JOHN CORCORAN

Truth-preservation, implication-preservation, and cognition-preservation.

This is one in a series of presentations designed to alert the philosophical community that claims made for the importance of truth-preservation are often overblown, frequently false, and—even when strictly speaking true--misleading and counter-productive. Truth-preservation per se is not nearly as important as implication-preservation; implication-preservation per se is not nearly as important as, cognition-preservation

1972. Weak and Strong Completeness in Sentential Logic, Logique et Analyse 59/60, 429–34.  MR0337476 (49 #2245)

This is another study illustrating the fruitfulness of thinking of “logics” as three-part systems composed of a language, a semantics [or model theory] and a deductive system [or proof theory]. For simplicity we take a fixed language for sentential logic [or propositional logics] with a fixed standard two-valued semantics, but we “vary” deductive systems. In the 1960s when this paper was written, it was still widely assumed that a “logic” was “determined” by its theorems, by what it could “prove”. The vague assumption was that once the theorems were fixed, it was somehow automatic which conclusions were derivable from given premises.
The purpose of this paper is to give a method which weakens deriving power of a sentential logic while leaving proving power unchanged. In particular if the method is applied to a strongly complete logic the resultant will not have strong completeness but it will retain weak completeness. Moreover, because of the nature of the method two added facts obtain.
First, weak completeness of the resultant is always an obvious corollary to weak completeness of the operand, so that no new completeness proof need be constructed. Second, failure of strong completeness in the resultant is always obvious, so no multi-valued matrices need be constructed. The method applies to all sentential logics formulated using only axiom schemes and schematically statable rules. These results reaffirm, generalize, and simplify a result of Hiz [Journal of Symbolic Logic, 24 (1959), pp. 193-202].

► JOHN CORCORAN, What syllogisms are: three views, eight centuries.
Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA
E-mail: corcoran@buffalo.edu
  At issue is the nature of “the syllogisms” in Prior Analytics [1]. For centuries from the 1200s, the dominant view was enshrined in the medieval mnemonic “Barbara-Celarent” [2]: syllogisms are certain valid premise-conclusion arguments commonly mislabeled “inferences”.  In the mid-1900s many modern logicians adopted a contrary view credited to Jan Łukasiewicz [3]: syllogisms are certain true universal propositions informally called “implications”. A fruitful two-sided debate ensued producing evidence weakening the Łukasiewicz case.
  Toward the last quarter of the 1900s, a third contender appeared. Independently, John Corcoran [4] and Timothy Smiley [5] took the class of syllogisms to exclude propositions while including not only the valid arguments recognized as syllogisms in medieval times but also deductions establishing validity. A deduction contains, over and above premises and conclusion of an argument, a chain of reasoning showing that the conclusion’s information is contained in that of the premises. The debate became three-sided for twenty years or more.
  The Łukasiewicz view lacks current defenders leaving the debate to the medieval and Corcoran-Smiley views.
  This study details all three views—emphasizing their differences. For example, the medieval syllogisms are valid but ‘true’ does not apply to them, though their premises and conclusions are all true or false. The Łukasiewicz syllogisms are all true but the term ‘valid’ as applied to arguments is inappropriate. The Corcoran-Smiley syllogisms cannot be said to be true in the sense applicable to propositions but they are all valid in the sense that their conclusions follow from their premises. Moreover, many of them containing chains of reasoning are “cogent” [6] in an epistemic sense not applicable either to the Łukasiewicz or to the medieval syllogisms.
  [1] ROBIN SMITH, Aristotle’s Prior Analytics, Hackett, 1989. 
  [2] JOHN CORCORAN, DANIEL NOVOTNÝ, AND KEVIN TRACY,  Deductions and Reductions Decoding Syllogistic Mnemonics, Entelekya Logico-Metaphysical Review,  vol.2 (2018), pp. 5–39.
  [3] JAN ŁUKASIEWICZ, Aristotle’s syllogistic, Oxford UP, 1951.
  [4] JOHN CORCORAN, Aristotle’s Prototype Rule-based Underlying Logic, Logica Universalis, vol. 2 (2018) pp. 9-35.
  [5] TIMOTHY SMILEY, What is a Syllogism?, Journal of Philosophical Logic,
vol. 2 (1973) pp. 136–154.
[6] JOHN CORCORAN, Argumentations and logic, Argumentation, vol. 3 (1989), pp. 17–43.

Tarski’s proof of the law of identity

  Tarski’s LEIBNIZ’S LAW [Introduction to Logic, Sect. 17] is the second-order sentence in variable-enhanced English:

For everything x, for everything y: x = y iff
x has every property y has and y has every property x has.

  The Law of Identity, Tarski’s LAW OF REFLEXIVITY FOR IDENTITY, is:

Everything is equal to itself. For everything x: x = x.

  Tarski says it can be “derived” from Leibniz’s Law. Indeed, the paragraph following its statement begins with the word “PROOF” and ends with the words “which was to be proved”. This paper analyzes Tarski’s argumentation. We found several errors, four noted below.

Boolean induction, Bulletin of Symbolic Logic. TBA XX (201X) XXX–YYY.


► JOHN CORCORAN, Boolean induction.
Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA
E-mail: corcoran@buffalo.edu
  George Boole (1815–1864), founder of modern logic according to C. I. Lewis [1], had many interesting ideas: some undeveloped [2]; some unrecognized until years after his death [3]; some still unexplored. One of the latter is Boolean induction: certain universal truths are latent in certain of their instances: knowledge of such a universal is obtainable from knowledge of an instance [4].
  Modifying Boole’s idea, a Boolean-universal is a universal sentence implied by an instance. Conversely, a Boolean-singular is a singular sentence that implies one of its universalizations.
  In standard first-order logics, Boolean-universals are [logically] equivalent to the instances that imply them. Conversely, Boolean-singulars are equivalent to the universals that they imply. For expository purposes first-order sentences are represented in variable-enhanced English interpreted in the integers [5].
  Every tautological universal, as ‘for every x, x = x’, is Boolean. Every inconsistent singular, as ‘0 ≠ 0’, is Boolean. A true non-tautological Boolean-universal is ‘for every x, for some y, x ≠ y’ and a true consistent Boolean-singular is ‘for some y, 0 ≠ y’. A false non-tautological Boolean-universal is ‘for every x, x = y, for every y’, and a false consistent Boolean-singular is ‘0 = y, for every y’.
  Boolean-universal and Boolean-singulars are abundant: it’s easily seen that every sentence equivalence-class contains infinitely many Boolean-universals, and thus also infinitely many Boolean-singulars. This paper studies these first-order-logic phenomena.
  [1] JOHN CORCORAN, C. I. Lewis: History and Philosophy of Logic, Transactions of the C. S. Peirce Society, vol. 42 (2006), pp.1–9.
  [2] JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol.  24 (2003), pp. 261–288.
  [3] JOHN CORCORAN, Revisiting Boole’s Principle of Wholistic Reference, this BULLETIN, vol. 12 (2006), pp. 515–6.
[4] GEORGE BOOLE, Laws of Thought, Introduction by J. Corcoran, Prometheus, 1854/2003. 
[5] JOHN CORCORAN 2016. Logic teaching in the 21st century. Quadripartita Ratio: Revista de Argumentación y Retórica, vol. 1 (2016) pp. 1–34.

Errata in Henkin’s 1950 type-theory completeness paper.
  The first paragraph of Leon Henkin’s influential and widely-read article [1] reads as follows.

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

  As it is written the second sentence says that each formula is a provable logical truth: a patent absurdity. Of course, what was intended was more like following, which moves the clause beginning ‘which becomes’ so it modifies ‘formula of the calculus’.

Roughly speaking, this proof demonstrates that each formula of the calculus which becomes a true sentence under every one of a certain intended class of interpretations of the formal system is a formal theorem.

  Taking more advantage of the disclaimer ‘roughly speaking’, he could say:

Roughly speaking, this proof demonstrates that each formula which is true under every interpretation is a theorem.

  Almost 20 years later, Jaakko Hintikka [2] reprinted [1] along with other important articles. Hintikka noted errata in another Henkin article but not in [1]. There are other mistakes in [1]. Moreover, it contains several passages that, if not actual mistakes, are regrettable. We list and analyse glitches in [1] in order to facilitate study of this landmark article. We also discuss the old question of whether, as logicians including Burton Dreben have charged [personal communication], [1]’s title is misleading in that it seems to contradict an easy consequence of Gödel’s 1931 incompleteness paper. The consequence in question is explicitly mentioned in the second paragraph of [1].
[1] LEON HENKIN, Completeness in the theory of types. Journal of Symbolic Logic, vol. 15 (1950) pp. 81–94.
[2] JAAKKO HINTIKKA, Philosophy of Mathematics, Oxford UP, 1969.
Acknowledgements: Mark Brown, William Demopoulos, Thomas Drucker, Rodolfo Ertola, William Frank, Richard Grandy, Idris Samawi Hamid, Allen Hazen, Leonard Jacuzzo, Miguel León, Kristo Miettinen, Joaquin Miller, Frango Nabrasa, David Plache, Sriram Nambiar, José Miguel Sagüillo, Michael Scanlan, Stewart Shapiro, James Smith, Roberto Torretti, Alasdair Urquhart, Albert Visser, George Weaver, George Williams, Stanley Ziewacz, and others—some of which disagree with some of my points or with my way of making them.

In particular, distinguished and accomplished logicians including Allen Hazen, Christopher Menzel, Alasdair Urquhart, and Albert Visser think that Henkin’s expression of completeness is correct as it is—awkwardness notwithstanding. To be clear, they believe and argue that I am wrong to imply that this is a mistake on Henkin’s part.
WHAT IS YOUR VIEW?

The syllogistic mnemonic known by its first two words Barbara Celarent introduced a constellation of terminology still used today. This concatenation of nineteen words in four lines of verse made its stunning and almost unprecedented appearance around the beginning of the thirteenth century—before or during the lifetimes of the logicians William of Sherwood and Peter of Spain, both of whom owe it their lasting places of honor in the history of syllogistic. The mnemonic, including the theory or theories it encoded, was prominent if not dominant in syllogistics for the next 700 years until a new paradigm was established in the 1950s by the great polymath Jan Łukasiewicz,

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. We give fresh new emphasis on the goal-directed nature of deduction and on evidence that Aristotle’s practice exemplifies goal-directedness even if his explicit theory does not deal with it. There is also new emphasis on the fact that the categorical syllogistic was intended as a prototype, like first-order monadic logic, and not as a “universal logic”, like Principia Mathematica. This paper adds to our appreciation of Aristotle penetrating insights into the nature of demonstrative knowledge while at the same time bringing to light more of the beginner’s oversights and omissions inevitable in original logical investigations.

This book is alleged to be a comprehensive but largely elementary description of mathematical logic including its historical development, its most important achievements and its implications for philosophy. Although the intended audience includes both mathematicians
and philosophers, 'the book presupposes no knowledge of logic and only high school mathematics". The author avows an aim "to stimulate long-lived interest by communicating some of the excitement and beauty of the subject". Details and techniques are eschewed in favor of emphasis on the development of an overview.
The book is also intended by the author to serve as an introduction to logic. His opinion of the shortcomings of most introductions to logic is perceptive and insightful.

Teaching course-of-values induction. Bulletin of Symbolic Logic.21 (2015) 101.

  Let P be a property that belongs to every number whose predecessors all have it. Clearly, P could be any property that belongs to every number: if P belongs to every number, then—a fortiori—P belongs to every number whose predecessors all have it.
  But is the converse true? Is it the case that if P belongs to every number whose predecessors all have it, then P belongs to every number?  A-fortiori reasoning is often non-reversible.

This expository paper on Aristotle's prototype underlying logic is intended for a broad audience that includes non-specialists. It requires as background a discussion of Aristotle's demonstrative logic. Demonstrative logic or apodictics is the study of demonstration as opposed to persuasion. It is the subject of Aristotle's two-volume Analytics, as its first sentence says. Many of his examples are geometrical. A typical geometrical demonstration requires a theorem that is to be demonstrated, known premises from which the theorem is to be deduced, and a deductive logic by which the steps of the deduction proceed. Every demonstration produces (or confirms) knowledge of (the truth of) its conclusion for every person who comprehends the demonstration. Aristotle presented a general truth-and-consequence theory of demonstration meant to apply to all demonstrations: a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining theory of deduction was meant to apply to all deductions: any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises. His deductions, both direct and indirect, were rule-based and not tautology-based. The idea of tautology-based deduction, which dominated modern logic in the early years of the 1900s, is nowhere to be found in Analytics. Rule-based (or " natural ") deduction was rediscovered by modern logicians. To illustrate his general theory of deduction, Aristotle presented a prototype: an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. With reference only to propositions of the four so-called categorical forms, he painstakingly worked out exactly what those immediately evident deductive steps are and how they are chained to complete deductions. In his specialized prototype theory, Aristotle explained how to deduce from a given categorical premise set, no matter how large, any categorical conclusion implied by the given set. He did not extend this treatment to non-categorical deductions, thus setting a program for future logicians. The prototype, categorical syllogistic, was seen by Boole as a " first approximation " to a comprehensive logic. Today, however it appears more as the first of the dozens of logics already created and as the first exemplification of a family that continues to expand. Mathematics Subject Classification. Primary 01A20; Secondary 00A25.
THIS HAS BEEN SUBMITTED. PROOFS TO CORRECT WILL COME IN 10 DAYS. PLEASE POST SENTENCES NEEDING ATTENTION. IT IS NOT NECESSARY TO SUGGEST A FIX OR EVEN SAY WHAT NEEDS A FIX.

There is something distressing in the fact that this book, coauthored
by a reputable logician, published by a reputable press and
favorably reviewed by reputable reviewers, is nevertheless so marred
that it cannot begin to serve its avowed purpose. The claims made by
the authors, publishers and reviewers amount to an elaborate and
cruel hoax. Where the reader is lead to expect clarity, insight and
rationality instead he finds himself in a largely indecipherable swamp
scattered with bizarre little islands of Kafkaesque puzzles and Alicein-
Wonderland meaning shifts.

CORCORAN ON NUMERAL-FREE MATHEMATICAL INDUCTION

1997–1998 WINTER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC
JOHN CORCORAN 1998. Mathematical induction and specific-case semantic omega properties. Bulletin of Symbolic Logic 4, 220–221.

Although the axiomatizations of second-order number theory preferred by mathematicians take as primitive entities (besides the universe of natural numbers beginning with zero or with one) the initial number and the successor function, there are axiomatizations whose languages do not involve numerals, i.e., terms composed of a symbol for the initial number preceded by zero or more symbols for the successor function.
  Taking the property of being initial (instead of the initial number) and taking the relation of [immediate] succession (instead of the successor function), the following property-universal proposition can serve as the principle of mathematical induction:

Every property that belongs to every initial number and to every number that succeeds a number to which it belongs also belongs to every number.

  In such contexts the usual (numeral-dependent) omega notions do not apply; instead a [semantic] theory—set of propositions closed under consequence—is defined to be omega complete (respectively, consistent) if and only if it contains (respectively, lacks the negation of) every number-universal proposition all of whose “specific cases” it contains. By the specific cases of a number-universal proposition, “every number has [property] P,” is meant the following: “every initial number has P”, “every number succeeding an initial number has P”, “every number succeeding a number succeeding an initial number has P”, etc.
  Presupposing a standard formalization of classical second-order logic, as found in the 1956
Church book, e.g., this paper shows that a theory is omega complete (respectively, consistent) if and only if it contains (respectively, lacks the negation of) the principle of mathematical induction.

The common noun ‘tautology’ (plural: ‘tautologies’) is used in the logic literature in many senses, some of which are precise and some vague. This revised dictionary entry restricts itself to one broad sense and one narrow sense
Although it is difficult to underestimate the importance of tautologies, Quine managed to do just that with his 1970 “definition of logic” as the systematic study of tautologies. With equal justification he might have defined logic as the systematic study of contradictions: after all in a sense the difference between tautologies and contradictions boils down to the presence or absence of one negation. Corcoran 1972. In some humanities contexts it would be more fruitful to define logic as the systematic study of argumentations as in Corcoran 1989. In some mathematics contexts it would be more fruitful to define logic as the systematic study of underlying logics as in Corcoran 1973g. Anyway, the quest for a “one-size-fits-all” definition of logic continues to be challenging and instructive.

This interesting and provocative book on the nature of mathematical thought is a joy. The quality of analysis, knowledge, and speculation represented in this work place it well above the standard fare in this area. Many mathematicians will place it in the tradition which includes such works as Descartes' Rules, J. Hadamard's The psychology of invention in the mathematical field, and G. H. Hardy's A mathematician's apology , but it goes beyond this category in two respects. First, Beth's part presents new contributions to the history of the philosophy of mathematics as well as a digestible and objective overview of the current viewpoints concerning the nature of mathematics. Second, Piaget's part presents the results of several fascinating experiments in mathematical learning together with a new philosophy of mathematics based on his genetic approach to the psychology of mathematical creativity.

INTRODUCING TARSKI'S 1983 LSM.docx
Editor's introduction revised edition.  Logic, Semantics, Metamathematics. Alfred Tarski, pages xv–xxvii.
I wish to express here my most genuine and cordial gratefulness to Professor John Corcoran for his work related to the publication of the present volume. The new edition of Logic, Semantics, Metamathematics is almost exclusively due to his initiative. In serving as the editor, he fulfilled his task with great fervor and enthusiasm. We collaborated in searching out and trying to remove various defects and weak spots of earlier texts. The new indices at the end of this volume, prepared by Corcoran, are much superior to those in the first edition and hopefully will facilitate the study and the use of the work. What is more valuable, he has provided the volume with his own introduction.--ALFRED TARSKI, Berkeley, California, January 1982
Corcoran’s 1983 edition of Tarski’s LOGIC, SEMANTICS, METAMATHEMATICS is available free. http://library1.org/_ads/DE3C6E8E30D380DC39D1057DB4BD8EE1
CORCORAN ON TARSKI
https://www.academia.edu/s/179246c7cd/corcoran-on-tarski-december-2016-pdf?source=link
Corcoran’s publications that interpret, criticize, explain, or build on Alfred Tarski’s theories and results.

We speak informally of " weakening " propositions (axioms or conjectures): e.g., axioms with undesirable consequences or conjectures with counterexamples. We also speak informally of " strengthening " propositions (theorems or hypotheses): e.g., if the theorem's proof reveals unnecessary restrictions or our evidence for the hypothesis suggests something stronger. This paper makes such language more formal. We consider certain weakenings and strengthenings of conditionals. Using the alternative constituent format [this Bulletin , vol. 15 (2009), p. 133], thirty-six hypotheses are presented, explained, and then finally settled—some are proved; some disproved. The result: thirty-six interrelated theorems about weakenings and strengthenings of conditionals. A weakening of a proposition is implied by but doesn't imply the proposition. A strengthening of a proposition implies but isn't implied by the proposition. Thus, a certain proposition is a weakening of a given proposition iff the latter is a strengthening of the former. A conserving of a given proposition implies and is implied by the given proposition.

Prior Analytics is often not clear whether a premise—either per se or of a syllogism—is a sentence, proposition, statement, fact, judgment, belief, or thought. To discuss the question we use numerically-indexed alternative-constituent sentences.

Abstract:

John Corcoran and Hassan Masoud. 2014. Existential import today: New metatheorems; historical, philosophical, and pedagogical misconceptions. History and Philosophy of Logic. 36: 39–61.

Ranked sixth on the “Most-read list” at History and Philosophy of Logic,
this demanding but self-contained and widely accessible paper refutes over a century of mistakes about existential import.

All terminology is not only explained but discussed. Many useful examples presented in usable form.

Central to our campaign is the fact that first-order logic has limited existential import: the universalized conditional implies its corresponding existentialized conjunction in some but not all cases. We prove the Existential-Import Equivalence: In any first-order logic, for a universalized conditional to imply the corresponding existentialized conjunction it is necessary and sufficient for the existentialization of the antecedent predicate to be tautological.
Research Interests: Logic, History of Logic, Philosophy of Logic, Mathematical Logic, C. I. Lewis, 1st-order logic, and Existential import
Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀x [S(x) → P(x)] implies its corresponding existentialized conjunction ∃x [S(x) & P(x)], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: ∀x [S(x) → P(x)] implies ∃x [S(x)& P(x)] iff ∃x S(x)is logically true. The antecedent S(x) of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction. A predicate is an open formula having only x free. An existential-import predicate Q(x) is one whose existentialization, ∃x Q(x), is logically true; otherwise, Q(x) is existential-import-free or simply import-free. How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential-import implications hold as widely as they fail. Existential import is not an isolated phenomenon. As documented below, these results correct false or misleading passages even in respected logic texts.

LOGICAL METHODOLOGY CHART
Charting a method for trying to determine the validity or invalidity of a given argument not known to be valid and not known to be invalid

METHOD OF DEDUCTION: An argument is valid iff its conclusion follows from its premise-set. Not every valid argument is known to be valid. To determine of a given valid argument not now known by you to be valid, you need a method that produces knowledge of validity. One method in use before Aristotle is to deduce the conclusion from the premises-set.
METHOD OF COUNTERARGUMENTATION: An argument is invalid iff its conclusion does not follow from its premise-set. Not every invalid argument is known to be invalid. To determine of a given invalid argument not now known by you to be valid, you need a method that produces knowledge of invalidity. One method, perhaps not in use before Aristotle, is to produce a known counterargument: an argument known to be in the same form, known to have a falsehood as its conclusion, and known to have all truths in its premises-set.

CORCORAN ON THE BIRTH OF LOGIC [IN ENGLISH]

The last two decades have witnessed a debate concerning whether Aristotle's syllogistic is a system of deductive discourses having epistemic import exemplifying an Aristotelian theory of deductive reasoning and justifying the claim that Aristotle is the founder of logic taken as the scientific study of proof or whether, on the contrary, the syllogistic is a system of true propositions of a theory of classes justifying the claim that Aristotle is the founder of logic is taken as the scientific study of formal relations such as class inclusion. […]
The epistemically-oriented interpretation permits the birth of logic as epistemic metascience to be located with Aristotle while deferring the birth of logic as ontic science to the modern period. In contrast, the ontically-oriented interpretation permits the birth of logic as ontic science to be located with Aristotle while deferring the birth of logic as epistemic metascience to the modern period.

CORCORAN ON THE BIRTH OF LOGIC [IN SPANISH]
CHAT https://www.academia.edu/s/a2af72ccfe/corcoran-on-the-birth-of-logic-in-spanish?source=link
TEXT https://www.academia.edu/29034139/CORCORAN_ON_THE_BIRTH_OF_LOGIC_IN_SPANISH_

John Corcoran. 1992. El Nacimiento de la Lógica (The Birth of Logic), Agora 11/2, 67–78, Spanish translation by C. Martinez-Vidal and J. M. Sagüillo of an expanded and revised version of an  unpublished English paper distributed to members by the Society for Ancient Greek Philosophy before being presented to the Society at its 1989 Chicago Meeting.

These works are listed in the following.
2015. Bibliography: John Corcoran’s Publications on Aristotle 1972–2015. Aporía · Revista Internacional de Investigaciones Filosóficas. 10 (2015) 73–118.

De Morgan, Augustus (1806 1871), prolific British mathematician, logician, philosopher of mathematics, philosopher of logic, remembered chiefly for several lasting contributions to logic and philosophy of logic including discovery and deployment of the concept of universe of discourse, co founding of relational logic, adaptation of what are now known as De Morgan’s Laws, and several terminological innovations including coining the expression ‘mathematical induction’. His main logical works, the monograph Formal Logic (1847) and the series of articles “On the syllogism” (1846 1862), demonstrate wide historical and philosophical learning, synoptic vision, penetrating originality, and disarming objectivity. His relational logic treated a wide variety of inferences involving propositions whose logical forms were significantly more complex than those treated in the traditional framework stemming from Aristotle, e.g., “Every ancestor of a parent of a person is a parent of an ancestor of the person”, “If every doctor is a teacher then every ancestor of a doctor is an ancestor of a teacher”. De Morgan’s conception of the infinite variety of logical forms of propositions vastly widens that of his predecessors and even that of his able contemporaries such as Boole, Hamilton, Mill, and Whately. De Morgan did as much as any of his contemporaries toward the creation of modern mathematical logic.

John Corcoran and Hassan Masoud. Three logical theories redux. Bulletin of Symbolic Logic. 22 (2016) 433.


The 1969 paper, “Three logical theories” [1], considers three logical systems all based on the same interpreted language and having the same semantics. At the time, terminology such as ‘truth-preservation’ had not yet become fashionable. Once logicians saw the usefulness of speaking of preserving truth, it became clear that rules preserve other properties as well and that rules could be classified according to which properties they preserve.
The 1969 paper presupposed audiences that accept deductive systems—natural-deduction—as epistemically fundamental and that regard logistic systems and consequence systems as technically sound but artificial constructs. However, this paper aims for wider audiences including logicians who regard logistic or consequence systems as epistemically fundamental and who take natural-deduction to be “psychological”, “heuristic”, or in some other way scientifically inferior, derivative, or even inadequate.
  Moreover, this paper also discusses epistemic foundations. How do proponents of logistic systems explain how knowledge of tautologousness is acquired? How do proponents of consequence systems explain how knowledge of consequence-preservation is acquired? How do proponents of deductive systems explain how knowledge of cogency-preservation is acquired?

In 1847 Boole described “laws of the mental processes” that “render logic possible” [1, pp. 4–6]. He considered “mental acts”—operations applying to classes and yielding respective subclasses:
Boole wrote: "Now the several mental operations […] are subject to peculiar laws. [Some concern] repetition of a given operation or the succession of different ones. […] These will […] be ranked as necessary truths […] probably they are noticed for the first time in this Essay."
  Our Boolean operation-class calculus [BOC] represents—probably for the first time in this lecture—Boole’s 1847 laws, but in modernized form.
This rich system is what Boole actually presented, only a distant ancestor of Boolean Algebra

Semiotic Triangles, also called semantic triads,  chart the the result of combing three distinctions that help students master language skills needed for critical thinking and effective writing: especially writing about writing. One is the expression-reference distinction that helps prevent use-mention fallacies. One is the expression-meaning distinction that helps prevent category-mistakes such as attributing to a word properties that are applicable only to meanings.  One is the meaning-denotation distinction that helps prevent category-mistakes such as attributing to a referent of a word properties that are applicable only to its meanings. Thus we have what may be called an expression-sense-reference distinction.
Acknowledgements: Aristotle, Augustine, Ockham, Peirce, Frege, Tarski, Quine, Church, Austin.

Employing second-order propositions and second-order reasoning in a natural way, this work illustrates the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework—not an artificial formalism created by specialists for technical purposes. Some of the main relationships between second-order logic and first-order logic are explained by introducing basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. Even though it lacks variables and quantification, basic logic is a genuine underlying logic. Although not a mere logic schema like propositional logic, it serves many of the pedagogical purposes the latter now serves. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving repudiation of a core part of our scientific tradition defended by traditionalists. But its rejection is also viewed as reactionary, analogous to the repudiation of symbolic logic by supporters of " Aristotelian " traditional logic. But even if " genuine " logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its essential importance for understanding the history of logic and mathematics will not be diminished. Second-order logic might some day be gone, but it should never be forgotten.

CORCORAN ON MATHEMATICAL OPINION AND KNOWLEDGE—draft 13 of a review for MATHEMATICAL REVIEWS of: Paseau, Alexander, Knowledge of mathematics without proof, British J. Philos. Sci. 66 (2015), no. 4, 775--799]
The article under review, “Knowledge of mathematics without proof”—hereafter KMWP—appears in a respected peer-reviewed journal. Its author, an Associate Professor at Oxford University, presented previous versions at conferences in Bergen, Cambridge, Dubrovnik, and Oxford. Its challenging subject, the nature of mathematical knowledge, has been studied and debated since ancient times.

We propose a mathematically precise definition of the intuitive relations of "being in the same logical form as" for formalized languages. Let L be an arbitrary first-order language. Any one-one function from the vocabulary (set of characters) of L onto itself is category-preserving, categorial, or a categorial, if it carries every non-logical constant to another of the same syntactic category and carries every other character to itself. Any categorial f extends naturally and uniquely to the set of sentences: if s is a sentence of L, then the n-th character-occurrence in fs is fk where k is the character occurring in the n-th place in s. Any [extended] categorial extends naturally and uniquely to sets P of sentences, to [premise-conclusion] arguments (P, c) where P is a set [premise-set] and c a sentence [conclusion], and to “discourses” which include formal deductions whether “axiomatic” or “natural”. A given sentence p is said to be formally similar to (or formally identical to or to have the same logical form as) a sentence q if there is a categorial f such that fp is q itself or an alphabetic variant of q. Formal similarity extends to sets, arguments, discourses, etc. Among the several obvious metatheorems are analogues of all of the traditional principles of form: every two sentences having the same form are both tautological or both informative (non-tautological), every two sets having the same form are both consistent or both inconsistent, every two arguments in the same form are both valid or both invalid, every two discourses in the same form are both formal deductions (in a given system) or neither is. This framework enables restatement of the Hilbert-Bernays-Quine theorem that, in a sufficiently rich interpreted language without identity, in order for a sentence to be consistent (respectively, tautological) it is necessary and sufficient for it to be in the same logical form as some true sentence (respectively, no false sentence).

We study several relations expressed by the two-place relational verb-phrases ‘X makes Y true’ and ‘X makes Y false’, or synonyms such as ‘X verifies Y’ and ‘X falsifies Y’ [3, pp. 180, 283]. This abstract gives three examples. Notation and terminology follow [2].
  Some usages mislead because the object “made true” or “made false” already had the truth-value in question. Other usages mislead because the object “made true” or “made false” doesn’t—and by nature can’t—have a truth-value. The object “made true” or “made false” isn’t true or false. Aristotle discussed related confusions in Categories xii, 14b16.
  In one usage, the number two makes the proposition “some number is even” true; three makes “every number is even” false. These have been called respectively proexample and counterexample relations [1].  Two is a proexample for “some number is even”; three is a counterexample for “every number is even”. Here we have propositions which by nature have truth-values; those “made true (or false)”didn’t come to have that truth-value.
  In another usage, the intended interpretation of number-theoretic English makes the sentence ‘some number is even’ true and it makes ‘every number is even’ false. Here we have uninterpreted English sentences which by nature cannot have truth-values: they are true (false) under the interpretations that “make them true (false)” [3, passim].
In still another usage, the fact that some number is even makes the proposition “some number is even” true; the fact that not every number is even makes “every number is even” false. Compare Aristotle, ibid.
[1] JOHN CORCORAN, Counterexamples and proexamples, this BULLETIN, vol.11 (2005), p. 460.
[2] JOHN CORCORAN, Sentence, Proposition, Judgment, Statement, and Fact, Many Sides of Logic, (Walter Carnielli et al., editors), College Publications, 2009.
  [3] WARREN GOLDFARB, Deductive logic, Hackett, 2003.

Predications in ancient logic. Bulletin of Symbolic Logic. 19 (2013) 132–3.

► JOHN CORCORAN AND COREY MCGRATH, Predications in ancient logic.
Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA
E-mail: corcoran@buffalo.edu
E-mail: cm267@buffalo.edu
  Our previous study “Predicates and predications”—abstracted in this BULLETIN, vol. 18 (2012), p. 999—identified several grammatical categories employing the word ‘predicate’: common nouns , two-place relation verbs taking linguistic subjects, three-place action verbs taking human subjects, and others.

‘True’ is a predicate.
‘True’ is a predicate of certain sentences.
Tarski predicates ‘true’ of sentences.

This more-limited study focuses on ancient logic taking examples from [1] as representative.
  Predication entered logic at the very beginning: the Greek title of Aristotle’s Categories is often translated predicates. The noun ‘predicate’ is used in Categories in a primary sense (as above) but also in a secondary sense as in the following.

The intransitive verb is not the only predicate Aristotle treats.
The complemented transitive verb is not the only predicate Aristotle treats.


In a primary sense, a noun denotes the individuals in its extension; in a secondary sense a noun denotes kinds, sorts, or types. In a primary sense of animal, Socrates is an animal. In a secondary sense, the tiger is an animal.
  Aristotle also uses the noun ‘predicate’ in a third sense in which every predicate carries the syntax of a common noun as opposed to a verb phrase ([1], pp. 109f, 118f, 122, 158f) as in the following.

Every predicate is a term.
Every subject is a predicate.

This lecture treats a neglected topic: the uses of the words ‘predicate’ and ‘predication’ and their synonyms, translations, and cognates in ancient logic. Authors discussed include Aristotle, Chrysippus, Galen, Porphyry, and Sextus Empiricus. It builds on lectures abstracted in this BULLETIN, vol. 11(2005) p. 117 and vol. 12(2006) pp. 353-4.
[1] JONATHAN BARNES, Truth, Etc.: Six Lectures on Ancient Logic, Oxford UP: Oxford, 2007.


OBSERVATIONS ADDED 090617

The tiger resembles the lion.
Tigers resemble lions.
A tiger resembles a lion.

The tiger is not a tiger.
No tiger is the tiger.
Every tiger is a tiger.
In the primary sense of the common noun ‘animal’, every tiger is an animal but the tiger isn’t an animal. In the secondary sense of the common noun ‘animal’, the tiger is an animal but no tiger is an animal.

A person who states that four is square predicates, or affirms, the property being square of the number four. Here the action verb ‘to predicate’ is used to express a three-place action: a person predicates a property of an entity.
  In the sentence ‘four is square’, the word ‘four’ is the [grammatical] subject and ‘is square’ is the predicate. Here the word ‘predicate’ is part of an expression for a two-place grammatical relation: a phrase is the [grammatical] predicate of a sentence. In a third sense, any phrase that occurs as the predicate of a sentence is a predicate, using the word as a non-relational common noun.
  A person who states that four is a number that is a square that is even predicates the property being a number that is a square that is even of the number four. The 10-word phrase

is a number that is a square that is even

is the grammatical predicate of the 11-word sentence ‘four is a number that is a square that is even’ although that sentence contains occurrences of five other predicates: ‘is a number that is a square’, ‘is a number’, ‘is a square that is even’, ‘is a square’ and ‘is even’. A person who uses the 11-word sentence to state that four is a number that is a square that is even makes one predication, but uses six predicates.
  This lecture is an introduction to a neglected topic: the uses of the words ‘predicate’ and ‘predication’ in logic. Authors discussed include Aristotle, Boole, DeMorgan, Peirce, Frege, Russell, Tarski, and Church. It builds on lectures abstracted in this BULLETIN vol. 11(2005) p. 117 and vol. 12(2006) pp. 353-4.