, its reference has not been
fixed in any way in our world, and though there may be a possible world (or
counterfactual situation), similar to ours in many respects, in which there
is a detective that performed the various exploits related by Conan Doyle
in the Sherlock Holmes stories, this is not a world (or situation) in which
this individual would have been Sherlock Holmes.
Similarly, Kripke argues that given that ‘unicorn’ is a mythical
natural-kind term, even if in other possible worlds (or counterfactual situations) there were animals that look like what unicorns are supposed to look
like, these animals would not be unicorns. And the reason is the same as
for the case of ‘Sherlock Holmes’; namely, that the description of unicorns
provided by the myth does not fix the reference of ‘unicorn’ in our world.
If the reference of ‘unicorn’ had been successfully fixed, there would have
been a natural kind with a certain biological structure, but the myth only
tells us what unicorns were supposed to look like. This fixes the reference
of such a predicate as, say, ‘looks like a unicorn’, through the description
‘looks like a small horse with a single horn on its forehead’, but doesn’t fix
the reference of ‘unicorn’16 .
In other worlds there could be many individuals who realize the
Sherlock Holmes stories, but nothing about the name ‘Sherlock Holmes’,
our world, or these worlds, would determine which one of them is Sherlock
Holmes. As opposed to, say, other worlds in which Conan Doyle had an
altogether different life than the one he actually had in our world, but in
which there was a man who had a life essentially similar to Conan Doyle’s
in our world. Facts about our world, including facts about naming and
reference, determine that such a man isn’t Conan Doyle. This depends on
Kripke’s view that names are rigid designators, and, therefore, that given
that ‘Conan Doyle’ designates Conan Doyle in our world, it must designate
Conan Doyle in every world in which he exists – no matter what his life
history, including his name, may be in that world17 .
Similarly, there could just as well be other worlds containing different species of animals that fit the description of unicorns provided by the
myth, without there being anything about the word ‘unicorn’, our world, or
these other worlds, which determines which among them are the unicorns.
Although Kripke considers this view about ‘unicorn’ to be rather
controversial, I have always found it strikingly persuasive. What I would
conclude from it is that the word ‘unicorn’, just like the name ‘Sherlock
Holmes’, simply does not have a reference. This is part of the point of
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Kripke’s remarks, but one can argue for it independently of any views concerning modality and possible worlds18 .
Kripke claims that his account of proper names is directly opposed
to Frege’s, in that it incorporates Mill’s view that proper names have denotation but do not have connotation (or express something like a sense),
and also holds, against both Mill and Frege, that natural kind terms do not
have connotation (or express something like a sense) either19 . While basically agreeing with Kripke on some main aspects of his picture of naming,
I disagree with him on this characterization of it. In fact, I think that his
picture involves the notion of sense and that it provides the basis for an
account of the senses expressed by proper names. I shall now argue for this
appealing to my earlier discussion of senses.
It is quite obvious, to begin with, that reference fixings for proper
names involve a manner of presentation which may or may not identify
an object. In a ceremonial baptism, the manner of presentation may be
something like the property ‘is the baby in front of me now’, or ‘is the
boat to which I am pointing’, or ‘is the building whose foundations we are
laying’, etc. Not that these words have to be spoken, but the very baptismal
situation involves a manner of presentation that can be characterized as such
a property; i.e., as a sense. In a reference fixing via a description, the sense
is the descriptive property. Thus, the reference of ‘Bloody’ is fixed by the
manner of presentation ‘is the mosquito that’s just been biting me’20 . In
less explicit baptisms the manner of presentation may not be so explicit
either, and may involve several presentations over a period of time, but, in
any case, there must be a manner of presentation in order for there to be
an intention to fix the reference of the name to some one thing.
When you fixed the reference of ‘Freddie’, you fixed the sense expressed by the name ‘Freddie’. Similarly, when I fixed the reference of
‘Bloody’, I fixed the sense expressed by the name ‘Bloody’. When the name
gets transmitted, it gets transmitted with the sense it expresses, though the
person acquiring the name may not know which sense this is. The transmission of the reference depends on this. I.e., if the sense expressed by the
name identifies an object, then, by transmitting the name with the sense
it expresses, the reference gets transmitted as well. If, on the other hand,
the sense expressed by the name does not identify an object, then there is
no reference to be transmitted; but that this is so depends on the name
expressing a sense. Just as the person acquiring the name may not know
which is the sense expressed by the name, he may not know which is the
reference of the name, if any – but he might.
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In fact, it is much more natural to say that what gets fixed in a
formal or informal baptism is the sense expressed by the name, rather than
its reference. Partly because the name may not have a reference, and partly
because in the case of fictional names there is no reference fixing at all. The
name ‘Sherlock Holmes’ was introduced into the language by Conan Doyle
in a certain specific context; namely, when writing the first Sherlock Holmes
story. Let’s suppose that he decided to write a story involving a fictional
detective whom he was going to call ‘Sherlock Holmes’. By this decision he
fixed a sense expressed by the name ‘Sherlock Holmes’; something like, say,
‘is the fictional detective who will be the main character in my story’. Again,
he may not have said such words, or even thought such a thought, but the
actual situation involves some kind of intention that can be characterized by
some such property. Since this property involves the qualification ‘fictional’,
it is part of the sense expressed by the name ‘Sherlock Holmes’ that it is not
the sense of an object. And even if, unknown to Conan Doyle, there was
a man whose name was ‘Sherlock Holmes’ who had performed exactly the
exploits that Conan Doyle wrote in his story, the name ‘Sherlock Holmes’
as introduced by Conan Doyle would not refer to this man.
When we fix the reference (or sense) of a name, and then go on to
use this name, we may associate many other senses with the name. Thus,
you may associate to the name ‘Freddie’ the sense ‘is the puppy that tore
up my pants’, or the sense ‘is the puppy that aunt Clara gave me’, or any
number of other senses. We may also associate to the name properties such
as ‘is a dog’, ‘barks very loudly’, etc. which aren’t senses. Some of these
senses and properties may be senses or properties of Freddie and some may
not. Suppose, for instance, that your son wanted to have a puppy, but
that he knew that you didn’t. He may convince you that aunt Clara gave
you the puppy and that you can’t refuse aunt Clara’s gift. In this way you
may associate a sense to the name ‘Freddie’ which is actually not a sense of
Freddie – i.e., in Kripke’s terminology, you may come to have a false belief
about the referent of ‘Freddie’. Similarly, when I acquire the name from you,
I may associate (immediately or over time) many senses and properties to
the name ‘Freddie’. Some of these senses and properties may be senses
or properties of Freddie and some may not. It is quite possible, in fact,
as Kripke maintains, that none of the senses or properties that I associate
with ‘Freddie’ are senses or properties of Freddie – i.e., that all my beliefs
about the referent of the name are false. The name still expresses its sense,
however, and it is because of this that if I use it with the intention to refer
to whatever you were referring to, I refer to Freddie.
We can distinguish, therefore, the sense expressed by a proper
name, which is fixed when the name is introduced into the language, from
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senses that a person (or social group) associates with the proper name at a
certain time, and from properties that a person (or social group) associates
with the proper name at a certain time. The totality of properties (including
senses) that a person or group associates with a proper name at a certain
time is what we may call the connotation of the name for that person or
group at that time. Thus, the connotation of the name ‘Aristotle’ for me
(and a certain group to which I belong) includes at this time such properties
as ‘was a philosopher’, ‘was a pupil of Plato’, ‘was born in Stagira’, etc.
Some of these properties may be combined into senses which I also associate
with the name ‘Aristotle’ at this time; senses such as ‘is the most important
philosopher who was a pupil of Plato’, ‘is the philosopher born in Stagira
who was a teacher of Alexander the Great’, etc. These properties and senses
may or may not be properties of Aristotle, though I believe they are, and it
could even happen that there is no referent of the name ‘Aristotle’, though
I believe there is. Whether this is so depends on the sense expressed by the
name ‘Aristotle’, as I am using it, which I simply do not know21 .
I think that this account of the senses expressed by proper names,
of senses associated to proper names, and of the connotation of proper
names, is compatible with much of what Kripke says in his lectures – though
not with his remarks concerning senses and connotation. And it is also
compatible with much of what Frege says about the senses expressed by, or
associated to, proper names22 . It is not quite compatible with what Russell
says, but even in Russell’s case it gives us a way of understanding some of
his intuitions as correct intuitions.
It seems quite plausible to me, for instance, that if I use a name,
then I generally associate some sense to it. Since a sense is a property of
the form is the F, it follows that when I use a name I generally associate
some descriptive property to it. Let me illustrate this with one of Kripke’s
examples. If I hear a conversation about someone called ‘Feynman’ – an
anecdote, perhaps – I may want to go on and use the name even if I have
no idea of whom Feynman may be. I may say: “I just heard a funny
story about Feynman”. In this case the sense that I associate to the name
‘Feynman’ is probably something like the property ‘is the person they were
talking about’. If they were indeed talking about Feynman, then this is a
sense of Feynman; but even if this is not the case, it doesn’t matter. Of
course, it doesn’t follow that the name ‘Feynman’ is an abbreviation for
the description ‘the person they were talking about’, even for me, and this
is Kripke’s main point against Russell. Nevertheless, Russell’s intuition
that there is a descriptive property associated with the name seems to me
essentially correct23 .
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Let me turn now to natural kind terms. It is clear that baptisms
of natural kinds also involve a manner of presentation. What may not be so
clear is what it is that is presented. Suppose that I am confronted with some
specimens of a new kind of animal and that I introduce the term ‘akiba’ to
refer to that kind. Suppose, moreover, that these animals are a natural kind.
To what am I fixing the reference of the term ‘akiba’ when I say (something
like): “I shall refer to this kind of animal as ‘akiba’”? I certainly do not
intend to fix the reference of the term to the actual animals in front of me,
or even to the totality of such animals in the world, but, rather, to that kind
of thing – as Kripke says24 . I would say that what I am fixing the reference
to is the common nature of these things, or to what these things have in
common. But what is it that these things have in common? Whatever it
is, it is some of what I’m referring to as a property. If I want to refer to
the actual animals, then I use a description operator. Thus, if I want to
say that the akibas in the field are grazing, I use the plural description ‘the
akibas in the field’ to refer to the totality of akibas in the field.
What confuses the issue a bit is that when I fix the reference of
‘akiba’ (or ‘tiger’, to take a real example) to a property expressing the
common nature of these animals, this property also has (or may have) an
extension. Since logical relations are generally extensional, it may seem
that when I use such terms I am referring to the extension of the property
(or to the actual things to which the property applies) rather than to the
property itself. Thus, if I say that tigers are mammals, it may seem that
I am predicating the term ‘mammal’ of the actual animals. Though in
a sense this is true, because the relation that I am asserting is usually
interpreted as a universal relation between the extensions of the two terms,
the relation is actually an extensional relation between the properties ‘tiger’
and ‘mammal’. To see this, consider the case where we find an animal that
looks very much like a tiger which upon examination we discover not to
be a mammal. We may say: “If this had been a tiger, it would have been
mammal.” Or: “Since it is not mammal, it isn’t a tiger.” It seems clear here
that by ‘tiger’ we are referring to the property, not to the extension. In fact,
it seems that such talk of tigers in other possible worlds (or counterfactual
situations) wouldn’t make sense if by ‘tiger’ we were to refer to the actual
animals that exist in our world.
Thus, just as we can distinguish the sense expressed by a proper
name from senses and properties that one or another person associates to
that proper name, we can also distinguish the sense expressed by a natural
kind term from senses and properties that are associated to that natural
kind term. The properties (and senses) that we associate with a natural kind
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term are actually properties of the property denoted by the natural kind
term. These are properties such as:
(11) All Z’s are mammals,
(12) No Z is trustworthy as a pet,
(13) Some Z’s are born with three legs,
etc. These properties are level 2 properties of level 1 properties and they
can be expressed by:
(11′ ) [∀x(Zx ⇒ x is mammal)](Z),
(12′ ) [¬∃x(Zx & x is trustworthy as a pet)](Z),
(13′ ) [∃x(Zx & x is born with three legs)](Z).
Also the property ‘is an animal which is a large feline with black stripes
natural of Asia’ is a property of kind (11). This property is part of the
connotation of ‘tiger’ but is not a sense25 .
As Kripke maintains, we may be mistaken in many (or all) our
beliefs concerning a natural kind term. And we may also be mistaken in
our belief that there is a natural kind to which the term refers. That this
is so depends on our having fixed the sense expressed by the natural kind
term in a certain way.
I conclude, therefore, that Kripke’s theory of reference fixing for
natural kind terms is also largely compatible with a theory of senses as I
have developed it. In fact, as in the case of proper names, I think that
his theory is more naturally interpreted as a theory of sense fixing than as
a theory of reference fixing. In both cases the fixing of the reference, if
any, depends on the fixing of the sense. And, as in the case of the name
‘Sherlock Holmes’, the points that Kripke wants to make about such terms
as ‘unicorn’, ‘centaur’, etc. are quite natural in this approach. If the term
‘unicorn’ was introduced in the context of a myth, then it is part of the sense
expressed by ‘unicorn’ that its purported reference is a mythical species;
and, hence, that there is no property to which ‘unicorn’ refers. The term
‘unicorn’, just like the name ‘Sherlock Holmes’, simply does not refer26 .
Also scientific terms like ‘phlogiston’ or ‘ether’, which were not introduced
in a mythical context, but which express a sense which does not identify
a property, do not refer to a property – and, therefore, do not have an
extension (not even empty). It does not follow, however, that such terms
cannot be used to form complex predicates that do refer to properties. Given
that Conan Doyle described in some detail various characteristics of his
fictional character Sherlock Holmes, we may have predicates like ‘looks like
Sherlock Holmes’, ‘reasons like Sherlock Holmes’, ‘plays violin like Sherlock
Holmes’, etc., that refer to properties that may or may not apply to one or
another person. Similarly for such predicates as ‘looks like a unicorn’, ‘is
analytical as a Vulcan’, ‘behaves like a faun’, etc27 .
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But let me now turn to some other issues concerning the notion of
sense.
As opposed to the ontological account of senses presented above, it
has been suggested that Frege’s notion of sense has a basic epistemological
motivation and is really an epistemological notion. The evidence for this
is that the puzzle about identity with which Frege starts “On Sense and
Reference” is stated partly as an epistemological puzzle. Since the notion of
sense is a main ingredient in Frege’s solution of the puzzle, it is concluded
that the notion of sense must be epistemological28 . This seems to me a non
sequitur, however.
If we look at the initial introduction of the notion of sense in pp.
26-27 of “On Sense and Reference”, it is quite clear that we are dealing with
an ontological notion. There is nothing epistemological in the fact that the
only even prime is the smallest natural number whose positive square root
is irrational. This is an ontological feature of the properties of being even,
being prime, being irrational, being a natural number, being a positive
square root, and being smaller than. It is in the nature of these properties
that there is only one number that is both prime and even; and it is also
in the nature of these properties that this number is the smallest natural
number whose positive square root is irrational. Where is the epistemology
in that? This is how the world is. Period. It does not follow, of course,
that how the world is isn’t relevant to epistemological issues; of course it
is. But even if Frege’s notion of sense is essential to his solution of a partly
epistemological puzzle, it doesn’t follow that it is an epistemological notion.
As a matter of fact, I think that Frege’s epistemological characterization of the puzzle was somewhat muddled by his appeal to a priority29 .
It is simply not true that statements of the form a = a can be established
a priori – though some of them can, as can some statements of the form a
= b. We cannot establish a priori that
(14) Homer = Homer,
for instance, because we do not know whether ‘Homer’ denotes. In fact,
it seems quite obvious that any way to establish (14) will have to involve
particular facts. And even leaving aside proper names, if we could establish
that
(15) the author of the Odyssey = the author of the Odyssey,
we would thereby establish that there was a unique author of the Odyssey,
which is an empirical discovery for which many scholars would be grateful.
And, in my view, to pooh-pooh these examples would be to show a total
lack of concern for the facts about language and about its connection to
reality.
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If, on the other hand, we assume as a general principle about names
that they do denote, then we can formulate Frege’s initial claims as follows.
Suppose that ‘a’ and ‘b’ stand for names. Then
(16) a = a
holds a priori, depending on the general principle about names, but
(17) a = b
does not generally hold a priori – although it might, as in the case of
mathematical identities30 . If ‘a’ and ‘b’ stand for names of ordinary objects,
then if (17) is true, it is a posteriori in that its justification depends on
particular facts. Given Frege’s characterization of the notions of a priori and
a posteriori, these are epistemological claims of a general nature concerning
the kinds of justification necessary to establish certain propositions. It
follows, however, that in order to know a specific a posteriori identity of
form (17), we must know some particular facts. Or, in other words, if a = b
is true a posteriori and we know that a = b, then we know some particular
facts31 .
Now the puzzle about identity is the following. If identity is simply
a relation between the objects designated by ‘a’ and ‘b’, then it would seem
that what we know when we know that a = b is that an object is selfidentical, which is precisely what we know when we know that a = a. So,
what are the additional particular facts that we know when we know an a
posteriori identity of the form a = b? If we say that these are facts about
the names ‘a’ and ‘b’, and think of these names as mere marks or labels
which have been arbitrarily attached to some objects, then the particular
fact in question is that the labels ‘a’ and ‘b’ have been attached to the
same object. Therefore, our knowledge would be knowledge of particular
arbitrary decisions to attach labels, which is not really knowledge about
the object to which these labels have been attached; it is knowledge about
labelling. If, for example, in a supermarket somebody goes around putting
different labels on various items, and maybe more than one on some items,
and I find an item with the labels ‘a’ and ‘b’ on it, then I know that the item
labelled by ‘a’ is the same as the item labelled by ‘b’, but this knowledge
tells me nothing about the item itself. So, although I do know something
beyond knowing that the item labelled ‘a’ is the same as the item labelled
‘a’, this is not knowledge about the item as such.
This problem is not a new problem that Frege is raising in “On
Sense and Reference”; it is precisely the problem that led to his characterization of identity in Begriffsschrift – and which he takes up again in
The Foundations of Arithmetic32 . He already had the notion of sense in
Begriffsschrift as the way in which something is determined. In fact, for
identity as a relation between objects, he already had the solution of the
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problem in Begriffsschrift; and it would have been exactly the same solution
he gave later in “On Sense and Reference”. What confused the issue was
that the signs of Begriffsschrift represented contents and that identity had
to hold for contents generally, including judgeable contents. If he identified
the content of a name with the way in which the reference of the name is
determined, which in a sense he does, then all identity statements would
be statements about contents, not about the objects determined by these
contents; so he also talks about content as the object that is determined
in a certain way. In the case of the content of a judgment, however, there
seems to be no object corresponding to the content. Therefore, in order to
have a uniform solution that made identity a relation between objects determined by the contents, he would have had to introduce objects determined
by the contents of judgments. Thus in Begriffsschrift he treats identity as
a relation between names – and gives a somewhat different explanation of
the relation between names, contents, objects, and ways of determining the
content (or object)33 .
Although in “On Sense and Reference” Frege does introduce clearly
the distinction between sense and denotation, there is nothing really new
in that as far as names are concerned; the main point of the paper is to
introduce the truth values; i.e., the objects determined by the contents of
judgments – and most of the paper is a justification for that. The fundamental aim of the paper concerns the notion of truth. Instead of ‘On Sense
and Reference’, he could have used ‘On Truth’ as the title; and it may have
been a better title.
Thoughts are the determinants of truth, and truth is objective.
What determines the truth of a thought is reality. How does reality do
this? Through the structure of the thought. This structure consists of logical relations between determinants of concepts and objects, which is what
reality is made of – and the logical relations themselves are determinants of
concepts; namely, logical concepts. What a thought determines in reality,
if anything, must be an object for Frege, because a thought is complete in
a fairly obvious sense. Since all objects are level 0 objects for Frege, and
since he didn’t see how to identify specific objects determined by specific
thoughts – partly because of the substitutivity argument – he came up with
the clever idea of the global objects the True and the False.
It seems to me, therefore, that the tendency to consider “On Sense
and Reference” as primarily concerned with epistemological issues is a misinterpretation of Frege, which gets compounded by the further confusion of
senses with meanings. I do agree, however, that Frege leaves room for this
interpretation through his handling of the notion of thought. The problem
is that there is no way to justify the introduction of truth values directly
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as the objects denoted by sentences. And he couldn’t just postulate them
because this would have had no credibility. This is precisely what happens
nowadays when people introduce truth values by fiat and then start flailing
about in search of an interpretation. So Frege appealed to thoughts as the
objective contents of sentences, and argued his way to truth values.
To deny that both the discussion in “On Sense and Reference”
and that the notion of sense itself are primarily epistemological, is not to
deny the epistemological importance of the paper and of the notion of sense.
Ontology is always fundamental to epistemology. Since senses are manners
of presentation they give us a road to the entities thus presented, if any. And,
as Frege claims, complete knowledge of an entity would involve knowledge
of all presentations of that entity by means of its properties. Even if one
claims that direct acquaintance with an entity does not involve senses, which
I would deny, it is still the case that many entities can only be known
through descriptive properties. The way we ordinarily “get” to things is
through their properties. If I tell you that I am lecturing in room 306 of
such-and-such building, I’m giving you a way to get to the room; if I tell you
that I’m lecturing in the room with the broken window pane, I’m giving you
a different way to get to the room. Senses need not be “epistemological” to
serve an important epistemological function.
Since senses are not linguistic I may discuss objects and other entities and their senses independently of any discussion of language – though it
is inevitable that the expression of senses is generally done linguistically34 .
Frege’s more systematic discussion of senses came later than “On Sense and
Reference” in some of his unpublished papers and in “Thoughts”. Although
there are many difficult issues that are raised by these discussions, I think
that the account of senses which I presented above is compatible with some
controversial points that Frege makes in these papers. Consider, for instance, his discussion of the indexical ‘I’ in “Thoughts”, which has led to
much debate.
Frege says that “everyone is presented to himself in a special and
primitive way, in which he is presented to no-one else”, and concludes that
thoughts expressed using ‘I’ with this sense are incommunicable35 . If I want
to communicate to you that I am in my office, I must do it by expressing
a thought that you can also grasp. Therefore, if I say to you ‘I am in my
office’ with the intent of communicating to you that I am in my office, then
I must use ‘I’ with a sense that can be grasped by you; something like,
says Frege, “he who is speaking to you at the moment”. This seems to
me quite plausible36 , but an objection that is raised against it is that such
incommunicable thoughts would not be the sort of objective entities which
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Frege held thoughts to be. How can it be objective if it cannot be grasped
by anyone else37 ?
One can certainly hold, however, that there are thoughts that are
so complex that they cannot be grasped by anyone at all; which wouldn’t
make them any the less objective, but merely ungraspable by human minds38 .
Similarly, there doesn’t seem to be anything absurd in maintaining that each
human mind has privileged access to certain objective thoughts which cannot be grasped by anyone else. From the fact that a thought is objective it
doesn’t follow that my consciousness cannot give me privileged access to it.
In fact, I don’t think that the questions that are raised by these
remarks of Frege’s are necessarily questions about demonstratives. If a sense
is, as I hold it to be, a manner of presentation, then there is no reason why
I cannot present an entity through identifying properties that involve my
private experiences. My appendix can be presented as the organ of my
body that is presently inflamed, and it can also be presented as the organ
of my body that is causing the pain I am feeling at this moment. The later
sense (‘is the organ of my body that is causing the pain I am feeling at this
moment’) is a property that involves a private experience, but it doesn’t
follow that such properties aren’t properties in an objective sense.
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Notes
1. See the last quote in note 1 of Chapter 9.
2. “Introduction to Logic” p. 192:
A sentence proper, in which a proper name occurs, expresses a singular thought, and in this we distinguish a complete part and an
unsaturated one. The former corresponds to the proper name, but
it is not the meaning of the proper name, but its sense. The unsaturated part of the thought we take to be a sense too: it is the sense
of the part of the sentence over and above the proper name. And it
is in line with these stipulations to take the thought itself as a sense,
namely the sense of the sentence. As the thought is the sense of the
whole sentence, so a part of the thought is the sense of part of the
sentence. Thus the thought appears the same in kind as the sense
of a proper name, but quite different from its meaning.
(In this passage ‘meaning’ is a translation of ‘Bedeutung’.)
3. See Searle “Proper Names” and Kripke’s discussion of description theories in
Naming and Necessity.
4. The idea that Frege was a description theorist is mostly based on the following
footnote in “On Sense and Reference” (p. 27):
In the case of an actual proper name such as ‘Aristotle’ opinions as
to the sense may differ. It might, for instance, be taken to be the
following: the pupil of Plato and teacher of Alexander the Great.
Anybody who does this will attach another sense to the sentence
‘Aristotle was born in Stagira’ than will a man who takes as the sense
of the name: the teacher of Alexander the Great who was born in
Stagira. So long as the reference remains the same, such variations
of sense may be tolerated, although they are to be avoided in the
theoretical structure of a demonstrative science and ought not to
occur in a perfect language.
It should be noticed, first, that the descriptions are not enclosed within quotes;
they are given after a colon. This can often be replaced by quotes but not in this
case. Why? Because Frege was careful in using quotation marks and if we put the
description within quotes, then the sense would be the phrase, which was certainly
not Frege’s idea. Second, we cannot interpret the description operator ‘the’ in the
usual sense it had for Frege, because if we do this the sense of ‘Aristotle’ would
be Aristotle; the man himself. Third, although we can interpret Frege as saying
that the sense of ‘Aristotle’ is the sense of the descriptive phrase, this raises some
problems. For, on the one hand, he doesn’t say that; and, on the other hand, this
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would just push back the question as to what the sense of the descriptive phrase
is. It would be rather problematic to think that the sense of the descriptive
phrase is given by the sense of another descriptive phrase, and so on, without
end. Though, of course, this is true of Fregean senses – that the description has
a sense, which also has a sense, and so on, indefinitely – this is not explanatory
of the notion of sense. (And the same would hold if one claims that in those
contexts the descriptive phrase refers to its customary sense.) So, in my view, the
most natural (re)interpretation of Frege is that what follows the colons should be
thought of as a predicate which denotes a property having the logical feature of
unicity. Frege, of course, does not put it this way, and I am not imputing the view
to him.
There is an interesting passage in “On Concept and Object” (pp. 202205) – and a long footnote in a preliminary version of this paper in Posthumous
Writings pp. 100-105 – where Frege discusses a distinction suggested by Kerry
between the expressions ‘the number 4’ and ‘ ‘the’ number 4’, with the first being
a concept and the second an object. Frege denies that there is such a distinction,
because of his views on the article ‘the’, but he does appeal to properties in order
to capture the distinction. Thus, he says (p. 203):
I should express as follows what Kerry is apparently trying to say:
‘The number 4 has those properties, and those alone, which are
marks of the concept: result of additively combining 3 and 1.’
In the note he says (p. 100):
. . . Kerry calls the content of the words ‘the result of additively combining 3 and 1’ a concept. I would agree with him if there was no
definite article before ‘result’. I think he will claim the content of
the quoted words, without the definite article, to be a concept too,
so that in this respect there will be complete agreement between us.
Now how, according to Kerry, would the concept which answers first
to the whole phrase be distinguished from that which answers, secondly, to this phrase with the definite article omitted? The definite
article does not add a new characteristic mark. What it does do is
to indicate:
1. That there is such a result.
2. That there is only one such.
That the definite article, in the form ‘is the’, can add a new characteristic mark
is seen from my formulation (1). Nevertheless, since the properties ‘is a result
of additively combining 3 and 1’ and ‘is the result of additively combining 3 and
1’ are necessarily equivalent, in this case the addition of the definite article does
not add a new characteristic mark. Which means, however, that the property
indicated by Frege is actually a sense of the number 4.
The problem of analyticity raised by Frege’s remarks, which was the
primary motivation of Searle’s paper and of other reformulations of Russell’s descriptive theory of names, is quite independent of the (re)interpretation of Frege
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that I am suggesting. The question still remains as to what an adequate theory of
the senses of proper names should be. Frege’s off-hand remarks about the sense,
or senses, of the name ‘Aristotle’ may be interpreted as suggesting a “descriptive”
theory of the senses of proper names, but there isn’t really much evidence either in
this passage or elsewhere that Frege thought that he was giving a theory of proper
names at all. In fact, it seems to me quite clear that Frege’s reflections about
ordinary language were merely intended to point the way for a theory of sense
and denotation for his logical language, and that he wasn’t particularly concerned
as to what a theory of the senses of ordinary proper names should be.
5. On these characteristics of senses see the second “Logic” manuscript in Frege’s
Posthumous Writings and “Thoughts”. That senses are ontological determinants
of reference doesn’t mean that whether the sense identifies an entity, and, if so,
which entity it identifies, is determined by the sense alone. For, as I argued in
Chapter 2, this determination is relative to the way the world is.
6. “On Sense and Reference”, p. 31. A little earlier in the article he says (pp.
27-28):
The regular connexion between a sign, its sense, and its reference
is of such a kind that to the sign there corresponds a definite sense
and to that in turn a definite reference, while to a given reference
(an object) there does not belong only a single sign. The same sense
has different expressions in different languages or even in the same
language. To be sure, exceptions to this regular behaviour occur.
To every expression belonging to a complete totality of signs, there
should certainly correspond a definite sense; but natural languages
often do not satisfy this condition, and one must be content if the
same word has the same sense in the same context. It may perhaps
be granted that every grammatical well-formed expression representing a proper name always has a sense. But this is not to say
that to the sense there also corresponds a reference. The words ‘the
celestial body most distant from the Earth’ have a sense, but it is
very doubtful if they also have a reference. The expression ‘the least
rapidly convergent series’ has a sense but demonstrably has no reference, since for every given convergent series, another convergent,
but less rapidly convergent, series can be found. In grasping a sense,
one is not certainly assured of a reference.
7. Part of Frege’s second argument concerning identity at the beginning of “On
Sense and Reference” goes as follows (p. 26):
If the sign ‘a’ is distinguished from the sign ‘b’ only as object (here,
by means of its shape), not as sign (i.e. not by the manner in
which it designates something), the cognitive value of a = a becomes
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essentially equal to that of a = b, provided a = b is true. A difference
can arise only if the difference between the signs corresponds to a
difference in the mode of presentation of that which is designated.
It is quite obvious from this that a sign is not a purely syntactic object for Frege.
8. The footnote I discussed in note 4 corresponds to the asterisk in the following
paragraph (Op. Cit., p. 27):
The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to
which it belongs;* but this serves to illuminate only a single aspect
of the reference, supposing it to have one. Comprehensive knowledge of the reference would require us to be able to say immediately
whether any given sense belongs to it. To such knowledge we never
attain.
If we have a syntactic code, and we know the code, then we can tell what every
syntactic string codes. Since ordinary language is a social affair which developed
over a long period of time, it is inevitable that none of us knows all of the code
and that we have different interpretations of many parts of it. As far as Frege’s
last remark is concerned, it is quite obvious that if we had complete knowledge
of all the hierarchy of objects and properties, then we would be able to say about
any entity whether any given sense belongs to it. A God should be able to do it,
for example. The closest we seem to get to this is in mathematics, which has often
been compared to a language of Gods; and constructivists try to make us into minor Gods by placing strong requirements of identification. They generally achieve
this, however, by imposing tremendous limitations on the scope of mathematics.
It is also fairly obvious that syntactic strings are not the only things that
do or can code senses and be signs. An expression (facial or corporal, including
gestures) can also express a sense. But Frege’s use of ‘sign’ should be distinguished
from the use according to which smoke is a sign of fire, clouds are a sign of rain, a
grimace is a sign of pain, etc. This is related to Grice’s distinction (in “Meaning”)
of natural and non-natural uses of ‘meaning’.
9. A description, as any other entity, has many senses. By “the primary sense” of
a sign I mean an intrinsic sense of the sign that gives its syntactic and semantic
role.
Besides the syntactic and semantic character of ‘ιxF x’ as sign, we can
also consider the use of the sign ‘ιxF x’ by the speakers of the language – which
I take to be a somewhat broader dimension related to the meaning of the sign.
It is generally the case that when used by a speaker in certain contexts – e.g.,
assertoric contexts – a sign ‘ιxF x’ is meant by the speaker to refer to an entity;
and, therefore, that in such contexts there is a presupposition that there is an
entity to which ‘ιxF x’ refers. But, according to Frege, this is not part of the sense
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expressed by the description – see the quotation in note 10 of Chapter 3. (And, I
would add, it is not part of the primary sense of the description.)
10. I raised a dog called ‘Freddie’, whose full name is ‘Frederick de Spook’, because
I thought that it was fun to call him that. I raised another one called ‘Rudy’
(‘Rudolph de Spook’) because when he was born he reminded me of Rudolph the
Red Nosed Rain Deer. Lady, in Lady and the Tramp, is called ‘Lady’ because she
“is a perfectly beautiful little lady”.
11. Naming and Necessity, pp. 91-97.
12. Ibid., pp. 116 ff.
13. You can name a robot, or a puppet, or anything you want, but you don’t
fix the reference to a robot if you intend to name a puppy. However, if after you
realize that it is a robot you continue to refer to it as ‘Freddie’, with the intention
of referring to it, then you may have re-fixed the reference of ‘Freddie’.
14. He says: ”. . . I may not have presented a theory, but I do think that I have
presented a better picture than that given by description theorists.” Ibid., p. 97.
15. One of the things that’s missing in Kripke’s picture is an account of intentionality, because his characterization of both baptisms and transmissions is heavily
dependent on this notion. Kripke concludes his outline by saying (Ibid., p. 97):
Notice that the preceding outline hardly eliminates the notion of
reference; on the contrary, it takes the notion of intending to use
the same reference as a given. There is also an appeal to an initial
baptism which is explained in terms either of fixing a reference by a
description, or ostension (if ostension is not to be subsumed under
the other category). (Perhaps there are other possibilities for initial
baptisms.)
I haven’t presented the basic ideas exactly as Kripke does, but I think
that I have followed the spirit of his presentation. I did emphasize the intentional
aspect of the reference fixings and transmissions because it is quite crucial for the
picture to get off the ground.
It is also important to emphasize that the notion of reference is a given
aside from the notion of intending to use the same reference. (Notice that intending to use the same reference does not presuppose that there is a referent; names
without reference get transmitted just as well as names with reference.) Kripke is
not proposing a definition of reference even relatively to the notion of intending
to use the same reference, but is giving an explanatory account of how reference
works in natural languages – or, at least, of some aspects of how it works.
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One may argue that the reference of a name (proper or natural kind)
N as used in a context . . . N . . . by a person X is defined by following back the
chain of acquisitions to an initial baptism, if any. If all the transmissions are
successful, and the initial baptism is successful, then the baptized object or kind
is the referent of N as used by X in . . . N . . .; otherwise, N as used by X in . . . N . . .
has no referent. This is a helpful picture but is not a definition because it would
yield no referent if the chain of transmissions is not quite well defined or if there
isn’t an identifiable initial baptism. Notice, by the way, as Kripke points out, that
the name N ′ used in the initial baptism, if any, need not be the same as N − N ′
may “transform into” N by a series of changes N ′′ , N ′′′ , etc., so that there may
be no syntactic similarity between N and N ′ .
There are many natural kind terms, such as ‘water’, for which it would
be rather farfetched to claim that there is anything even remotely like an initial
baptism, and for which the chain of transmissions is totally ill defined. When
a baby starts learning words and he learns ‘water’ by our pointing to samples
of water (and/or experiencing water) and by our saying ‘water’, is he acquiring
‘water’ from us, or is he fixing the reference of ‘water’ with our help? Is his
intention to use ‘water’ with the same referent as we do, or to use ‘water’ to refer
to that kind of thing that we are pointing at (and/or experiencing)? What if
the sample we chose to teach the baby is a bad one? What happens as the baby
goes on and the sample gets “corrected”? Or, even if the sample was a good
one, he applies ‘water’ to the wrong liquids and his use of ‘water’ gets corrected?
Moreover, he may have initially said ‘ahtah’ “referring” to some fruit next to
some water and we think that he meant ‘water’ and point happily in that general
direction. One could discuss this natural example for pages, and the conclusion
would be, I think, that in general there is no clear transmission, there are no clear
intentions, there is no clear initial baptism, etc. And this aside from the fact that
there is no clear kind either: why isn’t coca-cola the same kind of thing as dirty
and polluted water? Similar problems arise for proper names also.
Kripke is under no illusion about this, and it takes none of the merit
of his picture of reference. A theory, even a theory of how reference works, will
be considerably more complex and will have to involve elements from many other
pictures.
16. Ibid., pp. 156-158.
17. I don’t want to get into the intricacies of the notion of rigid designator here,
and it won’t be necessary for my purposes. Kripke characterizes it as follows (Ibid,
p. 48):
Let us call something a rigid designator if in every possible world
it designates the same object, a nonrigid or accidental designator if
that is not the case.
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(He discusses this notion of rigidity at some length in the preface, written after
the original lectures.) The intuition is that the referent of a name, once it has
been fixed, does not depend on properties that individuate that referent. This
is essentially the same as Russell’s intuition concerning logically proper names,
but Russell’s view of ordinary proper names derived from his intuitions about
descriptions. Kripke holds that descriptions designate but that they are typically
nonrigid, whereas proper names are typically rigid.
18. The traditional view is that words like ‘unicorn’, ‘centaur’, etc., have the
empty set as their extension. This is compatible with Kripke’s view; one just
adds to the traditional view that the extension of these terms is not merely empty,
but that it is necessarily empty. In fact, it is not clear from Kripke’s discussion
whether he wants to conclude that ‘unicorn’ does not refer, or that it necessarily
refers to the empty set. From my point of view, however, what is interesting
about Kripke’s arguments is that they show that the way in which these terms
are introduced into the language does not satisfy certain basic conditions that are
necessary conditions for a term having reference.
19. Kripke says (Ibid., pp. 134-135):
Mill, as I have recalled, held that although some ‘singular names’,
the definite descriptions, have both denotation and connotation, others, the genuine proper names, had denotation but not connotation.
Mill further maintained that ‘general names’, or general terms, had
connotation. . . .
The modern logical tradition, as represented by Frege and Russell,
disputed Mill on the issue of singular names, but endorsed him on
that of general names. Thus all terms, both singular and general,
have a ‘connotation’ or Fregean sense. More recent theorists have
followed Frege and Russell, modifying their views only by replacing
the notion of a sense as given by a particular conjunction of properties with that of a sense as given by a ‘cluster’ of properties, only
enough of which need apply. The present view, directly reversing
Frege and Russell, (more or less) endorses Mill’s view of singular
terms, but disputes his view of general terms.
20. Since Kripke does not distinguish descriptive properties from descriptions as
singular terms, he thinks of the sense as the conjunction of properties involved in
the descriptive singular term – as is clear in the quotation in the previous note.
But I have already argued that this is an incorrect way of thinking of senses.
Which shows also that the notion of sense is not like Mill’s notion of connotation.
21. Just as we can associate senses and properties to a proper name, we can
associate senses and properties to an object. Thus, besides the senses of an object,
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we may consider the senses and properties that a person associates to an object at
a certain time. We may refer to the totality of these properties as the connotation
of the object for that person at that time. Even if a name denotes an object,
the connotation of the name for a person at a certain time may be quite different
from the connotation of the object for that person at that time if the person has
a false belief about the denotation of the name.
The notion of connotation that I introduced in the text is essentially
Kripke’s thesis (1) in p. 64 of Naming and Necessity. This thesis is not rejected
by Kripke because, as he says, can be considered a definition – and this is how
I am using it. On the other hand, Kripke rejects the other theses (2)-(6) (pp.
64-65, 70ff.) that he associates to descriptive theories of names, and I agree with
his reasons for that.
22. I am not claiming, of course, that Frege makes the distinctions I introduced
in the text, but only that they are compatible with what he says and can be used
to interpret some of his remarks.
23. For Kripke’s remarks about the ‘Feynman’ example see Naming and Necessity
pp. 81-82, 91-92. As opposed to the case in the text, Kripke considers a case where
the person who uses the name may not know (or remember) how he acquired the
name. He says (pp. 91-92):
A speaker who is in the far end of this chain, who has heard about,
say Richard Feynman, in the market place or elsewhere, may be referring to Richard Feynman even though he can’t remember from
whom he first heard of Feynman or from whom he ever heard of
Feynman. He knows that Feynman was a famous physicist. A certain passage of communication reaching ultimately to the man himself does reach the speaker. He then is referring to Feynman even
though he can’t identify him uniquely. He doesn’t know what a
Feynman diagram is, he doesn’t know what the Feynman theory of
pair production and annihilation is. Not only that: he’d have trouble distinguishing between Gell-Mann and Feynman. So he doesn’t
have to know these things, but, instead, a chain of communication
going back to Feynman himself has been established, by virtue of his
membership in a community which passed the name on from link to
link, not by a ceremony that he makes in private in his study: ‘By
“Feynman” I shall mean the man who did such and such and such
and such’.
I quite agree with Kripke, but I think that Russell is right (see Chapter 3 note 33)
that a man in this situation probably associates to ‘Feynman’ something like the
descriptive property ‘is the famous physicist that I heard about called
‘Feynman’ ’. This may be a sense of Feynman, but even if it isn’t, the referent of the name as used by this man is the object (if any) determined by the sense
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expressed by the name ‘Feynman’ as he acquired it. The difference between this
man and Kripke is that the name ‘Feynman’ has a much richer connotation for
Kripke, and that Kripke can associate many other senses with ‘Feynman’.
Although there may be some question about this, I would in any case
defend the following weaker formulation of Russell’s basic intuition: if a person
uses (significantly) a proper name at a certain time, then the name must have
a connotation for that person at that time. In Kripke’s example (as discussed
above) the connotation would include the property ‘is a famous physicist’ – and
even if the person isn’t sure whether Feynman is a physicist, it would include
something like ‘is a person’, ‘is famous’, ‘is some kind of scientist’, etc.
More generally, I would say that when I acquire a name I associate some
properties, including senses, to it, whether or not these properties or senses are
properties or senses of the referent of the name (if any). My intention when I
use the name at a later time may not be to use it with the same reference as the
person(s) from whom I acquired it, but to use the name to which I associate(d)
such and such sense. Thus, I don’t remember at all how I acquired the name
‘Socrates’ to which I associate(d) the sense ‘is the Greek philosopher who was
Plato’s teacher’, and I don’t remember at all how I acquired the name ‘Socrates’
to which I associate(d) the property ‘is a soccer player who was in Brazil’s national
team in 1982’. These associated senses or properties help determine which sign I
am using, and thereby the reference, if any, determined by the sense expressed by
the sign.
24. Op. Cit., pp. 122 ff.
25. On the other hand, the property ‘is the kind of animal which is a large feline
with black stripes natural of Asia’ is a sense because it is a property [!ZGZ](Z)
of properties of objects that has unicity as a logical characteristic. If there is no
such kind, though we may think that there is, then this sense does not identify a
property.
In the connotation of a natural kind term I include both essential and
accidental properties and we could consider a more restricted notion that would
include only essential properties of the natural kind. It is only essential properties
that justify such counterfactuals as I used in the text.
26. This doesn’t mean that I cannot make statements using the predicate ‘is a
unicorn’. I can, but just like in the case of the proper name ‘Sherlock Holmes’,
such statements are neither true nor false. This is part of the motivation for the
semantics adopted in Chapter 6.
27. Although I have not given a general account of the senses expressed by all
signs in the language, it seems plausible to me that such an account could be
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developed along the lines suggested above. It also seems plausible to me that on
such an account Frege’s principles of substitutivity for sense and reference would
hold.
28. See, for example, T. Burge “Sinning Against Frege”. In p. 405 he says:
These passages dealing with the context dependence of indexical
constructions, including proper names, are but the most obvious
signs of the cognitive orientation of Frege’s notion of sense. That
orientation dominates his introduction of the notion in “On Sense
and Reference.” The paradox of identity, whose discussion opens the
essay, is a problem about information expressed through language.
Though I disagree with this view of senses, I do agree with Burge that the notion
of sense is quite relevant to Frege’s epistemological interests.
29. Frege characterizes this notion in pp. 3-4 of The Foundations of Arithmetic:
Now these distinctions between a priori and a posteriori, synthetic
and analytic, concern, as I see it, not the content of the judgment
but the justification for making the judgment. When there is no such
justification, the possibility of drawing the distinctions vanishes. An
a priori error is thus as complete a nonsense as, say, a blue concept.
When a proposition is called a posteriori or analytic in my sense, this
is not a judgment about the conditions, psychological, physiological
and physical, which have made it possible to form the content of the
proposition in our consciousness; nor is it a judgment about the way
in which some other man has come, perhaps erroneously, to believe
it true; rather, it is a judgment about the ultimate ground upon
which rests the justification for holding it to be true.
This means that the question is removed from the sphere of psychology, and assigned, if the truth concerned is a mathematical one,
to the sphere of mathematics. The problem becomes, in fact, that
of finding the proof of the proposition, and of following it up right
back to the primitive truths. If, in carrying out this process, we
come only on general logical laws and on definitions, then the truth
is an analytic one, bearing in mind that we must take account also of
all propositions upon which the admissibility of any of the propositions depends. If, however, it is impossible to give the proof without
making use of truths which are not of a general logical nature, but
belong to the sphere of some special science, then the proposition
is a synthetic one. For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to
facts, i.e., to truths which cannot be proved and are not general,
since they contain assertions about particular objects. But if, on
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the contrary, its proof can be derived exclusively from general laws,
which themselves neither need nor admit of proof, then the truth is
a priori.
30. Using the broader notion of consequence introduced in Chapter 6 (37) what
we can say independently of the assumption about names is that whereas
(16′ ) a |= a = a
can be established a priori,
(17′ ) a, b |= a = b
cannot generally be established a priori.
31. The opening of “On Sense and Reference” is actually the following:
Equality gives raise to challenging questions which are not altogether
easy to answer. Is it a relation? A relation between objects, or between names or signs of objects? In my Begriffsschrift I assumed
the latter. The reasons which seem to favour this are the following: a = a and a = b are obviously statements of differing cognitive
value; a = a holds a priori and, according to Kant, is to be labelled
analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a
priori. The discovery that the rising sun is not new every morning,
but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet
is not always a matter of course.
When Frege returns to the question at the end of the paper he says (p. 49):
When we found ‘a = a’ and ‘a = b’ to have different cognitive values,
the explanation is that for the purpose of knowledge, the sense of
the sentence, viz., the thought expressed by it, is no less relevant
than its reference, i.e. its truth value. If now a = b, then indeed the
reference of ‘b’ is the same as that of ‘a’, and hence the truth value
of ‘a = b’ is the same as that of ‘a = a’. In spite of this, the sense of
‘b’ may differ from that of ‘a’, and thereby the thought expressed in
‘a = b’ differs from that of ‘a = a’. In that case the two sentences do
not have the same cognitive value. If we understand by ‘judgment’
the advance from the thought to its truth value, as in the above
paper, we can also say that the judgments are different.
It is quite natural and legitimate to infer from these passages that senses are
relevant (and even essential) to the cognitive value of sentences; from which it
doesn’t follow that the notion of sense is an epistemological notion – nor is this
implied by what Frege says.
32. Pp. 78-79. See the quotation in Chapter 8, pp. 267-268.
404
33. Frege starts §8 of Begriffsschrift (pp. 20-21) with the following remarks:
Identity of content differs from conditionality and negation in that
it applies to names and not to contents. Whereas in other contexts
signs are merely representative of their content, so that every combination into which they enter expresses only a relation between their
respective contents, they suddenly display their own selves when
they are combined by means of the sign for identity of content; for it
expresses the circumstance that two names have the same content.
Hence the introduction of a sign for identity of content necessarily
produces a bifurcation in the meaning of all signs: they stand at
times for their content, at times for themselves. At first we have
the impression that what we are dealing with pertains merely to
the expression and not to the thought, that we do not need different
signs at all for the same content and hence no sign whatsoever for
identity of content. To show that this is an empty illusion I take the
following example from geometry.
He now gives an example which is similar to the example he gives in p. 26 of “On
Sense and Reference”, and which involves a point that is identified, or determined,
in two different ways. He then goes on:
To each of these ways of determining the point there corresponds
a particular name. Hence the need for a sign of identity of content
rests upon the following consideration: the same content can be completely determined in different ways; but that in a particular case
two ways of determining it really yield the same result is the content of a judgment. Before this judgment can be made, two distinct
names, corresponding to the two ways of determining the content,
must be assigned to what these ways determine. The judgment,
however, requires for its expression a sign for identity of content, a
sign that connects these two names. From this it follows that the existence of different names for the same content is not always merely
an irrelevant question of form; rather, that there are such names is
the very heart of the matter if each is associated with a different way
of determining the content. In that case the judgment that has the
identity of content as its object is synthetic, in the Kantian sense.
A more extrinsic reason for the introduction of a sign for identity
of content is that it is at times expedient to introduce an abbreviation for a lengthy expression. Then we must express the identity of
content that obtains between the abbreviation and the original form.
405
Now let
⊢ (A ≡ B)
mean that the sign A and the sign B have the same conceptual content, so that we can everywhere put B for A and conversely.
It is quite obvious that the doctrine of “On Sense and Reference” is contained
here, and that what is a little different is the solution. For Frege, signs were
always signs, not bunches of marks, and therefore he does not mean identity to be
a relation between names in the sense of expressions – as he makes perfectly clear
in the quotations above. So the senses are already here as ways of determination
embodied in the signs. In “On Sense and Reference” he gives the senses ontological
independence in relation to the signs that express them; and, since senses are
objects, he can talk of their identity and difference independently of the signs.
Therefore, he can treat identity as a relation between objects which, among other
objects, applies also to senses. The importance of “On Sense and Reference” has
to do directly with Frege’s ontology, and only derivatively with his epistemology.
34. Thus, in “Thoughts” Frege says (p. 66 note 6):
I am not here in the happy position of a mineralogist who shows his
audience a rock-crystal: I cannot put a thought in the hands of my
readers with the request that they should examine it from all sides.
Something in itself not perceptible by sense, the thought is presented
to the reader – and I must be content with that – wrapped up in a
perceptible linguistic form. The pictorial aspect of language presents
difficulties. The sensible always breaks in and makes expressions
pictorial and so improper. So one fights against language, and I
am compelled to occupy myself with language although it is not my
proper concern here. I hope I have succeeded in making clear to my
readers what I mean by ‘a thought’.
35. “Thoughts”, p. 66:
Now everyone is presented to himself in a special and primitive way,
in which he is presented to no-one else. So, when Dr. Lauben has
the thought that he was wounded, he will probably be basing it on
this primitive way in which he is presented to himself. And only Dr.
Lauben himself can grasp thoughts specified in this way. But now
he may want to communicate with others. He cannot communicate
a thought he alone can grasp. Therefore, if he now says ‘I was
wounded’, he must use ‘I’ in a sense which can be grasped by others,
perhaps in the sense of ‘he who is speaking to you at this moment’;
by doing this he makes the conditions accompanying his utterance
serve towards the expression of a thought.
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36. Especially if one distinguishes the sense expressed by ‘I’ in a context, which is
a reference to the speaker or writer (something like Frege’s “he who is speaking to
you at the moment”), from senses associated with ‘I’ in a context, which is how I
interpret Frege’s “primitive way in which [Dr. Lauben] is presented to himself”.
37. In “Frege on Demonstratives” Perry says (p. 474):
In “The Thought,” Frege briefly discusses sentences containing such
demonstratives as “today,” “here,” and ‘yesterday,” and then turns
to certain questions that he says are raised by the occurrence of
“I” in sentences . . .. He is led to say that, when one thinks about
oneself, one grasps thoughts that others cannot grasp, that cannot be communicated. Nothing could be more out of the spirit of
Frege’s account of sense and thought than an incommunicable, private thought. Demonstratives seem to have posed a severe difficulty
for Frege’s philosophy of language, to which his doctrine of incommunicable senses was a reaction.
38. In a long note in p. 136 of the second “Logic” manuscript, in which he is
discussing Dedekind’s proof that there is an infinite set, Frege comments:
Now if infinitely many thoughts have not yet been thought, the infinitely many thoughts φ(s) must comprise infinitely many thoughts
that are not thought, in which case it cannot be essential to a
thought that it should be thought. And this is precisely what I
am maintaining.
Although Frege does not say so, we may also reasonably conclude that among
these infinitely many thoughts there are infinitely many that are so complex that
they cannot be thought at all.
407
Chapter 12
Truth and Correspondence
I have been developing a realistic approach to truth and ontology
that largely derives from ideas of Plato, Aristotle, Frege, and Russell. My
view of truth is essentially Plato’s view; what is true is what is real. Obviously, not all that is real is true in this sense; we talk about truth in
connection with statements, or thoughts, or beliefs, or opinions, or propositions, or sentences. The predicate ‘true’ and the predicate ‘false’ classify
such things according as to whether they describe reality correctly or incorrectly. If we think of this describing as a correspondence between the
sentence (statement, proposition, etc., but I will talk about sentences) and
the world, then we must explain what sort of correspondence is that. In
Chapter 1 I suggested that the view of truth as identification of states of affairs respects the basic intuition of the correspondence view of truth. I now
want to discuss this question in a somewhat more general setting and make
some additional connections with the views of Frege, Russell, and Tarski.
In fact, there are a number of different questions involved here.
On any reasonable interpretation of the correspondence view, true
sentences correspond to aspects of the world which they delimit by articulating parts (at least some of) which are themselves connected to the world.
I will distinguish two main choices.
One is to confer ontological status to these aspects of the world
and to say that a sentence is true if and only if it corresponds to some such
aspect. For the sentence
(1) Frege taught Carnap,
this would read
(2) ‘Frege taught Carnap’ is true if and only if ∃x (‘Frege taught Carnap’
corresponds to x),
where ‘x’ ranges over states of affairs, or facts, or something like that.
The other choice is to say that a sentence is true if and only if
such and such conditions hold, where these conditions involve parts of the
sentence and whatever they connect to in the world – and maybe some other
things as well on either side. This transforms part of the problem into an
analysis of these connections, for which similar choices may arise. There are
several alternatives, but one of them could be, for our example,
(3) ‘Frege taught Carnap’ is true if and only if ∃y∃z∃w (‘Frege’ connects to
y, ‘taught’ connects to z, ‘Carnap’ connects to w, and z relates y to w).
409
We can generalize these approaches by introducing a notion of
analysis such that
(4) ‘Frege taught Carnap’ is true if and only if ∃p(‘Frege taught Carnap’
analyses as p and p is true),
where ‘p’ ranges over sentences in some sort of “normal form”. But for such
sentences the two choices come up again and we can either take ‘p is true’
as
(5) p is true if and only if ∃x(p corresponds to x),
or as
(6) p is true if and only if ∃y∃z∃w . . . C(y, z, w, . . .),
where the right hand side indicates conditions involving parts of p, aspects of
reality, the notion of connection, and maybe some other things as well. The
existentially quantified variables can range both over linguistic entities and
over other aspects of reality. These two alternatives are not incompatible,
of course, because we can hold that the x to which p corresponds in (5) has
a structure that is described by part of the right hand side of (6).
In all these cases we hold that it is reality that determines whether
a sentence is true or not, and our existential quantifications indicate not only
this but a specific view as to the structure of reality in terms of objects,
properties, facts, or whatever. Thus we must give some account of this
structuring, as well as of the notions ‘corresponds’, ‘connects’, and ‘analyses’
– or of some of them. Moreover, these approaches to truth do not necessarily
yield an account of falsity, because it is not at all clear that we can identify
‘false’ with ‘not true’. This leads to questions about falsity and negation.
The most natural suggestion, which I have been pursuing, is to say
that a sentence is false if and only if its predicate negation is true. But this
seems to leave us with many sentences that are neither true nor false. For
example, the sentences
(7) Sherlock Holmes is tall
and
(8) Sherlock Holmes is not tall.
There is nothing wrong with this conclusion, because it just emphasizes that
truth and falsity depend on there being certain connections with reality that
are missing in this case. For a number of reasons, however, many people
felt and feel that this approach should be avoided at all cost. There are two
main aspects of the approach that are seen as being problematic. One is
that it appears to go against the principle of excluded middle – and against
the laws of classical logic more generally. The other is that it is not clear
that it can be applied to sentences with more complex structure, especially
quantified sentences. I have argued that this is not so, but if one does want
410
to avoid the conclusion that (7) and (8) are neither true nor false, then there
are some alternatives.
The worst, it seems to me, is to postulate an existent but not quite
existent Sherlock Holmes who is either tall or not tall. Not only one muddles
reality with this move, but one can get into trouble with the principle of
contradiction.
The next to worst is to identify the ‘not’ in (8) with ‘not true’,
to identify ‘false’ with ‘not true’, and to claim that (8) is indeed false.
This muddles the distinction between a “really” false sentence, which does
describe reality but describes it incorrectly, and a sentence that has no
bearing on reality. In other words, a simple denial of the right hand side
does not explain the difference between, say,
(9) Frege is not tall
and (8).
Another alternative which I do not find appealing is to attach an
arbitrary connection to ‘Sherlock Holmes’. Partly because it doesn’t really
solve the problem, and partly because it makes truth and falsity for such
sentences rather arbitrary.
The best alternative, in my view, is Russell’s solution discussed in
Chapter 3. It identifies ‘false’ with ‘not true’, but distinguishes the ‘not’ in
(8) from a sentential ‘not’ and analyses both (7) and (8) into sentences that
do have a bearing on reality and describe it incorrectly. Thus, (7) and (8)
are “truly” false, and the principle of excluded middle works with respect
to the sentential ‘not’. This solution transfers a large part of the problem
to the account of ‘analyses’.
But Russell’s solution also suggests the simpler solution which consists in keeping (7) and (8) as neither true nor false and solving the problem
of excluded middle in a similar way through the distinction between the
two uses of ‘not’. Thus, whatever merits Russell’s solution may have, these
do not seem to hang on the question of classical logic. And the problem
with predicate negation can also be solved by means of a fairly conservative
reformulation of the usual logical notation, as I have tried to show in earlier
chapters. This reformulation also involves introducing a certain notion of
analysis, and it is conservative both in the sense of preserving the basic
characteristics of the standard notation and of appealing to a traditional
analysis of sentences.
Nevertheless, to say that false sentences are those whose predicate
negation is true also seems to raise problems in connection with the account
of reality and its structuring presupposed by the right hand side of the
various biconditionals. The right hand side of the biconditionals are those
411
existential claims, for any structure that the sentence in the left hand side
may have. So if there is a negation, then the right hand side must express
it in some way; which leads directly to Plato’s problem of non-being and to
his conclusion that negativeness, in one form or another, must be an aspect
of reality.
This is quite obvious for an approach like (2), because for the
sentence
(10) Carnap did not teach Frege,
it gives
(11) ‘Carnap did not teach Frege’ is true if and only if ∃x (‘Carnap did not
teach Frege’ corresponds to x),
where x can only be something like a “negative” aspect of reality; a negative
fact, for instance.
It may seem that a formulation along the lines of (3) avoids this
conclusion, because for (10) we can simply put ‘z does not relate y to w’
instead of ‘z relates y to w’. But what does the ‘not relate’ refer to? Isn’t
it an aspect of reality that the relation taught does not relate Carnap and
Frege? Even if we don’t want to say that it is the relation not-taught that
relates Carnap and Frege, we are still involved with an aspect of reality
that is real; because, in fact, Carnap did not teach Frege. To pass the buck
to language is merely to postpone the problem. We must distinguish in
language what connects with reality from what does not. The ‘not relates’
does connect with reality – it describes a feature of it – whereas ‘Sherlock
Holmes’ doesn’t.
If one is bothered by such talk of reality, however, one can try to
avoid it by eliminating the existential quantifiers from the right hand side
of the biconditionals. For true sentences this is very easy, because, since
they do describe reality, one can use them as the condition of truth. Thus,
(12) ‘Frege taught Carnap’ is true if and only if Frege taught Carnap.
This seems to avoid entirely the question of accounting for ‘corresponds’,
‘connects’, and the structure of reality, as Mates suggests in the passage I
quoted in Chapter 7 (pp. 236). And one can do it just as well for a negative
true sentence, and hence define falsity as truth of the (predicate) negation.
But it doesn’t work for sentences that have no bearing on reality, because
for such sentences the biconditionals are neither true nor false. Although
it seems clear to me that neither Tarski nor Mates considered the problem
of truth valueless sentences a relevant issue, I have tried to show that it is
a relevant issue and that it spoils the simple approach to the question of
truth through schema (T).
412
In relation to this I want to comment on the point I made in
connection with Mates’ remarks about Tarski’s semantic conception of truth
that the appeal to the world in this approach is vacuous. By following up
on one or another alternative that rules out truth valueless sentences, one
can claim that the appeal to reality is not vacuous. I.e., one can claim
that the point of Tarski’s approach is to avoid any specific commitment
as to what reality is like – which bears on the question of (metaphysical)
neutrality of Tarski’s semantic conception – but that there is an appeal to
reality nonetheless.
Obviously, if one develops the position along some line that rules
out truth valueless sentences, whether or not one does avoid specific commitments as to the nature of reality will depend on the details. The plausibility
of the claim for the original position is based on the fact that one is using
ordinary sentences, and that (almost) everyone, no matter what their metaphysical views, also uses ordinary sentences. But if one combines Tarski’s
approach with some aspects of Russell’s, for example, then it is not clear
that one can avoid metaphysical commitments. For as I argued in Chapter
3, Russell’s own view was metaphysical.
There is another side to this question, however. If the appeal to the
world is not vacuous, then one is appealing to the world. To what does this
appeal to the world amount to? Supposedly, it is determined in the world,
or by the world, that Frege taught Carnap. Whether anybody knows, or
believes, that Frege taught Carnap is entirely irrelevant for the truth of this
sentence. Even if it were inconsistent with everybody’s beliefs that Frege
taught Carnap, Frege would have taught Carnap nonetheless. This, and
much more, is so according to the classical conception of truth that appeals
to the world – and it is the basic intuition behind the correspondence view.
But this is already metaphysics, for it involves the view that truth and
falsity are determined independently of any of our beliefs, theories, etc.
Nevertheless, nothing is said explicitly about the world in a biconditional like (12). So let us suppose that the whole conception could be
formulated entirely by means of biconditionals like (12). Hence nothing is
said about the world; at least not explicitly. Then what is the basis for
saying that one is appealing to the world? It seems that I could interpret
the biconditionals in any way I want.
Now consider the following interpretation. I take ten (classical)
logicians and interpret the right hand side of a biconditional as expressing
the majority view. I.e., Frege taught Carnap if and only if at least six of
these logicians agree that Frege taught Carnap. This yields (in principle)
a classification of sentences into true, false, and neither. If I want to avoid
413
the neither, I make a rule that for each sentence everyone must either agree
with the sentence or agree with its predicate negation, whether they like
it or not. Moreover, I also make a rule that every individual verdict at a
time t must respect the laws of classical logic with respect to the sentences
classified prior to time t.
If we don’t analyze the biconditionals as in some way “grounded”
in reality, which would send us back to the earlier arguments, then there
seems to be nothing in Tarski’s conception of truth that would prevent this
interpretation of it. And there will be an indefinite number of other interpretations that are equally compatible with his account. So in what sense
does it capture the classical view of truth expressed in terms of correspondence? To capture the classical view we must interpret Tarski’s conception
realistically, with the biconditionals grounded in reality; but then the metaphysical issue is back. Thus my conclusion that either the appeal to the
world is not vacuous, in which case the account must have some metaphysical commitments, or else it is vacuous, in which case it is essentially a
syntactic account open to an indefinite number of interpretations1 .
Let’s now look at ‘corresponds’, ‘connects’, and ‘analyses’. These
are obviously interrelated, and one can try to build up some to simplify
the account of the others. Thus, one can build up ‘analyses’ so that the
‘corresponds’ or the ‘connects’ become very primitive and simple. The ‘corresponds’ may be something like a mirror image, or a mere pointing (without any content except to be a pointing). The ‘connects’ may be a form
of labelling, or again a pointing. This puts a big burden on the ‘analyses’,
because no (ordinary) sentence is quite like that. Besides, what happens to
fully analyzed false sentences?
The idea that the ‘corresponds’ and the ‘connects’ are so primitive
that there is no need to give an account of them can be understood as saying
that the content of a pointing is to point at something, that the content of
a label is to label something, that the content of a mirror image is to mirror
something; without the something all of these become contentless. But this
can only work for true sentences. A false sentence of this fully analyzed
sort would have to be a mirror image of nothing, which does not seem very
plausible. If one is to account for falsity, the mirror images cannot depend
for their content on something of which they are mirror images – unless one
really postulates non-being. And the same thing goes for pointings if their
only content is to point at something. Thus, one cannot place the entire
burden of explanation on ‘analyses’, for it doesn’t seem to eliminate the
need for an account of ‘corresponds’ and ‘connects’.
414
An alternative for pointings is to say that a fully analyzed false
sentence does not point at something but away from something. The distinction between pointings and awayings cannot be an intrinsic distinction,
however, because it is reality that determines whether a sentence points to
or away from it. In other words, it is impossible to distinguish a pointing
from an awaying independently of reality. But if the only content of a sentence is to point to or away from reality, then the distinction between a
sentence that points away from reality and a sentence that neither points
to reality nor points away from reality cannot be made in terms of reality
as well.
In target-shooting, for instance, one can (and must) distinguish
hits from misses in terms of the target, but one cannot distinguish misses
(that miss the target entirely) from misfires in terms of the target. The latter
distinction depends on additional considerations. Similarly, the distinction
between sentences that are awayings (misses) and sentences that are neither
pointings nor awayings (misfires) must depend on the content of sentences
and not simply on reality.
One way to approach these problems, as I suggested in connection
with Russell in Chapter 5, is to appeal again to predicate negation. An
awaying (miss) is a sentence whose predicate negation points to (hits) reality.
A sentence which neither points to reality nor away from reality is a misfire
(a mere noise, as Russell says).
But there are other possibilities, for one can try to place the burden
of explanation on ‘corresponds’. If the correspondence in question is some
sort of similarity relation, then either it is a direct reflection of reality (like
the mirror image idea), or it is a more abstract similarity. Something like a
map, maybe. But this image does not seem to me to get us any closer to a
solution to the problem of falsity.
Ordinary maps are of three kinds: correct maps, incorrect maps,
and fictitious maps. To say that a correct map is one that corresponds to
something does not distinguish the incorrect maps from the fictitious maps.
The natural suggestion is to make this distinction in terms of the connections
between the elements of the map and reality, but this just shifts the burden
of explanation to the account of ‘connects’. One can try to say that the
only content of ‘connects’ is to connect with something, and that, therefore,
maps made of elements (some of) which do not connect to anything are not
maps at all – i.e., the elements of a map must connect to something. But
then we are letting the content of ‘map’ depend on reality, which means that
the distinction between maps and non-maps is a transcendental distinction
just like the distinction between truth and falsity.
415
If the correspondence is not a similarity relation, then it may be
better to think of it as a connection rather than as a correspondence. This
is Russell’s idea in terms of pointings and awayings. But, as I argued before,
the content of a sentence must go beyond its being a pointing or an awaying.
One can claim, however, that sentences are articulations of other pointings
whose only content is to point at something. The content of sentences would
then derive from these elementary pointings and the manner in which they
are articulated. Sentences point to or away from facts, and the elementary
pointings point to other aspects of reality. But if the distinction between
contentual sentences and contentless sentences depends on whether their
elements (when fully analyzed) are indeed elementary pointings, and this
is determined by reality, then the distinction becomes like the distinction
between truth and falsity again. Moreover, for every contentual sentence to
be fully analyzable in this way one must hold that anything in reality that
is not a fact can be pointed at by an elementary pointing – or something
close to this.
This suggests that one should build up the contentual sentences
from the elementary pointings. An elementary pointing is a connection
to reality that is incorrigible – i.e., one can’t be wrong about something
being an elementary pointing. What there is, aside from facts, is what can
be pointed at in this way. The problem now is how to recover ordinary
sentences through this constructive process. Although Russell combines
both ideas, there is a certain tension between the constructive process and
the analytic (eliminative) process.
Frege’s denotational theory of truth also seems to me to be a version of the correspondence theory. ‘Corresponds’ and ‘connects’ give way to
‘denotes’, and the burden of explanation is spread out between the notions
of denotation, sense, judgment, the analysis of sentential form, and the objects the True and the False. The unintuitive part of the account has to do
only with this last bit. One can justify it on pragmatic grounds, perhaps.
On the correspondence view ‘true’ is a predicate but truth is not;
truth is essentially relational. Frege’s choice of the True and the False as
the denotations of sentences preserved this essential feature of truth and
allowed him to unify his ontology into the two fundamental categories of
functions and objects. In fact, the True was a reasonable candidate for the
denotation of true sentences. One can think of it as (a representation of)
Being, as Gödel suggests2 . The most natural interpretation for the False
seems then to be (as a representation of) Non-Being. But it was necessary
to have the False in order to get going and to preserve the connection with
reality. So Frege postulated the True and the False as objects reminiscent
of Being and Non-Being, respectively.
416
Although we don’t really know how Frege arrived at these ideas,
there is an alternative way of thinking about the True and the False which
makes more sense to me. The idea that the False represents Non-Being goes
together with the interpretation of Plato that suggests that Plato postulated
non-being in order to solve the problem of falsity. I think that in both cases
the problem was precisely how to account for falsity without appealing to
non-being.
Frege’s account of truth is a pointing version where pointings have
definite contents independently of whether they point at something or not.
Sentences are pointings which articulate parts which are also pointings.
Pointings that do not point at anything are pointings nonetheless, but if
some pointings that are part of a sentence (or expression more generally)
do not point at anything, then the sentence (or expression) itself does not
point at anything. When all pointings that are part of a sentence point at
something, then the sentence points at reality. But the sentence articulates
the various pointings in a specific way, which is its content, and what is
pointed at by the parts of the sentence may fit together in agreement with
this articulation or not; reality may accept or reject the proposed articulation. Sentences point at this acceptance (fit) or rejection (lack of fit) by
reality, and are accordingly true or false. The sentences that involve pointings that do not point at anything are of no concern to reality, so it doesn’t
bother with them – from which Frege concluded that he needn’t bother with
them either, which was a mistake.
Thus the True and the False do not stand for Being and NonBeing; they stand for different aspects of Being. The True represents the fit
(acceptance) and the False represents the lack of fit (rejection), but both fit
and lack of fit are in reality. Sentences (and thoughts), when they denote
at all, denote this fit and lack of fit that is an ineliminable contribution of
reality. On this interpretation Frege’s account is really a mixture of several
of the ideas I was discussing before. Even the states of affairs are there,
although they are not given independent ontological status.
My interpretation of the correspondence view of truth is very close
to this version of Frege’s view. But I bring in Plato (predicate negation),
Aristotle (combination and lack of combination), Russell (facts, distinction
of scope), and Frege again (quantification as predication and the hierarchy
of levels) – and I also use Tarski’s ideas, as in Chapter 6. The pointings
are senses, and senses are properties that purport to identify something;
i.e., they are properties [!xFx](x). Thus, the connection with reality is a
connection of purported identification, and both truth and denotation are
special cases of instantiation.
417
Of course, at the most abstract level this is not an account of truth
for sentences, or beliefs, or opinions, but an account for senses, including
propositions, that is entirely independent of language and mind. It is an account of certain connections between abstract aspects of reality with certain
other aspects of reality which need not have anything to do with us. However, as I briefly indicated in the last chapter, the account can be brought
to bear on language – although this is an aspect that has to be developed
much more fully.
Although I appeal to states of affairs, this is not essential. I can
combine my account with Tarski’s so that states of affairs are left out –
this can be seen from the beginning of Chapter 6. The interpretation of the
notion of correspondence is then rather similar to the interpretation I gave
above for Frege’s the True and the False. Except that I don’t need the False,
because false sentences are those whose predicate negation is true. And in
this case the True can be interpreted as being reality itself. True sentences
correspond to reality in the sense that they articulate parts which identify
various aspects of reality (properties and objects) which are combined in
agreement with the articulation of the sentence. False sentences do not
correspond to reality in the sense that they articulate parts which identify
various aspects of reality which are not combined in agreement with the
articulation of the sentence. Sentences that are neither true nor false do not
correspond to reality in the sense that they articulate parts some of which do
not identify aspects of reality – and therefore there aren’t aspects of reality
that can combine in agreement or disagreement with the articulation of the
sentence. Except, perhaps, for the last condition, this seems to me to be
very close to Aristotle’s view.
States of affairs seem a nice addition to the account, because
they are the combinations. Why shouldn’t the combinations themselves
be counted as aspects of reality? This is a natural view, which allows us to
say that the sentences themselves may be classified as true if they identify
such an aspect of reality. False sentences are those whose predicate negation
identifies an aspect of reality, and the others are neither true nor false. But
what are these aspects of reality?
Quine is quite right to say that they aren’t physical aspects of
reality, and that we may be mislead by a certain apparent bruteness of the
notion of fact. If a student is standing on my desk, and I point to that and
say “This is intolerable!”, it seems that I am pointing at physical reality.
There is the desk, there is the student standing on it: I point to that.
But the fact to which I am pointing cannot be the physical sum of student
and desk, even over some time. That’s just another physical object. Thus
Quine’s conclusion that facts are abstract, with which I agree. Although
418
Quine is not too fond of the abstract, he could let that part of it pass.
The problem for him is not so much that facts are abstract, but that we
don’t really know how to identify them. Hence the demand for a criterion
of identity. There are other objections, of course. Why do we need them?
And if we don’t really need them, then why bother? And, besides, aren’t
we confusing facts with propositions?
Although I have answered some of these objections3 , I haven’t
really answered Quine’s main objection. What is the criterion of identity
for facts, or states of affairs? Since I characterized states of affairs as a
combination of a property with some other entities which instantiate the
property, it seems that I can reduce the problem to the question of identity
for properties and to the question of identity for the other entities. Although
this is bad enough for Quine, at least insofar as properties is concerned, I
did argue in Chapter 10 that the question of identity for (pure) sets is
not really that much better off than the question of identity for properties.
In both cases identity is highly impredicative. And since Quine himself has
argued that material objects are also not very well off in terms of a criterion
of identity, at least for his standards, his earlier solution of having the sets
organized predicatively upon a basis of material objects as units is no longer
available to him.
But I agree that I haven’t answered Quine in the sense of having
provided a criterion of identity for states of affairs. And I further agree with
Quine that questions of identity are very difficult, as is demonstrated by the
development of his own work and by the work of everyone else who has dealt
with problems involving identity. Where I disagree with Quine is that this
is a knock-down argument against the existence of various kinds of entities:
properties, states of affairs, propositions, senses, meanings, etc. For if it is a
knock-down argument, then we seem to be left with the ridiculous position
that nothing is – not even sets. This is not to say that the question is not
worth asking and that we shouldn’t bother with it. I think that this sort
of questioning is often very helpful to point out confusions and unclarities
in our conception of one or another kind of entities – and in some cases
the confusions are so widespread that claims of legitimacy for such entities
must be rejected altogether. But I don’t think that this is the case for any
of the kinds of entities that I mentioned above.
Be this as it may, however, there are only two places in the previous chapters where states of affairs were used essentially. One was in
the characterization of propositions in Chapter 11, and the other was in
the characterization of sets as collections in Chapter 9. So I’ll make some
comments about these.
419
I think that the intuitive idea of the characterization of propositions as senses that purport to identify states of affairs is quite reasonable,
but since the account of truth can be formulated independently of states of
affairs (as suggested above), one may consider whether it is also possible to
reformulate the characterization of propositions independently of states of
affairs.
Let’s go back to Frege. If sentences denote truth values and a sense
is a manner of presentation, then the sense expressed by a sentence should
be the manner in which it presents its truth value. This is the idea that
the sense expressed by a sentence are the truth conditions for that sentence.
For the example that I used in Chapter 11 (6) we can write
(13) [[!ZGZ](Z) & [!xFx](x) & Zx](Z, x).
This is a property that (given my assumptions) determines a level 1 property (the property of being a philosopher) and a level 0 object (Quine). In
fact, given the character of this property, it has the logical feature that there
can be at most one instantiation of it. So it is a sense, though not a singular
sense. For the other example that I used in Chapter 11 (7) we get
(14) [[!ZKZ](Z) & [!xFx](x) & Zx](Z, x),
which is also a sense, although it is not instantiated. On the other hand,
(15) [[!ZKZ](Z) & [!xFx](x) & ¬Zx](Z, x)
is instantiated. Therefore we may characterize propositions as such properties and define a proposition as true if and only if it is instantiated. A
proposition is false if and only if its predicate negation is true – e.g., the
pair (14)-(15). Other propositions are neither true nor false. This is again
a completely general analysis, because every proposition has that form –
which gives its truth conditions. Since for a proposition to be instantiated
means to be instantiated in reality, we can say that true propositions are
those that correspond to reality, or whose identity conditions are fulfilled in
reality, or even those that denote reality. Thus I can fully recover a Fregean
account without appealing to states of affairs.
Let me now turn to the question about sets. As with propositions,
I find my account of the set {Frege, Russell} as the state of affairs that
Frege is different from Russell quite reasonable. But now I can use my new
account of propositions to solve this problem as well without appealing to
states of affairs. I can say that the set {Frege, Russell} is a proposition that
Frege is different from Russell. I.e., something like
(16) [[!xFx ](x ) & [!yRy](y) & x 6= y](x, y),
where [!xFx](x) and [!yRy](y) are senses of Frege and Russell, respectively.
The problem is that there are many such propositions, because each object
has many senses. I can say, however, that for each object there is a sense
which is simply the property of being that object; it has no other content.
420
This is something like Russell’s logical proper names, and I will use expressions like ‘is Frege’ and ‘is Russell’ instead of ‘this’ and ‘that’. Then instead
of the senses [!xFx](x) and [!yRy](y), I can take the senses [x is Frege](x) and
[y is Russell](y), yielding the proposition
(17) [[x is Frege](x ) & [y is Russell](y) & x 6= y](x, y).
So a set as a collection is a proposition; i.e., a property.
This solution is also quite reasonable, because as I explained in
Chapter 9 our conception of the set {Frege, Russell} is essentially that
Frege and Russell are treated as units that are elements of the set; and that
the only content of the set is to have these units as distinct elements. But to
treat Frege and Russell as units is to treat them as something abstract; for,
as Frege emphasized, the physical Frege and the physical Russell cannot be
said to be units4 . And if the only content of the set is to have these units
as distinct elements, then what the set does is to code this fact. This is
precisely the content of proposition (17).
One conclusion we can draw from this is that Quine’s idea to have
the sets arise predicatively from a basis of material objects is problematic
even aside from the problem of identity for material objects. For the material objects at the basis must be treated as units, and from Frege’s arguments
it seems to follow that as units they cannot be physical.
Another conclusion we can draw is that even the conception of sets
as collections is better understood in terms of properties.
Still another conclusion is that states of affairs themselves could
be identified with true propositions, after all. A state of affairs is a certain
kind of property that involves some aspects of reality in an ineliminable way.
Instead of saying that a sentence denotes a state of affairs, we could say that
a sentence expresses a proposition which, when true, is a state of affairs.
We may then say that true sentences are those that correspond to a state
of affairs. False sentences are those whose predicate negation corresponds
to a state of affairs. The others are neither true nor false.
Thus, in a Quinean spirit, we could reduce our ontology to properties and objects. We can still keep the states of affairs as something like
objects for heuristic purposes, but they may not really be needed – not so
far, anyway.
But what about properties, then? What are they? What are their
essential characteristics?
The way I think of properties is essentially as identity conditions – though not linguistically, of course, although they can be expressed
linguistically5 . My discussion of identity in chapters 9 and 10 was meant to
bring out some of the complexity of this notion and the general relativity
421
of criteria of identity. This suggests that there may be different kinds of
properties which are more or less independent of each other and of other
kinds of entities.
Sets can be conceived as the simplest kind of properties. The
identity condition that constitutes a set is essentially a listing of the elements
of the set. To be an element of the set {Frege, Russell} is to be identical to
Frege or to be identical to Russell; and the set has no other content aside
from this identity condition. Since sets are not conceived linguistically,
this idea can be projected to the infinite. To be an element of the set
{a, b, c, . . .} is to be a, or to be b, or to be c, etc. The idea that the set has
no additional content aside from this identity condition is what is expressed
by the standard version of extensionality, and the extensional character of
sets can be formulated in several different ways.
One way to formulate it is the following. Suppose we have properties, i.e., identity conditions that apply to certain entities but that could
also apply to other entities. Suppose that to each property [Fx](x) we associate a (set-)property {x : F x} which applies to the same entities as [Fx](x).
If two properties [Fx](x) and [Gx](x) apply to the same entities, then the
associated (set-)properties {x : F x} and {x : Gx} are the same. Another
characterization is that a set is an identity condition having the characteristic that if any “part” of the condition that identifies an entity is replaced
by another part that identifies the same entity, then the resulting identity
condition is the same.
Let me turn now to properties that aren’t sets in this sense. A
property applies to an entity because its identity condition is realized (or
exemplified) in that entity. If I have one or more entities that exemplify a
property, and if I think that other entities that exemplify the property must
have a fairly close structural similarity to the entities in question, then it
seems reasonable to characterize the property by appeal to that sample in
which the property is exemplified. This is precisely Kripke’s procedure for
characterizing natural kind properties. If I have a number of animals that
seem to me to exemplify a property, then I pick the property by saying:
is the property that these animals (the ones actually in front of me) have
in common in virtue of their biological structure (or of their nature). The
last clause (“in virtue of”) is quite essential, because the animals in front of
me may have many properties in common – they look alike, for one thing.
The point of the clause is to eliminate these other properties, and if the
clause does not apply – the animals just look alike, say – then I may fail
to pick a property by this procedure. I.e., in Kripke’s terminology again,
I may fail to fix the reference of the natural kind term. It is also for this
reason that Kripke doesn’t use this procedure to fix the reference of other
422
terms. What makes a chair a chair is not its structural similarity to other
chairs, but, rather, something like the function for which it is used in a
certain community. Kripke’s ideas contain an implicit criticism of Plato’s
procedure for introducing forms, since it is essentially the same procedure.
But Plato introduces both Man and Chair in the same way. This is not
necessarily incorrect, if we read the clause as formulated generally in terms
of “nature” rather than “biological structure”, but it may not be correct if
there is no common nature to chairs (as we use the term). Though Kripke
does not deny that there is a common nature to chairs, in this broader sense,
he may consider it rather unlikely.
From the fact, if it is a fact, that natural kind properties are identified in this way it doesn’t follow that a natural kind property depends for its
existence on the existence of any specific exemplifications of it – and it certainly does not generally depend on the existence of the actual sample that
is used. That’s why natural kind terms can also be introduced by description. If I (try to) identify a substance as the substance that has such and
such characteristics, which are such that they would characterize a natural
kind if anything, then I may be identifying a natural kind property even if
it doesn’t apply to anything. A description of some animals that would look
like what unicorns are supposed to look like by the mythical description,
but that describes these animals structurally down to the level of DNA, may
well identify a natural kind property with an empty extension6 . But even if
there were some such animals, they wouldn’t be unicorns – for the reasons
I discussed in Chapter 11. From this we can conclude that natural kind
properties, or at any rate some of them, not only do not depend on specific
exemplifications, but do not depend on there being any exemplifications at
all. Moreover, natural kind properties are quite different from sets, because
their identity does not generally depend on the entities to which they apply.
But there are properties whose existence does seem to depend on
one or more of the entities in which they are exemplified. This is the case
for sets as characterized earlier. Not only that, but these properties may
be temporal entities. For, as I suggested in Chapter 9 in connection with
states of affairs, the set {Frege, Russell} did not exist before both Frege
and Russell did. This doesn’t mean that there aren’t properties that apply
uniquely to Frege that did exist before Frege existed. A simple example
may be the property of being the first born son of Karl Alexander Frege –
assuming that Frege is indeed the first born son of Karl Alexander Frege.
And there may even be much more abstract properties of this kind. If for
each entity that is an element of a set there is a sense of that entity which
is not a temporal sense and that is necessarily a sense of that entity, then
a property combining these senses could be considered to be an atemporal
423
property that determines that set. These properties need not be unique,
but if for each set there is such a property, and there is some way in which
to each set a unique such property is associated, then we may have an
atemporal conception of sets as collections.
The question of temporality is interesting for several reasons. I
haven’t said anything in general about what a level 0 object is. In fact,
what I have done is to implicitly follow Frege’s characterization of level 0
objects as entities which are not properties (or states of affairs)7 . Although
I don’t have a theory of objects, I would add as a necessary condition for
level 0 objects that they be temporal. (Obviously this wouldn’t do for
Frege, because of the logical objects, but I shall comment on this question
below.) The previous condition is not sufficient to distinguish objects from
properties, however, because as suggested above also properties may be
temporal.
The difference between logic (and mathematics) and physics may
lie in that whereas logic and (at least part of) mathematics are essentially
atemporal, physics is essentially temporal. Besides being temporal, physics
is also spatial, but I don’t think that spatiality is a necessary condition for
something being a level 0 object, because I don’t think that mental objects
are spatial. In fact, temporality without spatiality may be a necessary
characteristic of the mental. It is not a sufficient characteristic, however; at
least not if some properties may be temporal without being mental. But
we may well hold that both temporal properties and mental objects depend
for their existence on spatial objects; i.e., that neither of these can exist
independently of the existence of (some) physical objects. Nevertheless,
temporal properties can exist independently of mental objects, because the
physical objects that are required for the existence of mental objects are
rather special.
We may distinguish several domains of inquiry, therefore. The
logico-mathematical domain, which is atemporal. The physical domain
which is spatio-temporal. The abstract temporal domain, which depends
only on there being some physical objects, and the mental temporal domain,
which depends on there being certain special physical objects. The objects
that we postulate at level 0 seem to me to reflect these distinctions. Also
the distinction between classical and intuitionistic logic can be seen in this
light; but let me first comment on Frege’s idea of atemporal logical objects.
Frege held that there is a fundamental distinction between concepts and objects, and that whenever we try to talk of a concept as a
(singular) subject we actually end up talking about an object. The latter is
related to his view about the article ‘the’ which I have criticized at various
424
points in previous chapters. I do agree with Frege that there is a fundamental distinction between properties and objects, however, and the view
I suggested above of properties as identity conditions is one way to express
that distinction. I also think that there is something to Frege’s intuition
concerning the objectification of properties, though I would formulate it in
a different way.
As Frege says, an object is never predicative whereas a property
is essentially predicative. Nevertheless, a property can be subject to other
(higher level) properties. Thus, a property can be both “subject” and “predicate”. We may have some situations where certain properties are only used
as subjects of predication of higher level properties. This may suggest that
these subjects, which are not used predicatively at all in that context, are
really objects. I think that this is one way to look at what happens in
arithmetic. The numbers 0, 1, 2, 3, etc., are treated as objects that are subjects of predications such as: is even, is prime, is divisible by, is smaller
than, etc. This suggested to Frege that numbers are objects, and since
he held that these objects cannot be either physical or mental, they must
be logical objects. By this move he “ontologized” the distinction between
subject position and predicate position for all subjects of a certain kind –
namely, those that are named by a proper name, definite description, or
sentence. But 0, 1, 2, 3, etc. can simply be, say, the level 2 properties Nullness, Oneness, Twoness, Threeness, etc. that in that particular discourse
are used only as subjects. Since they are used only as subjects one might
as well refer to them in the same ways that one refers to objects. In other
kinds of discourse, however, these properties are also used predicatively. In
this sense I agree with Frege. Where I disagree with Frege is that I see no
need to introduce an ontological distinction to capture these distinct uses
of property terms.
I shall comment now very briefly on one aspect of intuitionistic
logic and mathematics. From the point of view of Brouwer’s intuitionism,
on one reading of his remarks, there is a basic state (of reality) that is
neither spatial, nor temporal, nor logico-mathematical, but that is potentially temporal. This basic state of reality is characterized by Brouwer as
oscillating “slowly, will-lessly, and reversibly between stillness and sensation”, which can perhaps be understood as an oscillation between nothing
and something. Mind arises from sensation through a primitive phenomenon that Brouwer calls ‘a move of time’, and that can be conceived as
representing temporality. This phenomenon also leads to plurality and to a
distinction between subject and object. Brouwer then sees mathematics as
arising through mind by a process of abstraction by the subject with respect
to the primitive phenomenon ‘a move of time’. And then he goes on8 .
425
An alternative way of looking at Brouwer’s description is the following. By a move of time, mind, plurality, and logicality arise together.
What mind does “later” is to grasp the logical in the form of logical properties such as Diversity, Identity, etc. In this sense neither logic nor mathematics would “depend” on the subject, but would be objective – and recognized
as such by the subject. The development of logic and mathematics would
be carried out by the subject in a strictly mental way, but this wouldn’t
make them any the less objective.
It seems to me that such a position would be intermediate between
the classical position and Brouwer’s own position as I described it above.
The classical position, as I see it, holds that logical properties such as Identity, Diversity, Universality, Subordination, Unity, Duality, Plurality, etc.
are atemporal; and therefore independent of the existence of any temporal
(or spatial) entities. But as I pointed out in Chapter 9 (note 36), some of
Griss’ criticisms of Brouwer suggest that one cannot separate Identity and
Diversity from the primitive phenomenon of a move of time.
426
Notes
1. This does not make it uninteresting, however; on the contrary. What Tarski
did was to develop a theory that applies to any conception of truth that satisfies
certain conditions. This theory is particularly interesting in connection with the
question of definability of the extension of a predicate ‘true’ (satisfying those
conditions) for formalized theories – especially mathematical theories.
2. “Russell’s Mathematical Logic”, p. 214. (See the end of the quotation in Chapter 4 note 16.) Gödel does not make any suggestions about the False, however.
3. A recent book that raises a number of objections to facts (and to some other
views I have discussed) is Denyer Language, Thought and Falsehood in Ancient
Greek Philosophy. Although I found this book quite interesting, I think that these
objections suffer from the lack of a better conception of facts and of the issues
involved. Many of the objections (in Chapter 2) are directed at Euthydemus
(in Plato’s Euthydemus), but Denyer wants to generalize them to anyone who
appeals to facts as the denotation of sentences. I think that my discussion (in
earlier chapters) of the issues he raises may actually help to put some of the
problems in a better perspective – for example, the discussion of the statement
‘Nobody can jump more than ten feet into the air’ in p. 13. On my analysis this
statement speaks of the property ‘is a person who can jump more than ten feet
into the air’, and it asserts of that property that it does not apply to anything.
Therefore, the state of affairs denoted consists of a universal negative (which is
a second order property) and the first order property indicated above. But there
are two other issues on which I would like to comment.
One is his objection to Frege (p. 20):
. . . if the False is a genuine and nameable object, what is wrong
with assertions that name it? Why should assertions which name
the False be any worse that assertions which name the True?
If one doesn’t specify anything about the objects the True and the False, then this
is a real problem. On the other hand, if one does interpret the True and the False
as I did above, in terms of fit and lack of fit, then the answer is quite natural.
The other issue is the discussion of Menedemus’ views beginning in p.
37. As reported by Diogenes Laertius (quoted by Denyer), Menedemus “would
get rid of negative propositions, by making them positive.” On which Denyer
comments (p. 37): “To put his view in the paradoxical way it deserves, it is
the view that one should not say ‘not’.” This is related to Griss’ views, as I
mentioned in the Introduction and in Chapter 9 (note 37), which I think are rather
interesting views that are closely related to the problems that Denyer discusses in
his book. And as I have mentioned before, Griss (and Plato) had the insight that
distinguishability (or Otherness) is not really negative.
427
4. The Foundations of Arithmetic, section 22. See also Gödel’s remark quoted in
Chapter 9 note 17.
5. This conception of properties is not new, for it seems to be closely related to
Plato’s own conception of forms (see, e.g., Chapter 9 note 32). This can also be
seen from the following remarks by Allen in Plato’s ‘Euthyphro’ and the Earlier
Theory of Forms (pp. 71-72):
Epistemologically, Forms are standards for detecting their instances.
The ontological ground for this function is that instances have the
Form (5d), and that the Form is that by which its instances are what
they are (6d).
It is important to realize that Socrates’ assumption that Forms are
standards is as directly embedded in his ‘What is it?’ question as
the assumption that Forms are universals. The question, ‘What is
the holy?’ is prompted in the Euthyphro by a practical problem
of identification: it is important to find out what holiness is in order to know what sorts of action are holy and what sorts are not.
One cannot know whether a given action is holy without knowing
what holiness is (6e, 9a-c, 15d-e), any more than one can be certain
that two men are friends without knowing what friendship is (Lysis,
223b), or that a speech is beautiful without knowing what beauty is
(Hippias Major, 286c-d). Knowing the form is a condition for recognizing its instances: to ask what holiness is, is to ask for knowledge
of a criterion by which to distinguish things which are holy from
things which are not.
Evidently, one can think that forms contain identity conditions without being
identity conditions. Thus Plato’s struggle to understand the nature of forms. My
suggestion is to make this feature of forms, that they contain identity conditions,
their very essence. That is their nature. This is a rather abstract conception,
but it seems to me consonant with the abstract conception of sets developed by
Cantor and with the abstract conception of concepts developed by Frege. But, of
course, I haven’t said anything yet about the epistemological issues.
6. I think that Kripke is suggesting this when he says (Naming and Necessity, p.
157):
If we suppose, as I do, that the unicorns of the myth were supposed
to be a particular species, but that the myth provides insufficient
information about their internal structure to determine a unique
species, then there is no actual or possible species of which we can
say that it would have been the species of unicorns.
428
7. See the last quotation in note 1 of Chapter 9.
8. I am commenting on the beginning of “Consciousness, Philosophy, and Mathematics”, especially the following remarks (p. 1235):
First of all an account should be rendered of the phases consciousness has to pass through in its transition from its deepest home to
the exterior world in which we cooperate and seek mutual understanding. . . .
Consciousness in its deepest home seems to oscillate slowly, willlessly, and reversibly between stillness and sensation. And it seems
that only the status of sensation allows the initial phenomenon of
the said transition. This initial phenomenon is a move of time. By
a move of time a present sensation gives way to another present
sensation in such a way that consciousness retains the former one
as a past sensation, and moreover, through this distinction between
present and past, recedes from both and from stillness, and becomes
mind.
As mind it takes the function of a subject experiencing the present
as well as the past sensation as object. And by reiteration of this
two-ity-phenomenon, the object can extend to a world of sensations
of motley plurality.
In measure of the irreversibility with which the subject has receded
from an element of the object, this element loses its egoicity, i.e.
gets estranged from the subject, and in measure of this estrangement, mind becomes disposed to desire and apprehension, and consequently to positive or negative conative activity with respect to
the element in question.
And (p. 1237):
Mathematics comes into being, when the two-ity created by a move
of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic
intuition of mathematics, is left to an unlimited unfolding, creating
new mathematical entities in the shape of predeterminately or more
or less freely proceeding infinite sequences of mathematical entities
previously acquired, and in the shape of mathematical species i.e.
properties supposable for mathematical entities previously acquired
and satisfying the condition that if they are realized for a certain
mathematical entity, they are also realized for all mathematical entities which have been defined equal to it.
429
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Logical Forms Part I - Truth and Description
COLEÇÃO CLE, 2001
The main question treated in this book is the relation between statements (sentences, propositions) and reality. The book is the first volume of a larger work about the fundamental (ontological, epistemological, linguistic) character of logic. My viewpoint is a realistic and metaphysical view of logic and truth influenced to a very large extent by works of Plato, Aristotle, Frege, Russell, Gödel, and Hardy, among others. Hence the title "Logical Forms", inspired by Plato. I list below the titles of the chapters in lieu of an abstract.
Introduction. Logic, ontology, and epistemology.
1. Truth, description, and identification.
2. The True and the False.
3. Use, mention, and Russell's theory of descriptions.
4. Arguments for Frege's thesis.
5. Objections to facts.
6. Truth, denotation, and interpretations.
7. Tarski's semantic conception of truth.
8. The True and the False revisited: Frege's logic.
9.Structuring reality: properties, sets, and states of affairs.
10. Identity and extensionality.
11. Senses.
12. Truth and correspondence.
A translation to Spanish--Formas Lógicas I--was published by Eudeba in 2015 and I added some information and a link to Eudeba in TRANSLATIONS.
A translation to Portuguese--Lógica, ontologia e epistemologia --of the Introduction was published in 2006 and I added the pdf in TRANSLATIONS.
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