Modal Syllogisms Research Articles

Deductive Reasoning Based on the Aristotelian Modal Syllogism □AE◇O-2

Deductive Reasoning Based on the Aristotelian Modal Syllogism □AE◇O-2

Knowledge Mining Based on the Aristotelian Modal Syllogism ▢EAO-3

Knowledge Mining Based on the Aristotelian Modal Syllogism ▢EAO-3

Knowledge Representation and Knowledge Reasoning Based on the Aristotelian Modal Syllogism □AE◇E-4

Knowledge Representation and Knowledge Reasoning Based on the Aristotelian Modal Syllogism □AE◇E-4

The Deductibility of the Aristotelian Modal Syllogism E⼞I◇O-4 from the Perspective of Mathematical Structuralism

The Deductibility of the Aristotelian Modal Syllogism E⼞I◇O-4 from the Perspective of Mathematical Structuralism

The Validity of Generalized Modal Syllogisms Based on the Syllogism E□M◇O-1

The Validity of Generalized Modal Syllogisms Based on the Syllogism E□M◇O-1

The Validity of Generalized Modal Syllogisms with the Generalized Quantifiers in Square{most}

The Validity of Generalized Modal Syllogisms with the Generalized Quantifiers in Square{most}

Knowledge Representation Based on the Aristotelian Modal Syllogism □AE◇E-2

Knowledge Representation Based on the Aristotelian Modal Syllogism □AE◇E-2

Screening out All Valid Aristotelian Modal Syllogisms

It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.

The Analytical Perspective of Aristotle’s Categorical and Modal Syllogisms

What is meant under the genuine title of Aristotle’s ta Analytika is rarely properly understood. Presumably, his analytics was inspired by the method of geometric analysis. For Aristotle, this was a regressive or heuristic procedure, departing from a proposed conclusion (or prob­lem) and asking which premises could be found in order to syllogize, demonstrate or explain it. The terms that form categorical and modal propositions play a fundamental role in analytics. Aristotle introduces letters in lieu of the triples of terms (major – middle – minor) constitut­ing the propositions and the three syllogistic figures that schematize them. His formulation of the three syllogistic figures refers to a syntacti­cal and predicative order and position of the triples of terms, arranged in some diagrammed schemata, which, regrettably, are missing from the extant text of the Prior Analytics. Considering planar and graphic arrangements, both vertical and horizontal orders as well as the posi­tion of the three terms involved, we propose a reconstruction, at least to some extent, of these probable lettered diagrams. In such reconstructed diagrams, we can appropriately capture the definition of syllogism as a predicative connexion of terms, and easier survey a synoptic account of all valid predicative relations and transpositions, and also reduce the imperfect syllogisms into the moods of the first figure. Aristotle’s syllogistic is an analytical calculation of terms, understood as predicates and subjects within the categorical propositions, and more precisely of three terms schematized in three figures in predicative links such that, by means of a middle, follows from necessity a conclusion of the extreme terms. The necessity of the consequence is not based on the implication or inference of the propositions, but on a predictive transi­tivity through the middle term within the syllogistic figures. Syllogism must draw its conclusion through the way its terms are predicated of one another. Aristotle in his Prior Analytics (I 3, 8–22) developed also a complex account of modal syllogisms within necessity and possibility of belonging (predicating). This account involves also such an analyti­cal reduction to the syllogistic figures. In this analytical perspective, we try to throw some light on his modal syllogisms, although this difficult and nowadays thoroughly discussed topic would require a much wider treatment.

Is the Enhanced Indispensability Argument a Useful Tool in the Hands of Platonists?

Platonists in mathematics endeavour to prove the truthfulness of the proposal about the existence of mathematical objects. However, there have not been many explicit proofs of this proposal. One of the explicit ones is doubtlessly Baker’s Enhanced Indispensability Argument (EIA), formulated as a sort of modal syllogism. We aim at showing that the purpose of its creation – the defence of Platonist viewpoint – was not accomplished. Namely, the second premise of the Argument was imprecisely formulated, which gave space for various interpretations of the EIA. Moreover, it is not easy to perceive which of the more precise formulations of the above-mentioned premise would be acceptable. For all these reasons, it is disputable whether the EIA can be used to defend Platonist outlook. At the beginning of this century, Baker has shown that the so-called Quine-Putnam Indispensability Argument can not provide “full” platonism - a guarantee of the existence of all mathematical objects. It turns out, however, that the EIA has a similar disadvantage.

Term Kinds and the Formality of Aristotelian Modal Logic

Recent formalizations of Aristotle's modal syllogistic have made use of an interpretative assumption with precedent in traditional commentary: That Aristotle implicitly relies on a distinction between two classes of terms. I argue that the way Rini (2011. Aristotle's Modal Proofs: Prior Analytics A8–22 in Predicate Logic, Dordrecht: Springer) employs this distinction undermines her attempt to show that Aristotle gives valid proofs of his modal syllogisms. Rini does not establish that Aristotle gives valid proofs of the arguments which she takes to best represent Aristotle's modal syllogisms, nor that Aristotle's modal syllogisms are instances of any other system of schemata that could be used to define an alternative notion of validity. On the other hand, I argue, Robert Kilwardby's ca. 1240 commentary on the Prior Analytics makes use of a term-kind distinction so as to provide truth conditions for Aristotle's necessity propositions which render Aristotle's conversion rules and first figure modal syllogisms formally valid. I reconstruct a suppositio semantics for syllogistic necessity propositions based on Kilwardby's text, and yield a consequence relation which validates key results in the assertoric, pure necessity and mixed necessity-assertoric syllogistics.

Essence, Necessity, and Demonstration in Aristotle

Although the comments that follow are largely critical, I want to express at the outset my great admiration for Marko Malink’s excellent book. It makes a major contribution to our understanding of Aristotle’s logic and the broader philosophical commitments that underlie it. It is also a model of exegetical clarity and philosophical rigor the rest of us working in this field should strive to emulate. I want to explore some of the implications of Malink’s interpretation of the modal syllogistic in the Prior Analytics for Aristotle’s theory of demonstration in the Posterior Analytics. I’ll do so by focusing on Malink’s account of the connection between essence and necessity. I’ll argue that if Malink’s account of the modal syllogistic is correct, and in particular if his account of the connection between essence and necessity is correct, then a broad range of demonstrations Aristotle presents in the Posterior Analytics cannot be modal syllogisms. Malink may happily accept this as a consequence of his view. Still, it seems worth showing that this is a consequence of his view. I’ll also argue that Malink’s account of the connection between essence and necessity risks committing Aristotle to an unattractive form of essentialism.

A Formal Reconstruction of Buridan's Modal Syllogism

In this paper, we provide a historical exposition of John Buridan's theory of divided modal propositions. We then develop a semantic interpretation of Buridan's theory which pays particular attention to Buridan's ampliation of modal terms. We show that these semantics correctly capture his syllogistic reasoning.

Proof by Assumption of the Possible in Prior Analytics 1.15

In Prior Analytics 1.15 Aristotle undertakes to establish certain modal syllogisms of the form XQM. Although these syllogisms are central to his modal system, the proofs he offers for them are problematic. The precise structure of these proofs is disputed, and it is often thought that they are invalid. We propose an interpretation which resolves the main difficulties with them: the proofs are valid given a small number of intrinsically plausible assumptions, although they are in tension with some claims found elsewhere in Aristotle’s modal syllogistic. The proofs make use of a rule of propositional modal logic which we call the possibility rule. We investigate how this rule interacts and coheres with the core elements of the modal syllogistic.

The medieval theory of consequence

The recovery of Aristotle’s logic during the twelfth century was a great stimulus to medieval thinkers. Among their own theories developed to explain Aristotle’s theories of valid and invalid reasoning was a theory of consequence, of what arguments were valid, and why. By the fourteenth century, two main lines of thought had developed, one at Oxford, the other at Paris. Both schools distinguished formal from material consequence, but in very different ways. In Buridan and his followers in Paris, formal consequence was that preserved under uniform substitution. In Oxford, in contrast, formal consequence included analytic consequences such as ‘If it’s a man, then it’s an animal’. Aristotle’s notion of syllogistic consequence was subsumed under the treatment of formal consequence. Buridan developed a general theory embracing the assertoric syllogism, the modal syllogism and syllogisms with oblique terms. The result was a thoroughly systematic and extensive treatment of logical theory and logical consequence which repays investigation.

Hupo in the Prior Analytics: a note on Disamis XLL

This is a brief note that looks at the problem presented by the traditional rendering of the modal syllogism Disamis XLL. In two recent articles, I argue that we should not attribute Disamis XLL to Aristotle. The purpose of this note is to provide textual support for my claim.

Ibn Rušd et lesPremiers Analytiquesd'Aristote: Aaperçu sur un problème de syllogistique modale

Ibn Rušd devoted a certain number of works to Aristotle'sPrior Analytics. In a series ofopusculeswritten over a period of twenty years and following upon hisMiddle Commentary on Aristotle's Prior Analytics, he faced a problem particular to the modal syllogism – that of the mood of the conclusion in mixed syllogisms.The problem can be stated as follows: At the beginning of thePrior Analytics, Aristotle established a formal deductive principle – that of universal attribution (Pr. An.I.1.24b26–30). Applied to the modal syllogism, this principle is inadequate as stated. It is too general to be applied in a univocal manner in all modal syllogisms. To preserve a sense of coherence in Aristotle's declarations, the commentators had to interpret it. Presenting the interpretations of the commentators, primarily al-Fārābī and Alexander, on the basis of al-Fārābī'sLarge Commentary on Aristotle's Prior Analytics, Averroes criticizes them. Applied according to Alexander's interpretation, the principle of universal attribution is valid only for modal syllogisms one of whose premises is necessary and the other assertoric; according to al-Fārābī's interpretation, it is verified only when the minor premise is possible. Averroes proposes two preliminary solutions. Either this formal deductive principle must be applied differently according to the modal differences of the minor premises in mixed syllogisms (first solution) or would be used in two ways, generally or in keeping with each mood (second solution). These solutions are not satisfactory, for they call into question the unity and universality of the principle of universal attribution as established by Aristotle. What is the utility, Averroes asks, of a principle which does not hold for all modalities or does not apply to all the premises when thePr. An.ought to furnish formal and universal principles of deduction? And why did Aristotle define the principle of universal attribution without distinguishing its application according to each of the three modal premises? Returning at the end of his career to a literal exegesis of Aristotle's propositions and without harkening back to the earlier solutions, he proposes a theory of making the terms modal (fourth solution) in order to save Aristotle's declarations with respect to the principle of universal attribution and the mood of the conclusion of mixed syllogisms (Pr. An.I. 9.30al5–20). Though formally inadequate, this solution, which had a continued history, proposes a new way of looking at the classification of modal propositions.

Models for modal syllogisms.

A semantics is presented for Storrs McCalΓs separate axiomatiza- tions 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 hav- ing it accidentally. A completeness proof and a semantic decision procedure are given. In the opening chapters of (2) Lukasiewicz developed a nonmodal system of logic to illuminate Aristotle's discussion of nonmodal syllogisms. One of the distinctive features of his presentation is his syntactic treatment of the invalid syllogisms. In effect, invalid syllogisms are deduced in his system.1 Also, a deci- sion procedure for determining the validity and invalidity of Aristotelian non- modal syllogisms is given in purely syntactic terms. Though Lukasiewicz does not extend his treatment of Aristotelian nonmodal syllogisms to Aristotelian modal syllogisms, Storrs McCall in (3) does, by developing the system L-X-M, which treats syllogisms formed from assertoric and apodeictic propositions. The purpose of this paper is to provide a semantics for L-X-M. (McCall did not pro- vide a semantics for L-X-M.) I shall assume that L-X-M's relationship to Ar- istotle's modal syllogisms is accurately described in (3). So my primary interest is mathematical rather than historical. But my hope is that the semantics will provide the reader with an intuitive grasp of Aristotle's thinking about a sub- stantial fragment of Aristotelian syllogisms. Since modifications, though minor, will be made in McCalΓs L-X-M, we shall refer to the modified system as LXM, whose presentation will be self- contained. The syntax of LXM is as follows.

Statements about Events Modal and Tense Analysis in Medieval Logic

Now philosophers have thought about modal terms, modal propositions and modal syllogisms long before the notion of possible worlds was entertained. The question what interpretative model they used is a relevant one today. Jaako Hintikka and the circle of pupils and colleagues connected with him, in particular Simo Knuuttila, have tried to answer this question. They advance the following thesis1: underlying the modal analysis proposed by Aristotle and the scholastic

An interpretation of Aristotle's modal syllogism

An interpretation of Aristotle's modal syllogism