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)

Much of the last fifty years of scholarship on Aristotle’s syllogistic suggests a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory or formal ontology, a system concerned with general features of the world. In this paper, I will argue that this a misleading interpretative framework. The syllogistic is something sui generis: by our lights, it is neither clearly a logic, nor clearly a theory, (...) but rather exhibits certain characteristic marks of logics and certain characteristic marks of theories. In what follows, I will present a debate between a theoretical and a logical interpretation of the syllogistic. The debate centers on the interpretation of syllogisms as either implications or inferences. But the significance of this question has been taken to concern the nature and subject-matter of the syllogistic, and how it ought to be represented by modern techniques. For one might think that, if syllogisms are implications, propositions with conditional form, then the syllogistic, in so far as it is a systematic taxonomy of syllogisms, is a theory or a body of knowledge concerned with general features of the world. Furthermore, if the syllogistic is a theory, then it ought to be represented by an axiomatic system, a system deriving propositional theorems from axioms. On the other hand, if syllogisms are inferences, then the syllogistic is a logic, a system of inferential reasoning. And furthermore, it ought to be represented as a natural deduction system, a system deriving valid arguments by means of intuitively valid inferences. I will argue that one can disentangle these questions—are syllogisms inferences or implications, is the syllogistic a logic or a theory, is the syllogistic a body of worldly knowledge or a system of inferential reasoning, and ought we to represent the syllogistic as a natural deduction system or an axiomatic system—and that we must if we are to have a historically accurate understanding of Aristotle. (shrink)

Aristotle’s treatment of mixed, first-figure, problematic-assertoric syllogisms has generated a good deal of controversy among modern commentators.I argue that W.D.Ross’s criticism of A.Becker’s cr...

A connection between Aristotle's syllogistic and the calculus of relations is investigated. Aristotle's and Gergonne's syllogistics are considered as some algebraic structures. It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra. This isomorphism permits to evaluate Gergonne's syllogisms and also Aristotle's syllogisms, laws of conversion and relations in the "square of oppositions" by means of regular computations with Boolean matrices.

This book presents a (new) attempt to apply the notion of an underlying logic to Aristotle’s Organon and certain passages of the Metaphysics. The author situates his approach as part of a ‘deductio...

This paper is chiefly aimed at individuating some deep, but as yet almost unnoticed, similarities between Aristotle's syllogistic and the Stoic doctrine of conditionals, notably between Aristotle's metasyllogistic equimodality condition and truth-conditions for third type conditionals. In fact, as is shown in §1, Aristotle's condition amounts to introducing in his metasyllogistic a non-truthfunctional implicational arrow '', the truth-conditions of which turn out to be logically equivalent to truth-conditions of third type conditionals, according to which only the impossible follows from (...) the impossible. Moreover, Aristotle is given precisely this non-Scotian conditional logic in two so far overlooked passages of Themistius' Paraphrasis of De Caelo. Some further consequences of Aristotle's equimodality condition on his logic, and notably on his syllogistic, are pointed out and discussed at length. A extension of Aristotle's condition is also discussed, along with a full characterization of truth-conditions of fourth type conditionals. (shrink)

In this paper, we discuss the syllogisms from both fronts : Aristotle and Boethius. We mainly focus on the differences with respect to categorical and hypothetical syllogisms in Aristotle and Boethius. Regarding Aristotle’s works on logic, it is not unusual to claim that Aristotle extensively worked on categorical syllogisms. In Prior Analytics, Aristotle gave proofs for many valid moods. However we cannot find a similar treatment for hypothetical syllogism in his works. Thus, it might be a reason for his (...) students to improve Aristotle’s theory of syllogisms by filling the gaps in hypothetical syllogisms. As Boethius already emphasized in the Book I of De Hypotheticis Syllogismis, the Peripatetic School only initiated the start. The remaining and detailed treatise was due to Boethius himself. The impression we got from Boethius’s and Aristotle’s works on syllogisms is that the categorical syllogism is almost entirely developed by Aristotle himself, whereas the extensive and the detailed study of the hypothetical syllogism was carried out by Boethius, although he was not the first figure in the history who gave concise expositions on hypothetical syllogism. Boethius’s translations of syllogisms in medieval era were the primer resource and textbook for Aristotelian logic during the early Medieval ages. Therefore, the scholars of those times had no other chance but to work from Boethius’s commentaries. So, the evident properties of Aristotelian figures were not then visible. Boethius wrote several textbooks in logic. A detailed investigation might yield significant information on Boethius’s logic compared with the influence of Aristotle on Boethius. (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)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. (...) Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making necessary connections (...) between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

Richard Janko's acclaimed translation of Aristotle's _Poetics_ is accompanied by the most comprehensive commentary available in English that does not presume knowledge of the original Greek. Two other unique features are Janko's translations with notes of both the _Tractatus Coislinianus_, which is argued to be a summary of the lost second book of the Poetics, and fragments of Aristotle’s dialogue On Poets, including recently discovered texts about catharsis, which appear in English for the first time.

An interpretation of Aristotle's modal syllogistic is proposed which is intuitively graspable, if only formally correst. The individuals to which a term applies, and possibly-applies, are supposed to be determined in a uniform way by the set of individuals to which the term necessarily-applies.

A reprint of the first edition together with three new chapters and an enlarged index. The new chapters include some previously published material and discuss Aristotle's modal logic of propositions, Lukasiewicz' new system of modal logic, and Aristotle's modal syllogistic. The author relates his interpretation of Aristotle's modal logic to the philosophical issues raised by modern modal logics. -- J. J. C.

The paper shows that for any invalid polysyllogism there is a procedure for constructing a model with a domain with exactly three members and an interpretation that assigns non-empty, non-universal subsets of the domain to terms such that the model invalidates the polysyllogism.

A reprint of the first edition together with three new chapters and an enlarged index. The new chapters include some previously published material and discuss Aristotle's modal logic of propositions, Lukasiewicz' new system of modal logic, and Aristotle's modal syllogistic. The author relates his interpretation of Aristotle's modal logic to the philosophical issues raised by modern modal logics. -- J. J. C.

In the paper we examine the method of axiomatic rejection used to describe the set of nonvalid formulae of Aristotle's syllogistic. First we show that the condition which the system of syllogistic has to fulfil to be ompletely axiomatised, is identical to the condition for any first order theory to be used as a logic program. Than we study the connection between models used or refutation in a first order theory and rejected axioms for that theory. We show (...) that any formula of syllogistic enriched with classical connectives is decidable using models in the domain with three members. (shrink)

The author sets a goal to show the specificity of the formulation of universal prescriptive judgments about a virtuous act in the framework of Aristotelian ethical doctrine. To achieve this goal, Aristotle’s philosophy concept of practical wisdom is analyzed. It shows a necessity to distinguish the use of practical wisdom in a personal experience of the act and for forming the inter-subjective practical knowledge about making of a virtuous act. The specificity of ethics as practical knowledge and its difference (...) from individual moral experience are defined by means of the distinction of the use of practical wisdom. It also shows the limitations of practical syllogism as the main rational mechanism for the formation of inter-subjective practical knowledge. Additionally, the universal prescriptive judgments are divided into informative and functional: the former reveals the content of the action, the latter – its structure, that is, the former defines what a person must do, the latter defines what an action should be to comply with the right purpose, consequently, the right content. At the same time, the right content is recognized only in individual experience and can not be expressed universally, without losing its practical value. The author concludes that the formulation of informative universal prescriptive judgments is impossible within the framework of Aristotelian ethical doctrine. It is impeded by the unsolvable problem of the correlation between the general and the particular, the transition from descriptive judgments to prescriptive judgments. However, it is possible to formulate functional universal prescriptive judgments. They also constitute the methodological basis for criticism of the accepted in society moral norms, which are based on the past positive experience of actions. (shrink)

Aristotle was the founder not only of logic but also of modal logic. In the Prior Analytics he developed a complex system of modal syllogistic which, while influential, has been disputed since antiquity--and is today widely regarded as incoherent. Combining analytic rigor with keen sensitivity to historical context, Marko Malink makes clear that the modal syllogistic forms a consistent, integrated system of logic, one that is closely related to other areas of Aristotle's philosophy. Aristotle's modal syllogistic differs (...) significantly from modern modal logic. Malink considers the key to understanding the Aristotelian version to be the notion of predication discussed in the Topics--specifically, its theory of predicables and the ten categories. The predicables introduce a distinction between essential and nonessential predication. In contrast, the categories distinguish between substantial and nonsubstantial predication. Malink builds on these insights in developing a semantics for Aristotle's modal propositions, one that verifies the ancient philosopher's claims of the validity and invalidity of modal inferences. While it acknowledges some limitations of this reconstruction, Aristotle's Modal Syllogistic brims with bold ideas, richly supported by close readings of the Greek texts. (shrink)

In this paper an algebraic system of the new type is proposed (namely, a vectorial lattice). This algebraic system is a lattice for the language of Aristotle’s syllogistic and as well as a lattice for the language of Vasiľév’s syllogistic. A lattice for the language of Aristotle’s syllogistic is called a vectorial lattice on cap-semilattice and a lattice for the language of Vasiľév’s syllogistic is called a vectorial lattice on closure cap-semilattice. These constructions are (...) introduced for the first time. (shrink)