[full article, abstract in English; only abstract in Lithuanian] The paper deals with the problem of the “two Barbaras” in the Aristotelian modal syllogistic. The problem consists in Aristotle’s differing views on two at a first sight similar in nature syllogisms of mixed assertoric and necessary premises: Barbara LXL and Barbara XLL. The fact that Aristotle believed the first syllogism to be valid and the second one – not, has been received either 1) negatively, because both Barbaras have been held (...) to be invalid, or 2) negatively, because both Barbaras have been held to be valid, or 3) positively, by giving a reason why the two Barbaras differ. We commit ourselves to the position by proving that modal propositions for Aristotle have their modalities de dicto and that Aristotelian modal operators act according to their own separate rules where only the type of the modality of a major premise is relevant for the modal status of the conclusion. (shrink)

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the intended (...) schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency. (shrink)

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 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 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 developed also a complex account of modal syllogisms within necessity and possibility of belonging. 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. (shrink)

By a pure modal logic of names we mean a quantifier-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic :271–284, 1989; Thomason in J Philos Logic 22:111–128, 1993 and J Philos Logic 26:129–141, 1997. In both (...) cases we use Johnson-like models. We also analyze different kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the different versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidability” in the following sense: for every inference starting with a finite set of premises we can specify a finite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference. (shrink)

Fred's semantics for McCall's syntactic presentation of Aristotle's assertoric and apodeictic syllogistic is altered to free it from Thom's objections that it is unAristotelian. The altered semantics rejects Baroco-XLL and Bocardo-LXL, which Thom says Aristotle should have accepted. Aristotle's proofs that use ecthesis are formalized by using singular sentences. With one exception the (acceptance) axioms for McCall's system L-X-M are derivable. Formal proofs are shown to be sound.

One semantic and two syntactic decision procedures are given for determining the validity of Aristotelian assertoric and apodeictic syllogisms. Results are obtained by using the Aristotelian deductions that necessarily have an even number of premises.

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.

The following is a critical and historical account of Aristotelian Essentialism informed by recent work on Aristotle’s modal syllogistic. The semantics of the modal syllogistic are interpreted in a way that is motivated by Aristotle, and also make his validity claims in the Prior Analytics consistent to a higher degree than previously developed interpretative models. In Chapter One, ancient and contemporary objections to the Aristotelian modal syllogistic are discussed. A resolution to apparent inconsistencies in Aristotle’s modal syllogistic is proposed and (...) developed out of recent work by Patterson, Rini, and Malink. In particular, I argue that the semantics of negation is distinct in modal context from those of assertoric negative claims. Given my interpretive model of Aristotle’s semantics, in Chapter Two, I provide proofs for each of the mixed apodictic syllogisms, and propose a method of using Venn Diagrams to visualize the validity claims Aristotle makes in the Prior Analytics. Chapter Three explores how Aristotle’s syllogistic fits within Aristotle’s philosophy of science and demonstration, particularly within the context of the Posterior Analytics. Consideration is given to the Aristotelian understanding of the relationship among necessity, explanation, definition, and essence. Chapter Four applies Aristotelian modal logic in contemporary contexts. I contrast Aristotelian modality and essentialism with contemporary modalism based upon the semantics of possible worlds, e.g. Kripke and Putnam. I also develop an account of how Aristotelian modal logic can ground a sortal dependent theory of identity, as discussed by Wiggins. (shrink)