Croatian Journal of Philosophy
Vol. VI, No. 17, 2006
Logic, Reasoning
and the Logical Constants
PASCAL ENGEL
Université Paris Sorbonne
and Institut Jean Nicod
Introduction
Consider the three questions:
(a) how do we reason?
(b) how ought we to reason?
(c) what justifies the way we ought to reason?
Question (a) is a descriptive one: it calls for an account, presumably a
psychological one, of how humans reason, a theory of their reasoning
abilities and of their reasoning performances. Question (b) is normative:
it calls for a theory of how we evaluate whether a reasoning is good or
bad, correct or incorrect, valid or invalid. Usually this role is devoted to
logic. Question (c) is, so to say, meta-normative; it calls for the justification of our norms and standards of reasoning, possibly against other
norms.
On many accounts, the answers to these three questions differ widely, and on many accounts they must do so. On the classical view of logic
as a normative discipline, there is a strong distinction between the way
we reason actually and the way we should reason, and the attempt to
derive the latter from the former, is considered as the most obnoxious
fallacy about the nature of logic: psychologism. Today’s psychologists of
reasoning, however, guard themselves against this fallacy. They are all
the more suspicious of it that they find that there is a large gulf between
our actual reasoning practices and logic as the theory of valid reason-
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ing. So both on the classical philosophical view and on the contemporary view in psychology, (a) and (b) are independent. It is not clear either than question (c) is related to the others. What justifies our logical
norms is not related to the way we actually reason, for it would, again,
imply a form of psychologism. Presumably, questions (b) and (c) have
a more intimate link. But it is not clear that there is but one possible
justification of our logical norms, and that there is only one set of legitimate standards, either in deductive or in other sorts (e.g. probabilistic)
of reasoning. Although an ecumenist pluralism seems to reign in many
circles, the question: “Which logic is the right one?” still arises.
The received wisdom, therefore, is that the division of labour, which
leaves to the psychologists the answer to (a), to the logicians an answer
to (b), and to logicians and philosophers an answer to (c) is perfectly in
order as it stands. But is this wisdom correct? Some prima facie doubts
can be raised. If our logical norms had nothing to do with the way the
actually reason––if logic were totally irrelevant to reasoning––how
could they be prescriptive of our practice? How could we apply these
norms? How could we teach logic? And if our reasoning practices had
nothing to do with the way we evaluate reasoning, how could we simply describe the practices as a form of reasoning? In order to describe
certain thought processes as reasoning and inference, we need at least
some characterisation of what reasoning and inference are, and it is
unclear that this can be done without logic. I believe that we cannot
draw such a sharp line between our actual reasoning practice, which
the object of psychology and the study of valid reasoning, which is the
logician’s business. In this sense, I subscribe to a form of psychologism,
although quite weak, and I have argued elsewhere for the view that,
between the world of psychological processes (world 2 on Popper’s famous hierarchy of worlds, world 1 being that of physical objects) and
the world of abstract thought (world 3, Frege’s “Third Realm”), there is
room for an intermediary world 2, 5 situated midway between world 2
and world 3, between the muddy waters of psychology and the dry land
of the logical norms (Engel [1989], [1996]). I shall not try here to defend
this view in general, and I shall limit myself to deductive reasoning
with propositions, leaving aside the large body of work and controversy
about probabilistic reasoning.
I want to relate not only the questions (a)–(c) above, but also three
kinds of concerns: those of the psychologists of reasoning, who are interested in whether we reason through formal rules or through other
ways, in particular through the construction of mental models; those
of logicians, who are interested in the relationship between a semantic
and a proof-theoretic account of logical systems; and those of philosophers, who ask whether meaning is a matter of acceptance conditions
(or assertion conditions) or of truth conditions. My aim is to sketch – in
a quite programmatic way––what I take to be an integrated answer to
this set of questions and concerns. This answer will be strongly influenced by Peacocke ([1988], [1993], [1992]).
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This paper has three parts. In the first one I discuss some well known
objections to the relevance of logic to reasoning, and suggest some ways
in which this relevance can be assessed. In the second part, I propose a
general framework for answering to the above questions and concerns.
In the third part I deal, more briefly, with the relation between this
view and familiar arguments about the rationality of human thinking
and its justification through a “reflective equilibrium”, and indicate how
they differ from these arguments.
I. Is logic relevant to reasoning?
It is a commonplace that everyday reasoning does not conform to the
usual rules of classical logic, in so far as these are understood as descriptive of the practice of individuals. A large body of contemporary
psychological work shows that people make systematic mistakes and
fallacies. Philosophers and psychologists tend to agree on the fact that
in so far as logic tells us which arguments or proofs are valid, it does
not tell us anything about our reasoning as such, if by “reasoning” we
mean inferences from beliefs to other beliefs. Among philosophers, for instance, Harman [1986] has argued that reasoning, as “reasoned change
in view”, or as a process of belief revision, has nothing to do with logic.
Harman takes the familiar rule of modus ponens: P, and if P then Q,
therefore Q As he points out, this principle says nothing about our beliefs. It does not say that if one believes that P, and also believes that
if P then Q, then one can believe that Q. It does not even say that one
should, given the first and the second belief, have the third. Sometimes,
one should give up the belief that P, or the belief that if P then Q, instead. For instance, suppose you believe that if violets are red, then
daffodils are, and that you also believe that violets are red. Should you
believe that daffodils are red? No, you could drop the belief that violets
are red, or the belief that if violets are red, then daffodils are. Harman
considers also even more abstract logical principles, such as implication
or non contradiction. Logic tells us that if a proposition implies logically another then we should accept this proposition. But the case just
mentioned of the modus ponens shows that this need not be the case.
The notorious problem of logical omniscience shows that one should not
require, of a cognitive system, natural or artificial, that it believes or accepts everything that is implied logically by what it already believes or
accepts, for it would have to believe or accept an infinity of propositions,
which obviously finite minds such as us cannot do. And we should not
clutter our minds with trivial implications. Similarly logic is supposed
to prescribe us to avoid logical inconsistencies. But should we avoid any
inconsistency when we encounter one? Suppose that someone is confronted to the Preface paradox: he writes a book of which he thinks that
every sentence is true, but writes in his preface that he might well have
made a mistake somewhere. He contradicts himself. But should he stop
believing what his book says? This would be absurd. Harman concludes
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that logic, as a theory of validity, has no special role to play in an account of reasoning. It is merely “a body of truths”, a science, like physics
of chemistry (Harman [1984], 109). As such, it is not normative, it does
not tell us anything about what we may or should believe.
Another familiar philosophical argument about the normative irrelevance of logic to reasoning and inference is the famous Lewis Carroll’s
story of Achilles and the Tortoise (Carroll [1895]), which I have commented elsewhere (Engel [1998]). One of the main points of the story
is that explicit knowledge of a valid truth of logic or of a valid rule of
inference is insufficient to make the mind move, and there is always a
gulf between this knowledge and our putting it into practice.
Psychologists concur with philosophers in the irrelevance of logic to
everyday reasoning. Decades of work on Wason’s selection task seem
to confirm this. And one of the most obvious lessons of work in this
field seems to be that everyday reasoning is much more sensitive to the
content of particular statements than formal logic is. Content can affect
assessments of validity, and can impair good reasoning. In other cases,
it has facilitating effects. Content, however, just seems to be another
name for belief content If a content is believable, then it tends to be
believed, and its tends to be believed whether or not logic, or any sort
of normative standards, tells us to do so or not to do so. In particular,
even if an inference is invalid, but if its conclusion is highly believable,
it tends to be accepted. For instance people accept easily the invalid but
highly believable syllogism:
All of the Frenchmen were wine drinkers
Some or the wine drinkers are gourmets
∴ Some of the Frenchmen are gourmets
Studies in the “belief bias effect” strongly document this (Santamaria,
Garcia-Madruga & Johnson Laird [1998]; Torrens, Thomson & Cramer
[1999]). Another relevant field of study is work on “illusory inferences”
For instance Savary and Johnson-Laird [1998] have shown that the very
processes of construction of mental models which, according to them, is
the basic process of reasoning, leads people to consider as valid inferences which actually are invalid, such as:
If there is a king in the hand then there is an ace in the hand, or else
if there is queen in the hand then there is an ace in the hand.
There is king in the hand
∴ There is an ace in the hand
Other studies tends to show that features of the comprehension processes of the sentences involved in the tasks, or pragmatic features which
are extraneous with respect to the logical forms of the inferential tasks,
have more influence than the logical contents or logical forms of the
inferences in the drawing of conclusion (see e.g. Evans [1994], Sperber,
Cara and Girotto [1995]). The lesson of these studies seems clear: validity and believability fall apart, and it is one thing to asses the validity of
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an inference, and another thing to assess its believability and the way
people actually draw their conclusions. According to mental model theory, this is not surprising, since the assessment of validity often implies
the construction of a large number of mental models, and of sophisticated techniques for reducing these constructions to the relevant cases,
whereas the “natural” construction of models by untrained individuals
allows only a limited amount of processing and of imagination. People
tend to “satisfice” by taking up models which are readily available, neglecting others and failing to look further for more models which would
confirm or infirm a conclusion.
Other psychologists, who hold different views about the actual processes of reasoning (see e.g. Gigerenzer [1995]) tend nevertheless to
agree that it is wrong, when one assesses these processes, to look for
the “logically correct answer” and to contrast it from what is logically
right: it might be that subjects are right on other counts, and, in any
case, there is no real question, when one deals with reasoning of who is
“right” or “wrong” with respect to a given standard or norm. The question is rather, to see how people do actually deal with the tasks, and to
separate the descriptive issue from the normative one. On such a view
the norms of reasoning are not independent from our psychological capacities. There is not assessment of what is “rational” for an agent apart
from the way he actually performs a given task. In other words, there
are no other norms that those which emerge from the psychological capacities, studied in their social contexts. Norms have to “psychologized”
(Gigerenzer [1998]). This comes very close to full-blown psychologism,
and it is open to the current objections against such a view.1
As I said above, although this separation of questions (a) from question (b) seems to be the natural division of labor, and that the freeing
of the psychologist from the normative governance of the logician is in
itself a good thing, I do not think that this division is correct. Psychologists like Gigerenzer, in their boldest moments, seem to speak as if we
could completely separate off the issue of our normative standards of
validity form the issue of our actual practice of reasoning. I suspect that
mental model theorists can be tempted at times by this view. But it is
not clear that, on most psychological theories of reasoning, reasoning
has nothing to do at all with valid reasoning in the sense of an ideal
standard of what should follow from what. On theories according to
which people reason with formal rules of inference (e.g. Rips [1994])
indeed people use such valid rules. On mental model theory (henceforth
MMT), people do not use such rules, but their reasoning has something
to do with validity, at least on two counts:
a) in so far as validity is a semantic notion, MMT supposes that people
use semantic notions: indeed they use the most basic one, truth, for
1
Gigerenzer illustrates this mostly from probabilistic norms, but his view would
probably be parallel with deductive reasoning.
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they represent themselves the truth of the propositions they are
given, and each model is a situation in which a proposition is represented as true
b) the construction of models obeys the semantic principle of validity:
that an inference is valid if and only if there is no way in which its
premises could be true and its conclusion false. The construction of
models is guided by the search for counterexamples. Johnson Laird
and Byrne ([1991], 209) call this “the central core of rationality” In
this respect MMT presupposes that people have at least an implicit
understanding of validity, although the use of the semantic principle
of validity does not imply that people have a tacit repertory of valid
rules of inference, even less that they use such rules explicitly.
(Let us note, in passing, that what we may call the semantic core of
reasoning according to MMT can be found also in the quasi-Tractarian
theory of probability that it uses when it deals with probabilistic reasoning (see Girotto & Legrenzi [1998], 258).)
So even at the unreflective level of the representation of situations
where the state of affairs described by a proposition obtains, MMT presupposes that subjects have at least an implicit grasp of two normative notions: truth, and validity. The use of these notions, however, occurs also at a reflective level. As Johnson-Laird and Byrne ([1991], 147)
note:
Reasoners can know that they have made a valid deduction, and these metadeductive intuitions prepare the way for the development of self-conscious
methods of checking validity. without this higher-level, or meta-deductive,
ability, human beings would not have invented logic, they could not make
deductions about other people’s deductions, and they could not devise psychological theories of reasoning.
This tempers somewhat the radical claim that logic has nothing to do
with reasoning. For we can understand this claim not as a statement to
the effect that logic, as the reflective assessment of meta-deduction, is
not situated at the same level as our untutored inferential intuitions,
although it is necessarily connected with these intuitions: indeed we
would not even understand the rules of inference and the principles of
logical validity if they had absolutely no relationship with our intuitions
about inference and validity. It follows from these remarks that we
should distinguish two sorts of claims of irrelevance (and of relevance)
of logic to actual reasoning:
i) logic is irrelevant to reasoning because the usual logical laws or rules
of inference (of classical logic) do not describe the actual processes of
reasoning, and because subjects do not have an explicit representation of the laws
ii) logic is irrelevant to reasoning because the usual logic laws and rules
of inference (of classical logic) do not prescribe the way to reason, nor
guide implicitly our actual reasoning.
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From the few remarks above, I take (i) to be true, but (ii), although correct as it stands, is not obviously false. If by prescription, one means that
a subject could have an explicit representation of logical rules, and “follow” them consciously, it is clear that most psychological findings show
that people do not reason that way, at least when they are untrained. If
MMT is correct, and if and rule-based theories are incorrect, then it is
not even true that we are implicitly guided by formal logical rules. Now
from this it does not follow that usual logical rules and principles have
no normative role to play in reasoning, for people at least tacitly use the
principle of semantic validity in their construction of mental models,
and must have some grasp of the rules that they use if they are to have
any meta-deductive intuitions at all.
The distinction between (i) and (ii) seems to be close to Evans’s [1996]
distinction between two sorts of rationality: rationality1, the ability to
think in the service of a goal, and rationality2, the ability to reason according to a normative theory. But Evans considers that the two are
strongly independent, for about the same reasons as those given above.
His claim, therefore, looks very much like (i) as well as (ii) This is not my
claim. My claim is that even at the level of untutored intuition, people
have some grasp of rationality2. They grasp a normative theory, even
though they do not do this explicitly nor by consciously follow norms.
My aim is to suggest what grasp of a normative theory is involved
here. There are two dimensions here. One concerns the content of the
norms. The other one is the nature of the implicit understanding of
these norms. The two are not independent from each other.
Let us try first to characterise the content of the norms involved.
MM theorists have suggested that people have the grasp of two basic
norms, truth and validity. Truth indeed is very weak and general, for
according to what Johnson-Laird and his associates call “the principle
of truth”, people tend to minimise the load of working memory by representing explicitly only what is true, not what is false. As they say: “The
theory assumes that the mental footnotes [of the implicit models] about
false cases are rapidly forgotten an all but the simplest of cases. There
is a premium on truth in human reasoning. what is left out of models
is explicit information about possibilities and literals that are false.”
(Savary and Johnson-Laird [1998], ms 5). Now it seems odd to say that
people possess the concept of truth in their manipulation of propositions
if they do not take into account cases where the propositions which are
given to them are false. Grasp of truth as a norm should go with grasp of
the concept of falsity, and have an idea of the asymmetry between truth
and falsity. Now MMT predicts that people infer the false cases from
their knowledge of true cases. And in their application of the principle of
semantic validity, they do look for counterexamples. For instance, even
if they initially represent conditional statements through models where
the representation of the falsity of the antecedent is only implicit, they
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have a conception of what would falsify a conditional, namely a case
where the antecedent is true and the consequent false.
According to MMT, there is a least this minimal understanding of
the notion of truth and validity in reasoning. I have argued that it is
enough to temper the claim that logical normativity has nothing to do
with actual reasoning. Still, the amount of logicality involved is very
weak. The question is whether we can ascribe to the subjects a better
grasp of logical principles than this minimal understanding.
On the theory according to which people reason from their tacit
knowledge of a basic set of formal rules of a mental logic, subjects do
have more than this minimal understanding. They indeed are supposed
to master and to use a certain amount of rules similar to those of natural deduction. But such views grant too much logicality to subjects.
Against these views, MM theorists hold that only the minimal representations of models and the memory driven principles of construction
are necessary. The amount of “logic” involved in the basic models for the
propositional connectives postulated by MMT (see e.g. Johnson Laird
and Byrne [1991], 51) is very small. MMT is a parsimonious theory of
reasoning, and it owes its explanatory virtues to precisely this simplicity: when a theory explains more with less principles, the more powerful
it is.
I want, however, to argue that more than this minimal amount of
logicality is needed for reasoning, even though we do not need to suppose that people possess full blown formal rules. What is it, on MMT,
to understand the meaning of a propositional connective, such as “and”,
“or” or “if…then” ? It is to understand a basic set of mental models for
each connective, explicit and implicit. For instance
implicit explicit
A and B
ABAB
A or B (inclusive) A A ¬ B
B ¬A B
ABAB
It would be wrong to say that the meanings of connectives are explicitly
defined by they canonical mental models, since the theory postulates
that there are implicit definitions of these connectives. MM theorists
insist that the construction of mental models is not equivalent to the
use of rules. But even if we grant this, it is not obvious that subjects
cannot form inferences from their understanding of the mental models
associated with each connective. For instance a subject who is asked to
consider a conjunction “A and B” and who represents the mental model
A B above, is certainly able to assess the simple inference
A and B
therefore A
for A is indeed represented in his model. And the same for the inference
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A and B
therefore B
Just in the same way, MMT shows how the subjects who have the mental models for the conditional “ If A then B”
AB
…
can readily assess modus ponens forms of inference. As well known, the
above inferences for “and” and “if” are called, in natural deduction systems, the elimination rules for each connective. But these kinds of inference are not just some among those which are easily constructed from
the mental models. They are indeed partly constitutive of the meaning
of the connectives. A subject who would not agree that “A” follows from
“A and B” would simply show that he is not able to construct the appropriate mental model for the connective, and hence that he does not
know the meaning of “and”. In this sense, the meaning of a connective
is constituted by the role that it plays in reasoning. If one grants this, I
do not see why MMT would be incompatible with the view that there are
canonical kinds of inference attached to each connective. We could have
an indirect confirmation of this if we considered a spurious connective,
such as Prior’s [1960] tonk. Everyone knows the story about this odd
connective. How could we tell it in the vocabulary of MMT? I suppose
that it would go something like this. Imagine that I tell you that tonk is
a connective for which when
A tonk B is true,
A is true
and for which when A tonk B is true, B is true.
On MMT subjects should construct the model
AB
You will be led to identify tonk with “and”. But now suppose that I tell
you that it is also the case that when
A is true
A tonk B is true
(and similarly when B is true, A tonk B is true). You will then be unable
to represent the truth of A tonk B, for if you form a model
A
you obviously cannot form the model
AB
from it. However, the former model is consistent with
A
B
But here you get lost, for the two models are incompatible. They do not
represent the same state of affairs. In other terms, you cannot form
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mental models from the rules given to you about this connective tonk,
and this is why the connective has no role in one’s conceptual economy. This could be also formulated in terms of classical truth tables and
natural deduction rules. The introduction rule for tonk requires that
“A tonk B” is true when A is true and B is false. Its elimination rule
requires that in the same case “A tonk B” is false. Otherwise the rule
will lead from truth to falsity. Hence there is no coherent truth table for
tonk. I shall say more on this below, but for the moment I conclude from
this that a subject who understands the meaning of a connective understands some basic rules of inference associated to the mental models
that he is able to construct in inferences involving this connective, and
that these rules are normative for his inferential practice. The norms
attached to each connective are more substantial than the mere grasp
of the concept of truth and of semantic validity.
The second dimension of normativity concerns, as I said, the nature
of the implicit understanding of the norms involved. One of the reasons
why logic, as a set of valid rules or principles, seems to be irrelevant to
everyday reasoning, is that people do not have any explicit, reflective,
knowledge of these rules; moreover, many psychological experiments
show that even people trained in logic or in other normative theories
commit systematic errors. This was also part of Lewis Carroll’s point: explicit knowledge of a valid rule is of no help for reaching a given conclusion. This is why knowledge of the norms of reasoning must be implicit.
But the alternative view, that people obey to blind compulsions, and are
simply disposed to reach certain conclusions without any appreciation of
what rules they are following, is also unattractive: if they are to reason
at all, they must have some appreciation of the norms governing their
performance. If the foregoing remarks are correct, even when a number
experiments seem to show that people jump to conclusions because of
certain bias, memory constraints, and other causal factor, it does not follow that people fail to have reasons for inferring what they infer. So even
if their understanding of inference rules is not explicit, it is not simply
blind. (This remark has obvious ties with Wittgenstein’s discussion of
rule-following, but I cannot take it up here.2) An account is needed which
would make room both for the implicit (and hence not completely conscious) grasp of the norms and for the fact that people have at least some
knowledge of the reasons of their inferential activities.
II. Logical constants and conceptual role
Thus far I have only considered the relationship between questions (a)
and (b) above, and argued that reasoning is not a completely norm free
2
Wittgenstein argues both against the idea that we would have, in any rulegoverned activity the explicit knowledge of the rule set up in front of our mind (for
this would pose the threat of a regress, like in Carroll’s story) and against the idea
that following a rule is simply a matter of being disposed to act in certain ways.
See the interesting remarks of Brewer [1995].
P. Engel, Logic, Reasoning and the Logical Constants
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activity. But I have completely left out question (c), the question of the
justification of logical rules and principles. What justifies these rules?
Can they even be justified? Are they based on more primitive principles,
or are they themselves basic? Are they secure? Such questions (see e.g.
Dummett [1973]) are situated at an even more higher level of abstraction than the question whether are aware of the validity of principles of
logic and really use some valid principles, for they deal with the nature
of validity itself in this domain. Given that people do not use valid rules
of reasoning such as those that logicians investigate in their everyday
reasoning, why should they care about the very nature of validity? Of
course they don’t, unless they reach this highly sophisticated stage of
meta-deductive inquiry which comes when logicians raise questions
about the consistency, soundness and completeness of their rules. Nevertheless there is an actual link between the psychological problem of
logical competence and the metatheoretical problem of the justification
of deduction. It is this the following. To justify logical principles is to
give an account of the meanings of the logical expressions which determine a given set of sentences and of inferences as “logical”, i.e. of the
“logical constants” There are basically two styles of justification of the
principles of logic, or two ways of determining the meanings of the logical constants. One is proof-theoretic, and appeals to properties of the
systems of rules, independently of such semantical notions as truth or
validity. The other one is semantic, and appeals to such properties. The
two are not incompatible, of course, but they can become incompatible
if the truth or the validity of a given sentence (or set of sentences) is
understood in terms of the assertibility of this sentence. Those writers who favour an anti-realist conception of truth, according to which
there is no more to truth than assertibility, and who reject the idea that
there can be true but unverifiable statements, will tend to favor a purely proof-theoretic treatment (for this debate, see e.g. Tennant [1987],
Dummett [1991], Engel [1989]). Now there is a similar alternative between a proof-theoretic procedure of defining the logical constants and
a semantic procedure in the psychology of reasoning, although this is
not alternative about how these constants should be justified, but about
whether they are used at the psychological level: it is the familiar alternative between “formal rules” accounts and “mental model” accounts.
The former are clearly proof-theoretic, and the latter clearly semantic.
But apart from this analogy, what is the connection between the two?
The first alternative is a logical one, the second a psychological one. It
does seem to me that there is an important connection between the two,
and I want to argue that the very reasons which justify a certain account
of the meanings of logical constants are those which operate, at the psychological level, to rationalise the reasoning behaviour of subjects, and
these reasons are semantical.
Let us come back to Prior’s tonk. Prior invented this spurious connective in order to show that the falsity of the thesis according to which log-
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ical constants could be introduced by convention and through stipulated
inference rules. In his equally famous reply to Prior, Belnap pointed out
that what was wrong with tonk was not that its elimination and introduction rules were inconsistent, but that the requirement of conservative extension was not met: one should not introduce a new constant in
a language if that constant does not allow derivations which were not allowed in the language before the introduction of this new constant. Now
this conservative extension requirement is purely proof-theoretic, and it
is emphasised by philosophers would favour an anti-realist view of logical justification (see Dummett [1991], 217–220, Tennant [1987]). But let
us return to the remark about tonk above. The upshot of these remarks
was that there is not coherent semantic assignment of truth values for
tonk (and this why people cannot form mental models for tonk). So we
can formulate what is wrong with tonk in semantic terms, rather than
in proof-theoretic terms: there is no assignment which makes tonk inferences truth preserving. Indeed Peacocke ([1988], [1993]) has formulated
a requirement of conservative extension in semantic terms. The prooftheoretic requirement is (for a set of rules S, and where L is a system of
deduction for a language not containing the new logical constant):
S is deductively conservative over L iff for any sentences A1, …An, B
of L,
A1, …An |— L+S B only if A1, …,An |— L B
and the semantic requirement is:
S is semantically conservative over L iff for any sentences A1, …An, B
of L,
A1, …An |— L+S B only if A1, …,An |= L B (where ‘|=’ is the relation of
semantic consequence)
Peacocke then argues that conservative extension is a requirement
which is available for someone who favours a realist view of truth and
a semantic justification of deduction. I shall not develop here his argument (see Engel [1989], ch. XII). The important thing for us here is
that we can translate this framework for a theory of the justification
of deduction into a framework for a theory of the meaning and of the
understanding of logical constants.
This appropriate framework is that of a semantic in terms of “conceptual roles”. A conceptual role semantics is in general a theory of
meaning according to which to explain the meaning of an expression
is to specify its role in reasoning and inferences. There are a number
of versions of this view (see for instance Harman [1982], Block [1986])
but the one which interest us here is this: the conceptual role of an expression is constituted by what leads a subject to accept certain kinds
of sentences (the acceptance conditions of a sentence), and to infer certain kinds of sentences form others. In this sense it is close to a “use”
theory of meaning, and even to a verificationist view. In the case of the
meanings of the logical constants, a pure conceptual role theory of this
kind amount to the thesis that their role is entirely given by their elimi-
P. Engel, Logic, Reasoning and the Logical Constants
231
nation and introduction rules, without any appeal to semantic notions
such as truth or validity. But we have just seen that such a pure theory
can be formulated in semantic terms. Indeed, if we generalize from the
case of tonk, it can be argued that the acceptance conditions (or the assertion conditions) of a given connective determine its truth conditions.
In other terms, a conceptual genuinely determines a meaning for an
expression only if there is a semantic value of the appropriate category
for this expression which makes the transitions mentioned in the conceptual role always truth-preserving. (In Peacocke’s [1992] terminology,
this amount to saying that there is a “determination theory” for a given
concept.)
On Peacocke’s framework, which I follow here, understanding a logical constant is understanding these principles for it which makes them
truth-preserving. Peacocke holds also that this understanding is constituted by finding the introduction and elimination rules “primitively
obvious” For instance, in the case of negation, Peacocke says that
What is primitively obvious to anyone who understands negation is just that
A is incompatible with A. Unless the ordinary user of negation appreciates
that A and ¬ A cannot be both true, then he does not understand ¬. ([1988],
163)
Peacocke further hypotheses that in order to find this “obvious”, the
subject must have an “implicit conception” of negation. There is no incoherence in the idea that one can have an implicit or tacit knowledge of a
normative rule. Only the thought that the alternative must be between
conscious awareness of the rules and brute compulsion can lead us to
think the contrary. Thus we have three conditions:
(i) the rules for the basic connectives are “obvious”
(ii) they are normative, for they flow from their conceptual role
(iii) this conceptual determines truth conditions and validates the
connective introduction and elimination rule.
Now (i) is an answer to question (a) above, (ii) is an answer to question
(b), and (iii) is an answer to question (c). There is a parallelism with the
connectives that we find primitively obvious, the fact that they give us
reasons to draw a certain number of inferences, and the fact that the
inference rules of the connectives are semantically truth-preserving,
hence justified.
I anticipate the reaction: isn’t it too good to be true? Have I been
cheating? It might be objected also that in insisting, with Peacocke, on
the idea that the meanings of the logical connectives are given by their
basic inference rules and their conceptual roles, my view just amounts
to a revamping of the traditional conception of a mental logic of formal
rules. To the first question, I shall answer that my view implies that
there indeed a connection between the validity of a logical rule (the fact
that it is truth preserving) and the impression of validity that people
have. Otherwise, it seems to me, we could not recognise some rules,
such as modus ponens, or the elimination and introduction rules for
232
P. Engel, Logic, Reasoning and the Logical Constants
“and” as normative: we could not move from our implicit understanding
to an explicit understanding of the connectives. But of course, as logicians and psychologists alike insist, it is not because a rule seems to us
to be valid that it is valid, and it is not because a rule is valid that it
will seem to us to be so. If the former were the case, psychologism would
be true, and I agree that it is not, as an attempt to derive validity from
impression of validity. If the latter were the case, we would not observe
such a gap between logic and reasoning competence. But the relationship between impressions of validity and validity has nevertheless to be
close enough, if the meaning of logical words is supposed to be normative. To the second question, I answer that my view does not amount to
a mental logic view. Nothing in what I have said implies that subjects
have a full mastery of all the elimination and introduction rules for the
connectives that natural deduction logic postulates. But they must have
a least a partial grasp of it, and must be able to recover some of these
rules from (e.g. modus ponens) their intuitive understanding. Furthermore nothing in what I have said implies that there is some algorithm
which could allow subject to recover the truth conditions (and hence
the mental models by which we construct them, on MMT theory) from
their inferential rules and their conceptual roles. It does not, therefore,
seems to me that there an incompatibility between the present version
of conceptual role semantics and mental model theory.
III. Rationality and reflective equilibrium
I have claimed that there is a form of harmony between our reflective
understanding of logical rules and the norms of logic, in spite of the
fact that people do not reason according to logic rules. This position
may seem to be close to the position of those writers who have defended
a similar pre-established harmony under the name of “reflective equilibrium” (Cohen [1998]). According to this conception there should not
be any real discrepancy between the normative principles of reasoning
(logic) and human reasoning competence since these come from our intuitions about what constitutes good reasoning, which in turn comes
from our reasoning competence, since our normative principles are the
product of a series of weighted comparisons between these and our intuitions, until an equilibrium is reached. Reflective equilibrium thus
has been used as an argument for the claim that there is an a priori
presumption of rationality, a sort of virtuous circularity between our
intuitions and our normative standards. Other arguments for this claim
are the use of various principles of charity in interpretation which are
supposed to give us an a priori warrant that subjects cannot but be rational (for a study, see Engel [1993])
Reflective equilibrium arguments and similar arguments about the
necessary harmony between logical competence and the normative
rules of logic have been strongly criticized (see Stein [1996]). On the one
hand, what warrant do we have that the intuitions which are supposed
P. Engel, Logic, Reasoning and the Logical Constants
233
to serve as an input in the reflective equilibrium method are rational?
Indeed many of our intuitions are wrong. By what miracle should we
in the end come up with an harmony between these intuitions and the
normative principles? On the other hand, if the reflective equilibrium
method just supposes that in the end, after a series of appropriate adjustments we reach this harmony, then it seems to simply beg the question.
The present argument, however, does not depend upon the notion
of a reflective equilibrium, nor upon such arguments as those which
use the principle of charity in interpretation. It does not depend on the
first because, unlike the reflective equilibrium method, a conceptual
role theory of meaning is not strongly holistic. It is holistic in this sense
that it supposes that the meaning of an expression is determined by its
inferential connections with other expressions, but it is not strongly holistic in the sense that it would have to suppose that all the inferential
links of an expression matter in fixing its conceptual role. This thesis is
often called “meaning holism”, and the implication just mentioned is not
found, to say the least, very attractive by many.3 The thesis defended
here, with respect to logical constants, is that a delimited set of inferential roles is characteristic of a logical constant, and serves to define
it implicitly. Hence this thesis grants, against strong holists such as
Quine, that there is a legitimate distinction between a set of meaning
giving principles which are analytic and other meaning link which are
synthetic. This is not the place to argue for this here, but it seems to me
to be an implication of this view. For just the same reason, the present
view does not depend upon a method of radical interpretation based on
the principle of charity, and hence it does not allow any argument to the
effect that there is a general presumption of rationality.
Conclusion
I have been far too sketchy. But let me try to summarise nevertheless
what I have attempted to propose. First, in spite of the fact that logic,
as a normative discipline made up of valid principles and rules, is not
relevant to everyday reasoning, it is not true that it is totally irrelevant.
Even psychologists who, like the MM theorists, claim that reasoning
goes by construction of models and various bias and not by formal rules,
admit that we could not have meta-deductive intuitions if everyday reasoning did not at least allow people to have a grasp of the normative
concepts of truth and semantic validity. Second, it seems to me that we
can go further, and argue that people have an implicit understanding
of some normative logical principles, at least a subset of the inferential
rules for the logical connectives, and that this understanding is based
on a semantic principle, according to which a logical connective is legiti3
See Stein’s (1996) comments p.124-127 on the relationship between the principle
of charity and holism.
234
P. Engel, Logic, Reasoning and the Logical Constants
mate if and only if it is truth preserving. There is thus a link between
a realist view of truth (truth is not assertibility), a semantic justification of logic, and our implicit understanding of normative principles.
It seems to me that there is nothing here with which semantically inspired researchers on reasoning could disagree. The gap between logic
and the psychology of reasoning is not, on my view, as large as it is often
claimed to be.4*
References
Belnap, N. [1961–2] “Tonk, Plonk and Plink”, Analysis, XXXII, 130–134.
Block, N. [1986], “Advertisement for a Semantics for Psychology”, Midwest
Studies in Philosophy, X.
Brewer, B. [1995], “Compulsion by Reason”, Proceedings of the Aristotelian
Society, sup. vol. LXIX, 237–254.
Caroll, L. [1895], “What the tortoise said to Achilles”, Mind, 4.
Castellani, F. & Montecucco, L. [1998], a cura di, Normativita logica e ragionamento di senso commune (Bologna: Il Mulino).
Cohen, L. J. [1981], “Can Human Irrationality be Experimentally Demonstrated?”, Behavioral and Brain Sciences, 4, 317–331.
Dummett, M. [1973], “The Justification of Deduction”, in Truth and Other
Enigmas (Cambridge, Mass.: Harvard University Press).
_______[1991], The Logical Basis of Metaphysics (Cambridge, Mass.: Harvard University Press).
Engel, P. [1989], La norme du vrai (Paris: Gallimard), revised English tr.
The Norm of Truth (Toronto: Harvester Wheatsheaf and University of
Toronto Press, 1991).
_______[1993], “Logique, raisonnement et rationalité”, in Houdé, O. et
Miéville, D., Pensée Logico-mathématique, nouveaux objects interdisciplinaires (Paris: P.UF.), 205–228.
_______[1996], Philosophie et psychologie (Paris: Gallimard, Folio), Italian
translation Filosofia e psicologia (Torino: Einaudi, 1999).
_______[1998], “La logique peut-elle mouvoir l’esprit”, Dialogue¸ XXXVII,
35–54.
4*
This paper was read at the conference “Reasoning: the logical and the
psychological perspectives”, at the University of Padova, in May 1999. But it is
actually a descendent of a paper, on Lewis Carroll’s paradox, read two years earlier in
the same place at the conference “Modeli mentali” in honor of Philip Johnson Laird,
which was published in a French version as Engel 1998. In both occasions, Philip
Johnson Laird’s comments have been enormously useful, and my debt to him goes
further. He is the one who put me to read more of the psychological literature and
his open mindedness to philosophical issues is unique. I thank also for their remarks
Marco Santambrogio, my commentator, Vittorio Girotto, Paolo Legrenzi, Daniele
Giaretta, Giorgio Samblin, Paolo Leonardi, and Carlo Filotico; for their hospitality
on both occasions Alberto Mazocco, Paolo Flores d’Arcais, and Daniele Giaretta, and
Luisa Montecucco for all her encouragements. An Italian version of the text has been
published as “ Logica, ragionamento e constanti logische”, ch. 3 de P. Cherubini, P.
Giaretta et A. Mazzocco (eds.), Ragionamento, psicologia e logica (Firenze: Giunti,
2001), 108–127.
P. Engel, Logic, Reasoning and the Logical Constants
235
Evans, J. St. B. T. [1993], “Bias and Rationality” in J. Manktelow & D.
Over, Rationality (London: Routledge).
_______[1994], “Relevance and reasoning”, in S. Newstead & J.St. B Evans, eds., Current Directions in Thinking and Reasoning (Erlbaum: Hilsdale).
Gigerenzer, G. [1995], “The taming of content: some thoughts about domains and modules”, Thinking and Reasoning, vol. 1, 289–400.
_______[1998], “Psychological challenges for normative models”, in D. Gabbay and P. Smets, eds., Handbook of Defeasible Reasoning and Uncertainty Managements Systems, vol. 1, 441–467.
Girotto, V. [1994], Il ragionamento (Bologna: Il Mulino).
_______ & P. Legrenzi [1998], “Logica, probabilitŕ e ragionamento ingenuo”,
in Castellani & Montecucco [1998], 241–260.
Harman, G. [1982], “Conceptual Role Semantics, Notre Dame Journal of
Formal Logic, 23, 242–56.
_______[1986], Change in View (Cambridge, Mass.: MIT Press).
Johnson-Laird, P. & R. Byrne [1991], Deduction (Hillsdale, NJ: Lawrence
Elbaum).
_______ & F. Savary [1999], “Illusory inferences: a novel class of erroneous
deductions”, Cognition, 71, 191–229.
Montecucco, L. [1998], “Normativita e descrittivitta nello studio del ragionamento,” in Luisa Montecucco, Contesti filosofici della scienza (Brescia:
La scuola).
Peacocke, C. [1988], “Understanding Logical Constants”, Proceedings of the
British Academy, 73, 153–200.
_______[1992], A Study of concepts (Cambridge, Mass.: MIT Press).
_______[1993], “Proof and Truth,” in J. Haldane and C. Wright, Realism and
Reason (Oxford: Oxford University Press).
Prior, A. [1960], “The Runabout inference ticket”, Analysis, XXI, 38–39.
Rips, L. [1993], The Psychology of Proof (Cambridge, Mass.: MIT Press).
Santamaria, C., J. Garcia-Madruga, & P. Johnson-Laird [1998], “Reasoning
from double conditionals: the effects of logical Structure and Believability”, Thinking and Reasoning, 4, 97–122.
Sperber, D., F. Cara & V. Girotto [1995], “Relevance Theory Explains the
Selection Task”, Cognition, 31, 31–95.
Stein, E. [1996], Without Food Reason: The Rationality Debate in Philosophy and Cognitive Science (Oxford: Oxford University Press).