Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaśkowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks—the standard conduits of (...) the method to a generation of philosophers—with an eye to determining what the ‘essential characteristics’ of natural deduction are. (shrink)

I consider the proper interpretation of the process of ecthesis which Aristotle uses several times in the Prior analytics for completing a syllogistic mood, i.e., showing how to produce a deduction of a conclusion of a certain form from premisses of certain forms. I consider two interpretations of the process which have been advocated by recent scholars and show that one seems better suited to most passages while the other best fits a single remaining passage. I also argue that ecthesis (...) for Aristotle really means ?setting out? the case to be proved using letters. Aristotle?s remarks about the use of letters in mathematical proofs suggest that he had some understanding of rules equivalent to universal generalization and existential instantiation; the ?proofs through ecthesis? are so called because they rest on the latter rule, with which use of letters is involved in a special way. (shrink)

Since McCall (1966), the heterodox principle of propositional logic that it is impossible for a proposition to be entailed by its own negation—in symbols, ¬(¬φ→φ)—has gone by the name of Aristotle’s thesis, since Aristotle apparently endorses it in Prior Analytics 2.4, 57b3–14. Scholars have contested whether Aristotle did endorse his eponymous thesis, whether he could do so consistently, and for what purpose he endorsed it if he did. In this article, I reconstruct Aristotle’s argument from this passage and show that (...) he accepts this thesis. Further, I show that the argument he gives is, making plausible assumptions, a correct proof in a consistent fragmentary nonclassical metalogic for a metatheorem he previously states concerning his assertoric syllogistic. In this way, Aristotle’s argument emerges as a fascinating case study in the use of a nonclassical metalogic to prove a result about a nonclassical object system. (shrink)

In this paper we provide an interpretation of Aristotle's rule for the universal quantifier in Topics Θ 157a34–37 and 160b1–6 in terms of Paul Lorenzen's dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in (...) the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen's dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato's dialogues, thus showing that Aris... (shrink)

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

In 1928, Friedrich Solmsen argued that Aristotle'sPosterior Analyticswas largely composed before thePrior Analytics. Ross rejected Solmsen's position in 1939, and a rather lengthy series of rebuttals and counter-attacks between the two scholars followed. Quite recently, Jonathan Barnes has revived this issue with arguments in favour of something very close to Solmsen's thesis: that Aristotle first developed a theory of demonstration (‘apodeictic’) before he had worked out the syllogistic, and that thePosterior Analyticswas originally conceived against this background. Subsequently, when Aristotle formulated (...) a syllogistic, he is supposed by Barnes to have revised or added to the contents of thePosterior Analyticsso as to make syllogistic the logic of Aristotelian science. Thus, Barnes says: ‘the syllogism is in fact an incidental adjunct to the theory of demonstration: the theory can be formulated without reference, explicit or implicit, to Syllogistic, and itcould havebeen discovered by someone who knew nothing whatever about the Syllogism’ (pp. 33–4). (shrink)

By considering the new notion of theinversesof syllogisms such asBarbaraandCelarent, we show how the rule ofIndirect Proof, in the form (no multiple or vacuous discharges) used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. After this, Aristotle compares this method with Plato’s method of division, a procedure designed to find essences of natural kinds through systematic classification. This critical comparison in APr I.31 raises an interpretive puzzle: how can Aristotle reasonably juxtapose two methods that differ so much in their aims and approach? What can be gained by doing so? Previous interpreters have failed (...) to show how this comparison is legitimate or what important point Aristotle is making. The goal of this paper is to resolve the puzzle. In resolving this puzzle we not only learn more about the relationship between division and the syllogistic in Aristotle. We will also learn something about the motivation for the syllogistic itself, by seeing the role that it plays in his philosophy of science. (shrink)

After presenting the rules of Eleatic antilogic, i.e., dialectic, I argue that Zeno was a practitioner, and, on the basis of key passages from Plato’s Parmenides (127e-128e and 135d-136c), that his paradoxes of divisibility and movement were notreductio ad absurdum, but simple derivation of impossibilities (adunaton) meant to ridicule Parmenides’ adversaries. Thus, Zeno did not try to prove that there is no motion, but simply derived this consequence from premises held by his opponents. I argue further that these paradoxes were (...) devised, in accordance with Eleatic antilogic, following a scheme that included hypotheses and their contradictories, within which the subject is to be treated both “in relation to itself,” and “in relation to other things”. (shrink)

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa-omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

I reconstruct Aristotle’s analytical procedure in Prior Analytics I.45 and its metalogical implications. Aristotle’s analysis unfolds three groups of syllogisms: symmetrically analysable, asymmetrically analysable, and non-analysable syllogisms. From the first and the third group could be extracted 27 combinations of the two mutually non-derivable deductive rules. Aristotle’s reduced deductive system in APr. I.7 with the two moods in the first figure (traditionally called Barbara and Celarent) follows this pattern. I demonstrate that the deductive system with Barbara and Celarent is just (...) one of 27 possible complete deductive systems with the two mutually non-derivable syllogistic rules. On top of that, given the fact that there are two combinations of mutually non-derivable conversion rules, I conclude that Prior Analytics hides in itself 54 complete deductive systems with a minimal number of mutually non-derivable rules. Finally, it is commented what might be the role of analysis; to what extent Aristotle could be aware of possibilities presented here; why he still prefers only one deductive system to many others with the same deductive power. (shrink)

This paper (1) criticizes Patzig's explanation of Aristotle's reason for calling his first figure syllogisms perfect syllogisms, i.e. the transitivity relation: it can only be used for Barbara, not for the other three moods. The paper offers (2) an alternative interpretation: It is only in the case of the (perfect) first figure moods that we can move from the subject term of the minor premiss, taken to be a predicate of an individual, to the predicate term of the major premiss. (...) This contention is supported (i) by Aristotle's wording of the dictum de omni et nullo and (ii) by Aristotle's use of a formula which puts the minor term in the first position when he first states Barbara and Celarent. (shrink)

We begin with an introductory overview of contributions made by more than twenty scholars associated with the Philosophy Department at the University of Buffalo during the last half-century to our understanding and evaluation of Aristotle's logic. More well-known developments are merely mentioned in..

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order (...) to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

JUNE 2015 UPDATE: A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015 By John Corcoran -/- This presentation includes a complete bibliography of John Corcoran’s publications relevant to his research on Aristotle’s logic. Sections I, II, III, and IV list 21 articles, 44 abstracts, 3 books, and 11 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article from Corcoran’s Philadelphia period that antedates his Aristotle studies and the Journal of Symbolic Logic article from his (...) Buffalo period first reporting his original results; it ends with works published in 2015. A few of the items are annotated as listed or with endnotes connecting them with other work and pointing out passages that in-retrospect are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 23 in the 1980s, 42 in the 1990s, 56 in the 2000s, and 69 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally, of the Lukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. -/- REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. Also, let me know of passages I should comment on. (shrink)

A formal analysis is offered of Pseudo-Scotus's theory of the conversion of (i) propositions containing singular terms (including propositions with a singular term as predicate): and (ii) propositions with a quantified predicate. An attempt is made to steer a middle course between using the Aristotelian logic as a framework for the analysis, and using a Fregean framework.

This presentation includes a complete bibliography of John Corcoran’s publications relevant on Aristotle’s logic. The Sections I, II, III, and IV list respectively 23 articles, 44 abstracts, 3 books, and 11 reviews. Section I starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article—from Corcoran’s Philadelphia period that antedates his discovery of Aristotle’s natural deduction system—and the Journal of Symbolic Logic article—from his Buffalo period first reporting his original results. It ends with works published in 2015. (...) Some items are annotated as listed or with endnotes connecting them with other work and pointing out passages that, in retrospect, are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 10 items published in the 1970s, 24 in the 1980s, 42 in the 1990s, 60 in the 2000s, and 70 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally in some cases, even of the Łukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably George Boger, Charles Kahn, John Mulhern, Mary Mulhern, Anthony Preus, Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. (shrink)

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order (...) to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. (shrink)

This essay addresses the collapse/incoherence problem for normative frameworks that contain both fundamental values and rules for promoting those values. The problem is that in some cases, we would bring about more of the fundamental value by violating the framework’s rules than by following them. In such cases, if the framework requires us to follow the rules anyway, then it appears to be incoherent; but if it allows us to make exceptions to the rules, then the framework “collapses” into one (...) that doesn’t make use of rules in the first place. The chapter begins with an examination of happiness and truth as fundamental values in Mill’s work, which lead into parallel versions of the collapse/incoherence problem in ethics and epistemology. It then sets out the collapse problem for rule-consequentialist approaches in ethics, truth-directed accounts of justification in epistemology, and epistemological approaches to argument cogency. The chapter closes with discussion of two potential solutions to the problem. (shrink)

RésuméDans cette étude j'examine la théorie des propositions conditionnelles d'Alfarabi et son système des syllogismes conditionnels. J'établis qu'Alfarabi a formulé sa théorie des propositions conditionnelles et syllogismes conditionnels comme une extension d'une théorie de langue dans laquelle le contexte dialectique demeure au centre de l'analyse des propositions et des syllogismes. Je démontre que selon l'avis d'Alfarabi les propositions conditionnelles ont conditions de vérité. Je fournis des conditions de vérité conjecturales et des conditions de validité conjecturales. Je suggère que ces conditions (...) de vérité et ces conditions de validité sont sensibles aux conditions pragmatiques dans lesquelles les prémisses conditionnelles et les arguments conditionnels sont utilisés. Je conclus que le pragmatisme logique d'Alfarabi est une conséquence de son adoption d'une théorie logique sensible au contexte dialectique d'antiquité tardive plutôt qu'une conséquence de développement de la théorie syllogistique formelle d'Aristote dans lesPremiers Analytiques. (shrink)

Hypersequents are a natural generalization of ordinary sequents which turn out to be a very suitable tool for presenting cut-free Gentzent-type formulations for diverse logics. In this paper, an alternative way of formulating hypersequent calculi (by introducing meta-variables for formulas, sequents and hypersequents in the object language) is presented. A suitable category of hypersequent calculi with their morphisms is defined and both types of fibring (constrained and unconstrained) are introduced. The introduced morphisms induce a novel notion of translation between logics (...) which preserves metaproperties in a strong sense. Finally, some preservation features are explored. (shrink)

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

Avec l’article « Aristotle’s natural deduction system », publié en 1974, J. Corcoran a contribué à diffuser une nouvelle perspective sur les écrits logiques d’Aristote et sur la théorie du syllogisme en particulier. Dans cet article, Corcoran affirme que, dans les premiers chapitres des Premiers Analytiques, Aristote ne propose pas un système axiomatique, qui supposerait une logique sous-jacente, ainsi que le pensait Łukasiewicz, mais plutôt un système de déduction naturelle, avec des dimensions métalogiques. Notre propos est ici basé sur une (...) courte monographie de Kurt Ebbinghaus, intitulée Ein formales Model der Syllogistik des Aristoteles (1964), où est fixé le canon de cette nouvelle perspective mentionnée plus haut et qui a été développé dans le cadre conceptuel de la « logique opérative » de Paul Lorenzen. Ebbinghaus développe une reconstruction formelle montrant que l’approche d’Aristote relève de la « théorie de la preuve », non seulement pour ce qui concerne le système d’inférence sous-jacent, mais aussi pour ce qui concerne les éléments métalogiques. C’est notamment à travers ce dernier aspect que se manifeste une différence majeure par rapport à la reconstruction de Corcoran. Alors que celui-ci pose que le système d’inférence d’Aristote est enraciné dans une sémantique de théorie des modèles (élaborée par Corcoran lui-même), Ebbinghaus comprend que la théorie du syllogisme a été développée à partir d’une approche de la signification par des règles (« rule based »), semblables aux règles d’un jeu. En fait, la reconstruction d’Ebbinghaus offre une lecture pragmatiste de la syllogistique d’Aristote, qui, tel est notre propos, paraît non seulement beaucoup plus proche du point de vue d’Aristote que ne l’est la sémantique de théorie des modèles proposée par Corcoran, mais qui, de plus, permet de saisir l’unité systématique de la théorie du syllogisme. (shrink)