Building on previous scholarly work on the mathematical roots of assertoric syllogistic we submit that for Aristotle, the semantic value of the copula in universal affirmative propositions is the relation of divisibility on positive integers. The adequacy of this interpretation, labeled here ‘arithmetical dictum’, is assessed both theoretically and textually with respect to the existing interpretations, especially the so-called ‘mereological dictum’.

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)

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort (...) of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system. (shrink)

In his 1910 book On the principle of contradiction in Aristotle, Jan Łukasiewicz claims that syllogistic is independent of the principle of contradiction . He also argues that Aristotle would have defended such a thesis in the Posterior Analytics. In this paper, we first show that Łukasiewicz's arguments for these two claims have to be rejected. Then, we show that the thesis of the independence of assertoric syllogistic vis-à-vis PC is nevertheless true. For that purpose, we first establish that there (...) is no possibility to account for Aristotle's original views on syllogistic without using per impossibile deductions. At last, following an idea due to Smith 1983, we prove that there is a complete reconstruction of assertoric syllogistic using ecthesis instead of reductio ad absurdum. Contrary to what is claimed in Smith 1983, we show that Smith's system SE has to be improved for such a reconstruction. This results in an ecthetic system which is complete with respect to syllogistic arguments, not explosi.. (shrink)

In this paper we first show that Robin Smith’s ecthetic system SE for Aristotle’s assertoric syllogistic is not complete, despite what is claimed by Smith. SE is then not adequate to establish that ecthesis allows one to dispense with indirect or per impossibile deductions in Aristotle’s assertoric logic. As an alternative to SE, we then present a stronger system EC which is adequate for this purpose. EC is a nonexplosive ecthetic system which is shown to be sound and complete with (...) respect to all valid syllogistic arguments with a consistent set of premises. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

John Corcoran’s natural deduction system for Aristotle’s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz’s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role assigned by tradition and Lukasiewicz (...) to Barbara and Celarent is retained.The two ‘ perfect syllogisms’ serve as ‘basic elements’ in the construction of an inductively defined set of valid syllogisms.The proposal departs from Lukasiewicz, and follows Corcoran, however, in construing the construction as one in natural deduction.The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions.To compare the theory to Corcoran’s, his original is reformulated in current terms and generalized.It is shown to be equivalent to the proposed sequent system, and several variations are discussed.For all systems mentioned, a method of Henkin‐style completeness proofs is given that is more direct and intuitive than Corcoran’s original. (shrink)

One way to determine the quality and pace of change in a science as it undergoes a major transition is to follow some feature of it which remains relatively stable throughout the process. Following the chosen item as it goes through reinterpretation permits conclusions to be drawn about the nature and scope of the broader change in question. In what follows, this device is applied to the change which took place in logic in the mid-nineteenth century. The feature chosen as (...) the focal point is the categorical syllogism. (shrink)

In this work, attention is drawn to the abundant use of metaphor and analogy in works of logic. I argue that pervasiveness of figurative language is to be counted among the features that characterize logic and distinguish it from other sciences. This characteristic feature reflects the creativity that is inherent in logic and indeed has been demonstrated to be a necessary part of logic. The goal of this paper, in short, is to provide specific examples of figurative language used in (...) logic that yield insights into the nature of the subject. I encourage the reader to take metaphors seriously, and to accept that they are not mere embellishments but key elements in our understanding of logic. (shrink)

The aim of this paper is to provide a complete Natural Kind Semantics for an Essentialist Theory of Kinds. The theory is formulated in two-sorted first order monadic modal logic with identity. The natural kind semantics is based on Rudolf Willes Theory of Concept Lattices. The semantics is then used to explain several consequences of the theory, including results about the specificity (species–genus) relations between kinds, the definitions of kinds in terms of genera and specific differences and the existence of (...) negative kinds. First, I show under which conditions the Hierarchy principle, which has been subjected to counterexamples in the literature, holds. I also show that a different principle about the species–genus relations between kinds, namely Kant’s Law, follows from the essentialist theory. Second, I introduce two new operations for kinds and show that they can be used to provide traditional definitions of kinds in terms of genera and specific differences. Finally, I show that these operations of specific difference induce, for each kind, a uniquely specified contrary kind and a uniquely specified subcontrary kind, which can be used as semantic values for non-classical predicate negations of kind terms. (shrink)

Since Freges terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this questionBegriffes syntaxhighercorresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcorans theory via predicate logic.

Three distinctly different interpretations of Aristotle’s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle’s remarkable degree of metalogical (...) sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ‘sullogimos’ to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle’s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction. (shrink)

A semantics is presented for Storrs McCall's separate axiomatizations of Aristotle's accepted and rejected polysyllogisms. The polysyllogisms under discussion are made up of either assertoric or apodeictic propositions. The semantics is given by associating a property with a pair of sets: one set consists of things having the property essentially and the other of things having it accidentally. A completeness proof and a semantic decision procedure are given.

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...) of the natural deduction counterpart ? that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. (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.

Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite (...) sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument. (shrink)

In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle’s Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle’s work within the context of dialectic and uses Lorenzen’s dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as ‘immanent reasoning’. We then provide rules of syllogistic for the latter, and provide proofs of (...) e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle’s text these proofs remain. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S 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 is one whose existentialization, ∃ x Q, is logically true; otherwise, Q 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.. (shrink)

First-order logic has limited existential import: the universalized conditional ∀x [S → P] implies its corresponding existentialized conjunction ∃x [S & P] in some but not all cases. We prove the Existential-Import Equivalence:∀x [S → P] implies ∃x [S & P] iff ∃x S is logically true.The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import: implies its corresponding existentialized conjunction.A predicate is a formula having only x free. An existential-import predicate Q is one (...) whose existentialization, ∃x Q, is logically true; otherwise, Q is existential-import-free or simply import-free. Existential-import predicates are also said to be import-carrying.How widespread is existential import? How widespread are import-carrying predicates in themselves or in comparison to import-free predicates? To answer, let L be any first-order language with any interpretation INT in any [sc. nonempty] universe U. A subset S of U is definable in L under INT iff for some predicate Q in L, S is the truth-set of Q under INT. S is import-carrying definable iff S is the truth-set of an import-carrying predicate. S is import-free definable iff S is the truth-set of an import-free predicate.Existential-Importance Theorem: Let L, INT, and U be arbitrary. Every nonempty definable subset of U is both import-carrying definable and import-free definable.Import-carrying predicates are quite abundant, and no less so than import-free predicates. Existential-import implications hold as widely as they fail.A particular conclusion cannot be validly drawn from a universal premise, or from any number of universal premises.—Lewis-Langford, 1932, p. 62. (shrink)

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 Aristotle’s 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 knowledge 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 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. (shrink)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...) by deductively evident steps that its conclusion is a consequence of its premises. In particular, 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 conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

Prior Analytics by the Greek philosopher Aristotle and Laws of Thought by the English mathematician George Boole are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important aspects (...) of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.Paul Rosenbloom 1950. (shrink)

Studies of Aristotle’s syllogistic system, since Corcoran’s deductionist interpretation supplanted Łukasiewicz’ axiomaticist interpretation, misrepresent Aristotle’s logic in two important respects. Following Corcoran, they take indirect deduction to occur only once in a deduction discourse; they then obviate the system having a reductio rule. Second, they represent reduction as a deductive process for deriving ‘imperfect’ syllogisms from ‘perfect’ syllogisms to impose an axiomatic interpretation on the logic. Denying that Aristotle's logic admits of a reductio rule results from this misrepresentation of reduction. (...) This discussion shows that the defects imputed to Aristotle's logic, and systems devised to resolve them, result from misunderstanding reduction, which itself results from misapprehending Prior Analytics expressly to identify in a metadiscourse deduction rules and not deductions per se. (shrink)

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran's deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the (...) goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel's famous 'squeezing argument'; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure. (shrink)

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. It consists of a deductive system (I.4-22), meta-logical results (I.23-26), and a method for finding and giving deductions (I.27-29) that can apply in “any art or science whatsoever” (I.30). 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 be- tween 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)

The goal of this paper is to present a new reconstruction of Aristotle's assertoric logic as he develops it in Prior Analytics, A1-7. This reconstruction will be much closer to Aristotle's original...

Aristotelian causal theories incorporate some philosophically important features of the concept of cause, including necessity and essential character. The proposed formalization is restricted to one-place predicates and a finite domain of attributes (without individuals). Semantics is based on a labeled tree structure, with truth defined by means of tree paths. A relatively simple causal prefixing mechanism is defined, by means of which causes of propositions and reasoning with causes are made explicit. The distinction of causal and factual explanation are elaborated, (...) and examples of cyclic and convergent causation are given. Soundness and completeness proofs are sketched. (shrink)

Discussion of the Aristotelian syllogistic over the last sixty years has arguably centered on the question whether syllogisms are inferences or implications. But the significance of this debate at times has been taken to concern whether the syllogistic is a logic or a theory, and how it ought to be represented by modern systems. Largely missing from this discussion has been a study of the few passages in the Prior Analytics where Aristotle provides explicit guidance on how to individuate syllogisms. (...) I argue that syllogisms are individuated by unordered premise pairs and so may be fruitfully thought of as multi-succedent sequents. I conclude by drawing a few consequences for the representation of the syllogistic. (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)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument -/- The book is devoted to the ancient logical theories, reconstruction of their semantic proprieties and possibilities of their interpretation by modern logical tools. The Ancient arguments are frequently misunderstood in modern interpretations since authors usually have tendency to ignore their historical proprieties and theoretical background what usually leads to a quite inappropriate picture of the argument’s original form and mission. Author’s primary intention was to draw attention to the (...) complexity of some historical arguments and to the theoretical context in which arguments were created, circulated, developed, and finally tuned. Four well-known ancient arguments – with a common central subject related to the future contingencies problem – are reconstructed from available historical sources: “The Sea Battle”, which is drawn from Aristotle’s treatise De Interpretatione; two arguments, usually ascribed to the Stoics, “The Lazy Argument” and “The Reaper”; “The Master Argument” of the Megarian philosopher Diodorus. Arguments are linguistically and semantically detaily analyzed, formally presented by reflecting some relevant corresponding hypotheses based on physical or logical theories of their ancient authors, and finally covered by appropriate logical tools familiar to a modern reader. Two appendices are added at the closing part of the book. One covering some assumptions relevant for understanding of rival streams in ancient theories of meaning related to the nature of names and naming; the other is devoted to the ancient understanding of logical proposition and attempts to find an adequate Latin translation of the Greek delicate philosophical term “ἀξίωμα”. (shrink)

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...) and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege. (shrink)

It has been said that there is no scholarly consensus as to why Aristotle’s logics of proof and refutation would have borne the title _Analytics._ But if we consulted Tarski’s (Introduction to logic and the methodology of deductive sciences, Oxford University Press, New York, 1941) graduate-level primer, we would have the perfect title for them: _Introduction to logic and to the methodology of deductive sciences._ There are two strings to Aristotle’s bow. The methodological string is the founding work on the (...) epistemology of science, and the logical string sets down conditions on the proofs that bring this knowledge about. The logic of proof presents a difficulty whose solution exceeds its theoretical reach. The logic of refutation takes the problem on board, and advances a solution whose execution is framed by fallacy-avoidance at the beginning and fallacy-adoption at the end. But with a difference: the avoidance-fallacies are of Aristotle’s own conception, whereas the adoption-fallacies, so judged in the modern tradition, aren’t fallacies at all for Aristotle. The avoidance-fallacies are begging the question and _ignoratio elenchi_, and the adoption-fallacies, fallacies in name only, are the _ad hominem_ and _ad ignorantiam_, an inductive turning in the first instance, and an abductive finish in the second. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one (...) syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (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)

How does Aristotle think about sentences like ‘Every x is y’ in the Prior Analytics? A recently popular answer conceives of these sentences as expressing a mereological relationship between x and y: the sentence is true just in case x is, in some sense, a part of y. I argue that the motivations for this interpretation have so far not been compelling. I provide a new justification for the mereological interpretation. First, I prove a very general algebraic soundness and completeness (...) result that unifies the most important soundness and completeness results to date. Then I argue that this result vindicates the mereological interpretation. In contrast to previous interpretations, this argument shows how Aristotle’s conception of predication in mereological terms can do important logical work. (shrink)

This article is primarily concerned with Aristotle’s theory of the syllogistic, and the investigation of the hypothesis that logical symbolism and methodology were in these early stages of a geometrical nature; with the gradual algebraization that occurred historically being one of the main reasons that some of the earlier passages on logic may often appear enigmatic. The article begins with a brief introduction that underlines the importance of geometric thought in ancient Greek science, and continues with a short exposition of (...) Aristotle’s views and methods in regard to logic. We then offer an interpretation of syllogisms, as well as of the main proof methods that are utilized by Aristotle, in terms of diagrams that can be seen as analogous to those of Euclidean geometry; where the various proofs proceed by appropriately manipulating an initially drawn diagram, in a way parallel to constructions that occur in Euclid’s Elements. In this way, logic is presented as following, to a large degree, methodological tenets established by the practice of geometry. Finally, we present a diagrammatic decision procedure that allows one to directly read off the validity of syllogisms from a corresponding diagram, in a way such that no further manipulation of it is necessitated. (shrink)

The purpose of this paper is to contribute to the natural logic program which invents logics in natural language. This study presents two logics: a logical system called d R containing transitive verbs and a more expressive logical system R containing both transitive verbs and intersective adjectives. The paper offers three different set-theoretic semantics which are equivalent for the logics.

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form “Most A are B”, where A and B are plural nouns and the interpretations of A and B are infinite subsets of \. There are two widely used semantics for Most A are B: \ > C \) and \ > \dfrac{C}{2} \), where C denotes (...) the cardinality of a given finite set X. Although is more descriptive than, it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of \, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit. (shrink)

The aim of this paper is to provide a contribution to the natural logic program which explores logics in natural language. The paper offers two logics called \ \) and \ \) for dealing with inference involving simple sentences with transitive verbs and ditransitive verbs and quantified noun phrases in subject and object position. With this purpose, the relational logics are introduced and a model-theoretic proof of decidability for they are presented. In the present paper we develop algebraic semantics of (...) the logics using congruence theory. (shrink)

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is (...) a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (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)

Professor Girle suggests that the ancient Athenian interest in Aristotle’s syllogistic flowed from a preoccupation with debate in the form of a dialogue game. But other cultures, especially in India, also had a preoccupation with debate that could be characterized in the same way. This kind of explanation seems to us to ignore the elephant in the room: the fact that, in ancient Athens, dialogue and debate were not merely a game. They were the life and death of the state. (...) Many Athenians felt that foolish policies had cost the state dearly, especially during the Peloponnesian War, and many of them blamed those policies on shoddy political reasoning in the Assembly. (Plato hammers at this theme relentlessly.) And tellingly, it is only in the wake of such tragedies that Aristotle’s syllogistic finally emerged. We think that any credible explanation of the ancient Greek interest in formal logic needs to include this grim political reality. (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.