John Corcoran

(1937 - 2021)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a logical point of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.]

The primary sense of the word ‘hypothesis’ in modern colloquial English includes “proposition not yet settled” or “open question”. Its opposite is ‘fact’ in the sense of “proposition widely known to be true”. People are amazed that Plato [1, p. 1684] and Aristotle [Post. An. I.2 72a14–24, quoted below] used the Greek form of the word for indemonstrable first principles [sc. axioms] in general or for certain kinds of axioms. These two facts create the paradoxical situation that in many cases it is impossible to translate the Greek form of the word using the English form: the primary sense of the word ‘hypothesis’ in modern colloquial English is diametrically opposed to one sense used by Plato and by his most accomplished student Given current colloquial English usage it is impossible to get the word hypothesis to carry the connotation of “settled truth” much less “axiomatic truth”. The ‘hypo-’ [under] in the Plato-Aristotle use of ‘hypothesis’ might carry the sense of “basis” or “foundational” as opposed to “less than usual or normal”. This paradox parallels the one pointed out by Robin Smith: it is impossible for the English word ‘syllogism’ to carry the meaning of its Greek form Aristotle intended. There are other cases as well: it is impossible for the English biological term ‘genus’ to carry the meaning of its Greek form the Greek genos refers to family as in our ‘genealogy’, not to “higher species” as in our ‘generic’.

Corcoran, J. 2007. Syntactics, American Philosophy: an Encyclopedia. 2007. Eds. John Lachs and Robert Talisse. New York: Routledge. pp.745-6. Syntactics, semantics, and pragmatics are the three levels of investigation into semiotics, or the comprehensive study of systems of communication, as described in 1938 by the American philosopher Charles Morris (1903-1979). Syntactics studies signs themselves and their interrelations in abstraction from their meanings and from their uses and users. Semantics studies signs in relation to their meanings, but still in abstraction from their uses and users. Pragmatics studies signs as meaningful entities used in various ways by humans. Taking current written English as the system of communication under investigation, it is a matter of syntactics that the two four-character strings ‘tact’ and ‘tics’ both occur in the ten-character string ‘syntactics’. It is a matter of semantics that the ten-character string ‘syntactics’ has only one sense and, in that sense, it denotes a branch of semiotics. It is a matter of pragmatics that the ten-character string ‘syntactics’ was not used as an English word before 1937 and that it is sometimes confused with the much older six-character string ‘syntax’. Syntactics is the simplest and most abstract branch of semiotics. At the same time, it is the most basic. Pragmatics presupposes semantics and syntactics; semantics presupposes syntactics. The basic terms of syntactics include the following: ‘character’ as alphabetic letters, numeric digits, and punctuation marks; ‘string’ as sign composed of a concatenation of characters; ‘occur’ as ‘t’ and ‘c’ both occur twice in ‘syntactics’. However, perhaps the most basic terms of syntactics are ‘type’ and ‘token’ in the senses introduced by Charles Sanders Peirce (1839-1914), America’s greatest logician, who could be considered the grandfather of syntactics, if not the father. These are explained below.

History witnesses alternative approaches to “the proposition”. The proposition has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes, that which can be said to be believed, disbelieved, understood, etc. It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions, that which can be said to be grasped, judged true or false, assumed for reasoning purposes, etc. The proposition has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties, that which can be said to be true, false, tautological, informative, inconsistent,etc. It has also been taken as the subject and object of logical relations, e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. Prima facie, such properties and relations are non-mental and objective. It has also been taken to be the resultants or products of propositional operations, usually mental or linguistic; e.g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences. Finally,propositions have been taken to be meanings of certain declarative sentences. This essay is an informal, selective, and incomplete survey of alternative approaches to “the proposition” with special attention to the views of the late American philosopher Peter Hare (1935–2008)and of those who influenced him. DOI: 10.5007/1808-1711.2011v15n1p51

The word ‘equality’ often requires disambiguation, which is provided by context or by an explicit modifier. For each sort of magnitude, there is at least one sense of ‘equals’ with its correlated senses of ‘is greater than’ and ‘is less than’. Given any two magnitudes of the same sort—two line segments, two plane figures, two solids, two time intervals, two temperature intervals, two amounts of money in a single currency, and the like—the one equals the other or the one is greater than the other or the one is greater than the other [sc. in appropriate correlated senses of ‘equals’, ‘is greater than’ and ‘is less than’]. In case there are two or more appropriate senses of ‘equals’, the one intended is often indicated by an adverb. For example, one plane figure may be said to be equal in area to another and, in certain cases, one plane figure may be said to be equal in length to another. Each sense of ‘equality’ is tied to a specific domain and is therefore non-logical. Notice that in every cases ‘equality’ is definable in terms of ‘is greater than’ and also in terms of ‘is less than’ both of which are routinely considered domain specific, non-logical. The word ‘identity’ in the logical sense does not require disambiguation. Moreover, it is not correlated ‘is greater than’ and ‘is less than’. If it is not the case that a certain designated triangle is [sc. is identical to] an otherwise designated triangle, it is not necessary for the one to be greater than or less than the other. Moreover, if two magnitudes are equal then a unit of measure can be chosen and, no matter what unit is chosen, each magnitude is the same multiple of the unit that the other is. But identity does not require units. In this regard, congruence is like identity and unlike equality. In arithmetic, the logical concept of identity is coextensive with the arithmetic concept of equality. The logical concept of identity admits of an analytically adequate definition in terms of logical concepts: given any number x and any number y, x is y iff x has every property that y has. The arithmetical concept of equality admits of an analytically adequate definition in terms of arithmetical concepts: given any number x and any number y, x equals y iff x is neither less than nor greater than y. As Aristotle told us and as Frege retold us, just because one relation is coextensive with another is no reason to conclude that they are one.

English translation of an entry on pages 137–42 of the Spanish-language dictionary of logic: Luis Vega, Ed. Compendio de Lógica, Argumentación, y Retórica. Madrid: Trotta. DEDICATION: To my friend and collaborator Kevin Tracy. This short essay—containing careful definitions of ‘counterargument’ and ‘counterexample’—is not an easy read but it is one you’ll be glad you struggled through. It contains some carefully chosen examples suitable for classroom discussion. Using the word ‘counterexample’ instead of ‘counterargument’ in connection with Aristotle’s invalidity proofs leads to a misinterpretation of Aristotle’s intentions. The two relations expressed by these two words are drastically different—as explained in this essay. There is however a subtle overlap: Every argument that is a counterargument for a given argument is a counterexample to the universal proposition that every argument in the same form as the given argument is valid. The word ‘counterexample’ was coined in the 1800s. The word ‘counterargument’ was given this sense in the 1960s. Saying that one argument is a counterexample for another argument is bad English—taking the word ‘counterexample’ in its Standard English sense. The word ‘counterinstance’ is an exact synonym for ‘counterexample’. Which logic books define ‘counterinstance’ or ‘counterexample’? Who started using ‘counterinstance’ or ‘counterexample’ in the sense of ‘counterargument’? Did they note that they were not using the word in the Standard sense? Why did it catch on? Also see: https://www.academia.edu/8840909/Argumentations_and_logic_1989_

►JOHN CORCORAN AND IDRIS SAMAWI HAMID, Two-method errors: having it both ways. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] Philosophy, Colorado State University, Fort Collins, CO 80523-1781 USA E-mail: [email protected] Where two methods produce similar results, mixing the two sometimes creates errors we call two-method errors, TMEs: in style, syntax, semantics, pragmatics, implicature, logic, or action. This lecture analyzes examples found in technical and in non-technical contexts. One can say “Abe knows whether Ben draws” in two other ways: ‘Abe knows whether or not Ben draws’ or ‘Abe knows whether Ben draws or not’. But a stylistic TME occurs in ‘Abe knows whether or not Ben draws or not’. One can say “Abe knows how Ben looks” using ‘Abe knows what Ben looks like’. But syntactical TMEs are in ‘Abe knows what Ben looks’ and in ‘Abe knows how Ben looks like’. One can deny that Abe knows Ben by prefixing ‘It isn’t that’ or by interpolating ‘doesn’t’. But a pragmatic TME occurs in trying to deny that Abe knows Ben by using ‘It isn’t that Abe doesn’t know Ben’. There are several standard ways of defining truth using sequences. Quine’s discussions in the 1970 first printing of Philosophy of logic [3] and in previous lectures were vitiated by mixing two [1, p. 98]. The logical TME in [3], which eluded Quine’s colleagues, was corrected in the 1978 sixth printing [2]. But Quine never explicitly acknowledged, described, or even mentioned the error. This lecture presents and analyses two-method errors in the logic literature. [1] JOHN CORCORAN, Review of Quine’s 1970 Philosophy of Logic. In Philosophy of Science, vol. 39 (1972), pp. 97–99. [2] JOHN CORCORAN, Review of sixth printing of Quine’s 1970 Philosophy of Logic. In Mathematical Reviews MR0469684 (1979): 57 #9465. [3] WILLARD VAN ORMAN QUINE, Philosophy of logic, Harvard, 1970/1986.

CORCORAN REVIEWS THE 4 VOLUMES OF TARSKI’S COLLECTED PAPERS Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about logic, whereas virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82.

A truth-preservation fallacy is using the concept of truth-preservation where some other concept is needed. For example, in certain contexts saying that consequences can be deduced from premises using truth-preserving deduction rules is a fallacy if it suggests that all truth-preserving rules are consequence-preserving. The arithmetic additive-associativity rule that yields 6 = (3 + (2 + 1)) from 6 = ((3 + 2) + 1) is truth-preserving but not consequence-preserving. As noted in James Gasser’s dissertation, Leibniz has been criticized for using that rule in attempting to show that arithmetic equations are consequences of definitions. A system of deductions is truth-preserving if each of its deductions having true premises has a true conclusion—and consequence-preserving if, for any given set of sentences, each deduction having premises that are consequences of that set has a conclusion that is a consequence of that set. Consequence-preserving amounts to: in each of its deductions the conclusion is a consequence of the premises. The same definitions apply to deduction rules considered as systems of deductions. Every consequence-preserving system is truth-preserving. It is not as well-known that the converse fails: not every truth-preserving system is consequence-preserving. Likewise for rules: not every truth-preserving rule is consequence-preserving. There are many famous examples. In ordinary first-order Peano-Arithmetic, the induction rule yields the conclusion ‘every number x is such that: x is zero or x is a successor’—which is not a consequence of the null set—from two tautological premises, which are consequences of the null set, of course. The arithmetic induction rule is truth-preserving but not consequence-preserving. Truth-preserving rules that are not consequence-preserving are non-logical or extra-logical rules. Such rules are unacceptable to persons espousing traditional truth-and-consequence conceptions of demonstration: a demonstration shows its conclusion is true by showing that its conclusion is a consequence of premises already known to be true. The 1965 Preface in Benson Mates (1972, vii) contains the first occurrence of truth-preservation fallacies in the book.

Imagine an equilateral triangle “pointing upward”—its horizontal base under its apex angle. A semiotic triangle has the following three “vertexes”: (apex) an expression, (lower-left) one of the expression’s conceptual meanings or senses, and (lower-right) the referent or denotation determined by the sense [1, pp. 88ff]. One example: the eight-letter string ‘coleslaw’ (apex), the concept “coleslaw” (lower-left), and the salad coleslaw (lower-right) [1, p. 84f]. Using Church’s terminology [2, pp. 6, 41]—modifying Frege’s—the word ‘coleslaw’ expresses the concept “coleslaw”, the word ‘coleslaw’ denotes or names the salad coleslaw, and the concept “coleslaw” determines the salad coleslaw—recalling Frege’s principle that sense determines denotation. Church [2, p. 6] wrote: -/- We shall say that a name denotes or names its denotation and expresses its sense. […] Of the sense we say that it determines its denotation, or is a concept of the denotation. -/- Aristotle seems cognizant of distinctions going beyond those in semiotic triangles. The expression Aristotle’s semiotic pyramids seem warranted by Aristotle’s Categories, 1a1: -/- When [two] things have a name (onoma) in common and the concept (logos) of being (ousia) which corresponds to the name in each case is different, they are called same-named (homonuma). Thus, for example, both a man and a picture [of an animal] are called animals. These have only a name in common. In each case the name’s concept of being [an animal] is different; for if one says what being an animal is for each of them, one will give two distinct concepts. -/- Semiotic triangles and pyramids in Aristotle’s logic are compared to those in Church’s [2]. [1] JOHN CORCORAN, Sentence, proposition, judgment, statement, and fact, Many Sides of Logic, College Publications, 2009. [2] ALONZO CHURCH, Introduction to Mathematical Logic, Princeton, 1956. -/- The semiotic pyramid in Categories, 1a1 has a square base under the vertex ‘animal’. On the corners of the square are: the concept “animal”, the concept “animal picture”, the animals, and the animal pictures. The animals are homonymous with the animal pictures. People find Aristotle’s example far-fetched or inept even if the experience of pointing to a picture while saying “That is Tarski” is familiar. Imagine looking at a painting while thinking “That is an animal”. Without putting too fine a point on this, notice that in Aristotle’s sense it is individual things that are homonymous, not words. It would be natural to say also in his sense that two things are homonyms if one is homonymous with the other. In contrast, we use the words homonym and homonymous to relate words that are spelled the same and pronounced the same but have different meaning. Consider the noun ‘center’ and the verb ‘center’. Consider the noun ‘smell’ and the verb ‘smell’. The spelling of two homonyms is an ambiguity, or an ambiguous spelling. We need appropriate adjectives to distinguish the Categorical senses of ‘homonym’ and ‘homonymous’ from the current English sense just mentioned. I propose ‘ontological’ for the sense relating things and ‘linguistic’ for that relating words. Given that all words are things but not all things are words, we ask are words that are linguistically homonymous also ontologically homonymous? END OF POST ABSTRACT.

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the parallel postulate. Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. There are other differences that are or might become important. Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content.

This paper will annoy modern logicians who follow Bertrand Russell in taking pleasure in denigrating Aristotle for [allegedly] being ignorant of relational propositions. To be sure this paper does not clear Aristotle of the charge. On the contrary, it shows that such ignorance, which seems unforgivable in the current century, still dominated the thinking of one of the greatest modern logicians as late as 1831. Today it is difficult to accept the proposition that Aristotle was blind to the fact that, for example, incommensurability is a relation and not a property: that the proposition “In every square, the diagonals are incommensurable with the sides” is relational and not categorical. This paper asks the reader to do something more difficult: to accept the proposition that as late as 1831 De Morgan was blind to the same fact. This paper shows conclusively that in 1831, De Morgan was still in the grips of the allegedly Aristotelian paradigm.

We discuss misinformation about “the liar antinomy” with special reference to Tarski’s 1933 truth-definition paper [1]. Lies are speech-acts, not merely sentences or propositions. Roughly, lies are statements of propositions not believed by their speakers. Speakers who state their false beliefs are often not lying. And speakers who state true propositions that they don’t believe are often lying—regardless of whether the non-belief is disbelief. Persons who state propositions on which they have no opinion are lying as much as those who state propositions they believe to be false. Not all lies are statements of false propositions—some lies are true; some have no truth-value. People who only occasionally lie are not liars: roughly, liars repeatedly and habitually lie. Some half-truths are statements intended to mislead even though the speakers “interpret” the sentences used as expressing true propositions. Others are statements of propositions believed by the speakers to be questionable but without revealing their supposed problematic nature. The two “formulations” of “the antinomy of the liar” in [1], pp.157–8 and 161–2, have nothing to do with lying or liars. The first focuses on an “expression” Tarski calls ‘c’, namely the following. c is not a true sentence The second focuses on another “expression”, also called ‘c’, namely the following. for all p, if c is identical with the sentence ‘p’, then not p Without argumentation or even discussion, Tarski implies that these strange “expressions” are English sentences. [1] Alfred Tarski, The concept of truth in formalized languages, pp. 152–278, Logic, Semantics, Metamathematics, papers from 1923 to 1938, ed. John Corcoran, Hackett, Indianapolis 1983. https://www.academia.edu/12525833/Sentence_Proposition_Judgment_Statement_and_Fact_Speaking_about_the_Written_English_Used_in_Logic

This review places this translation and commentary on Book A of Prior Analytics in historical, logical, and philosophical perspective. In particular, it details the author’s positions on current controversies. The author of this translation and commentary is a prolific and respected scholar, a leading figure in a large and still rapidly growing area of scholarship: Prior Analytics studies PAS. PAS treats many aspects of Aristotle’s Prior Analytics: historical context, previous writings that influenced it, preservation and transmission of its manuscripts, editions of its manuscripts, interpretations, commentaries, translations, and its influence on subsequent logic, philosophy, and mathematics. All this attention is warranted because Prior Analytics marks the origin of logic: the field that, among other things, asks of a given proposition whether it follows from a given set of propositions; and, if it follows, how we determine that it follows; and, if it does not follow, how we determine that it does not follow. This translation and commentary is not suitable for use in an undergraduate course. It has too many quirks that the teacher would want to warn against. A copy editor should have dealt with these things and with other matters such as incorrect punctuation and improper end-of-line divisions. The prose is heavily laden with glaring clichés. The one-page preface contains “longer than I care to remember”, “more than I can possibly list here”, “first and foremost”, and “last and by no means least”—a sentence later is devoted to thanking the “incredibly meticulous and helpful copy-editor”. A few pages later the translator reveals the need “to find a path between the Scylla … and the Charybdis …”. Moreover, the index is far from meeting the needs of undergraduate students. The attention to scholarly detail is not what one hoped for from Oxford University Press. At 26b10-15, this translation reads “let swan and white be chosen as white things” for what Smith correctly translates “let swan and snow be selected from among those white things”. At 41b16, “angles AB and CD” should read “angles AC and BD”. Despite this book’s flaws, it will be found useful if not indispensable for those currently engaged in Prior Analytics studies. The alternatives suggested to Robin Smith’s translation choices are often worth consideration. It is to be emphasized, however, that this book is unsuitable for those entering Prior Analytics studies.

Contrary to dictionaries, a non sequitur isn’t “any statement that doesn’t follow logically from previous statements”. Otherwise, every opening statement would be a non sequitur: a non sequitur is a statement claimed to follow from previous statements but that doesn’t follow. If the sentence making a given statement doesn’t contain ‘thus’, ‘so’, ‘hence’, ‘therefore’, or something else indicating an implication claim, the statement isn’t a non sequitur in this sense. But this is only one of several senses of that expression, which, contrary to appearances, is a unitary common noun more properly written as one word: non-sequitur or nonsequitur. Above a nonsequitur was, roughly speaking, a conclusion of an invalid premise- conclusion argument, but not every such, only those that were stated to follow from premises that had been stated. Every informative proposition is the conclusion of an invalid argument. Anyway, nonsequiturs were not arguments. However, the expression ‘nonsequitur’ also means “invalid premise- conclusion argument”. It also means something like “fallacious reasoning” as in ‘Abe’s proof of the big-bang hypothesis is a nonsequitur’. The English noun is also used for an irrelevancy, an abrupt change of topic, or for any of many other expressions that are gratuitous, inappropriate, incongruous, or in some other way out of place. The English noun is derived from the Latin expression ‘non sequitur’, which is not a common noun but a sentence variously meaning “it isn’t in sequence ”, “it doesn’t follow”, “it is out of order”, etc. A statement that begins ‘in the second place’ is a nonsequitur if the previous sentence lacked implication of something like ‘in the first place’ or if it contained something similar to ‘in the third place’. In fact, a true implication claim can be a nonsequitur if there is no reason for making it. If Ben says ‘Hence, Abe swims and Abe swims’ in reply to Carl’s statement ‘Abe swims‘, then Ben may well be committing a nonsequitur. Many conversation stoppers are nonsequiturs. This lecture surveys and analyses a wide selection of uses of ‘nonsequitur’ in logic and logic-related literature. -/- END OF DRAFT 4 -/- NOTE 091215: When Rick Perry was discussing possible withdrawal from the race—unfortunately not the human race but the Republican race—a reporter reminded him that just last week Donald Trump predicted the Perry withdrawal. Perry interrupted with “A broken clock is right once a day”. -/- If you did not know Perry you might think his reply was a brilliant diversionary nonsequitur. If you know Perry you might think his reply was a stupid nonsequitur. -/- For the record, he did say ‘once a day’, not ‘twice a day’. BTW, my interjection ‘unfortunately not the human race but the Republican race’ was also a nonsequitur

The word pro-tasis is etymologically a near equivalent of pre-mise, pro-position, and ante-cedent—all having positional, relational connotations now totally absent in contemporary use of proposition. Taking protasis for premise, Aristotle’s statement (24a16) A protasis is a sentence affirming or denying something of something…. is not a definition of premise—intensionally: the relational feature is absent. Likewise, it is not a general definition of proposition—extensionally: it is too narrow. This paper explores recent literature on these issues.

This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point of the quotation is to make clear how murky, incompetent, and badly written the paper is. There is no way to determine from the article whether the editor or referees suggests improvements.

Iffication, Preiffication, Qualiffication, Reiffication, and Deiffication. Roughly, iffication is the speech-act in which—by appending a suitable if-clause—the speaker qualifies a previous statement. The clause following if is called the qualiffication. In many cases, the intention is to retract part of the previous statement—called the preiffication. I can retract part of “I will buy three” by appending “if I have money”. This initial study focuses on logical relations among propositional contents of speech-acts—not their full conversational implicatures, which will be treated elsewhere. The modified statement—called the iffication—is never stronger than the preiffication. A degenerate iffication is one logically equivalent to its preiffication. There are limiting cases of degenerate iffications. In one, the qualiffication is tautological, as “I will buy three if three is three”. In another, the negation of the qualiffication implies the preiffication, as “I will buy three if I will not buy three”. Reiffication is iffication of an iffication. “I will buy three if I have money” is reifficated by appending “if there are three left”. Deiffication is the speech-act in which—by appending a suitable and-clause—the effect of an iffication is cancelled so that the result implies the preiffication. “I will buy three if I have money” is deifficated by appending “and I have money”. All further examples come from standard (one-sorted, tenseless, non-modal) first-order arithmetic. All theorems are about first-order arithmetic propositions. An easy theorem, hinted above, is that an iffication is degenerate if and only if the negation of the qualiffication implies the preiffication. The iffication of a conjunction using one of the conjuncts as qualiffication need not imply the other conjunct: “Two is an even square if two is square” does not imply “Two is even”. END OF PRINTED ABSTRACT. Some find that the last statement provides a surprise in logic. https://www.academia.edu/s/a5a4386b75?source=link Acknowledgements: Robert Barnes, William Frank, Amanda Hicks, David Hitchcock, Leonard Jacuzzo, Edward Keenan, Mary Mulhern, Frango Nabrasa, and Roberto Torretti.

JOHN CORCORAN AND WAGNER SANZ, Disbelief Logic Complements Belief Logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 USA E-mail: [email protected] Filosofia, Universidade Federal de Goiás, Goiás, GO 74001-970 Brazil E-mail: [email protected] Consider two doxastic states belief and disbelief. Belief is taking a proposition to be true and disbelief taking it to be false. Judging also dichotomizes: accepting a proposition results in belief and rejecting in disbelief. Stating follows suit: asserting a proposition conveys belief and denying conveys disbelief. Traditional logic implicitly focused on logical relations and processes needed in expanding and organizing systems of beliefs. Deducing a conclusion from beliefs results in belief of the conclusion. Deduction presupposes consequence: one proposition is a consequence of a set of a propositions if the latter logically implies the former. The role of consequence depends on its being truth-preserving: every consequence of a set of truths is true. This paper, which builds on previous work by the second author, explores roles of logic in expanding and organizing systems of disbeliefs. Aducing a conclusion from disbeliefs results in disbelief of the conclusion. Aduction presupposes contrequence: one proposition is a contrequence of a set of propositions if the set of negations or contradictory opposites of the latter logically implies that of the former. The role of contrequence depends on its being falsity-preserving: every contrequence of a set of falsehoods is false. A system of aductions that includes, for every contrequence of a given set, an aduction of the contrequence from the set is said to be complete. Historical and philosophical discussion is illustrated and enriched by presenting complete systems of aductions constructed by the second author. One such, a natural aduction system for Aristotelian categorical propositions, is based on a natural deduction system attributed to Aristotle by the first author and others. ADDED NOTE: Wagner Sanz reconstructed Aristotle’s logic the way it would have been had Aristole focused on constructing “anti-sciences” instead of sciences: more generally, on systems of disbeliefs

In its strongest, unqualified form the principle of wholistic reference is that each and every proposition refers to the whole universe of discourse as such, regardless how limited the referents of its non-logical or content terms. Even though Boole changed from a monistic fixed-universe framework in his earlier works of 1847 and 1848 to a pluralistic multiple-universe framework in his mature treatise of 1854, he never wavered in his frank avowal of the principle of wholistic reference, possibly in a slightly weaker form. Indeed, he took it as an essential accompaniment to his theory of concept formation and proposition formation. Similar views are found in later logicians, and some of the most recent formulations of standard, one-sorted first-order logic seem to be in accord with a form of it, if they do not actually imply the principle itself.Em sua forma mais forte e geral, o princípio da referência universalista afirma que toda proposição refere-se ao universo completo do discurso propriamente dito, independentemente de quão limitados sejam os referentes de seus termos não-lógicos ou com conteúdo. Embora Boole mudasse de um modelo monístico de universo-fixo nos seus trabalhos iniciais de 1847 e 1848 para um modelo pluralista de universo-múltiplo no seu tratado maduro de 1854, ele nunca hesitou em sua aceitação franca do princípio da referência universalista, possivelmente em uma forma ligeiramente mais fraca. De fato, ele considerou este princípio como um acompanhamento essencial para a sua teoria de formação de conceitos e de proposições. Visões semelhantes são encontradas em lógicos posteriores, e algumas das mais recentes formulações da lógica clássica de primeira ordem parecem estar de acordo com uma forma deste, se é que elas não implicam, de fato, o próprio princípio.

It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy.

ABSTRACT This part of the series has a dual purpose. In the first place we will discuss two kinds of theories of proof. The first kind will be called a theory of linear proof. The second has been called a theory of suppositional proof. The term "natural deduction" has often and correctly been used to refer to the second kind of theory, but I shall not do so here because many of the theories so-called are not of the second kind--they must be thought of either as disguised linear theories or theories of a third kind (see postscript below). The second purpose of this part is 25 to develop some of the main ideas needed in constructing a comprehensive theory of proof. The reason for choosing the linear and suppositional theories for this purpose is because the linear theory includes only rules of a very simple nature, and the suppositional theory can be seen as the result of making the linear theory more comprehensive. CORRECTION: At the time these articles were written the word ‘proof’ especially in the phrase ‘proof from hypotheses’ was widely used to refer to what were earlier and are now called deductions. I ask your forgiveness. I have forgiven Church and Henkin who misled me.

This self-contained lecture examines uses and misuses of the adverb conversely with special attention to logic and logic-related fields. Sometimes adding conversely after a conjunction such as and signals redundantly that a converse of what preceded will follow. (1) Tarski read Church and, conversely, Church read Tarski. In such cases, conversely serves as an extrapropositional constituent of the sentence in which it occurs: deleting conversely doesn’t change the proposition expressed. Nevertheless it does introduce new implicatures: a speaker would implicate belief that the second sentence expresses a converse of what the first expresses. Perhaps because such usage is familiar, the word conversely can be used as “sentential pronoun”—or prosentence—representing a sentence expressing a converse of what the preceding sentence expresses. (2) Tarski read Church and conversely. This would be understood as expressing the proposition expressed by (1). Prosentential usage introduces ambiguity when the initial proposition has more than one converse. Confusion can occur if the initial proposition has non-equivalent converses. Every proposition that is the negation of a false proposition is true and conversely. One sense implies that every proposition that is the negation of a true proposition is false, which is true of course. But another sense, probably more likely, implies that every proposition that is true is the negation of a false proposition, which is false: the proposition that one precedes two is not a negation and thus is not the negation of a false proposition. The above also applies to synonyms of conversely such as vice versa. Although prosentence has no synonym, extrapropositional constituents are sometimes called redundant rhetoric, filler, or expletive. Authors discussed include Aristotle, Boole, De Morgan, Peirce, Frege, Russell, Tarski, and Church. END OF PUBLISHED ABSTRACT See also: Corcoran, John. 2015. Converses, inner and outer. 2015. Cambridge Dictionary of Philosophy, third edition, Robert Audi (editor). Cambridge: Cambridge UP. https://www.academia.edu/10396347/Corcoran_s_27_entries_in_the_1999_second_edition_Audi_s_Cambridge_Dictionary_of_Philosophy

Cosmic Justice Hypotheses. This applied-logic lecture builds on [1] arguing that character traits fostered by logic serve clarity and understanding in ethics, confirming hopeful views of Alfred Tarski [2, Preface, and personal communication]. Hypotheses in one strict usage are propositions not known to be true and not known to be false or—more loosely—propositions so considered for discussion purposes [1, p. 38]. Logic studies hypotheses by determining their implications (propositions they imply) and their implicants (propositions that imply them). Logic also studies hypotheses by seeing how variations affect implications and implicants. People versed in logical methods are more inclined to enjoy working with hypotheses and less inclined to dismiss them or to accept them without sufficient evidence. Cosmic Justice Hypotheses (CJHs), such as “in the fullness of time every act will be rewarded or punished in exact proportion to its goodness or badness”, have been entertained by intelligent thinkers. Absolute CJHs, ACHJs, imply that it is pointless to make sacrifices, make pilgrimages, or ask divine forgiveness: once acts are done, doers must ready themselves for the inevitable payback, since the cosmos works inexorably toward justice. Ceteris Paribus CJHs, CPCJHs, on the other hand, such as “in the fullness of time every act will be rewarded or punished in exact proportion to its goodness or badness—other things being equal”, leave room for exceptions. For example, some people subscribing to Ceteris Paribus CJHs think that certain bad acts can be performed with impunity as long as certain procedures are carried out previous to, or simultaneous with, or even after the acts. Belief Ceteris Paribus CJHs has been exploited by unscrupulous “spiritual leaders” who claim to have power to grant exceptions. In opposition to belief in CPCJHs are CJHs that hold belief in CPCJHs to be inherently wrong and subject to punishment. Other variants of CJHs are Cumulative Cosmic Justice Hypotheses, such as “in the fullness of time every person will be rewarded or punished in exact proportion to the net goodness or badness of their acts”. Still other variants include the Hereditary Cumulative Cosmic Justice Hypotheses, such as “in the fullness of time every person will be rewarded or punished in exact proportion to the net goodness or badness of their ancestors’ acts”. [1] JOHN CORCORAN, Inseparability of Logic and Ethics, Free Inquiry, S. 1989, pp. 37–40. [2] ALFRED TARSKI, Introduction to Logic, Dover, 1995.

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching!

As noted in 1962 by Timothy Smiley, if Aristotle’s logic is faithfully translated into modern symbolic logic, the fit is exact. If categorical sentences are translated into many-sorted logic MSL according to Smiley’s method or the two other methods presented here, an argument with arbitrarily many premises is valid according to Aristotle’s system if and only if its translation is valid according to modern standard many-sorted logic. As William Parry observed in 1973, this result can be proved using my 1972 proof of the completeness of Aristotle’s syllogistic.

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early development of algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems).