Models for Modal Syllogisms

A semantics is presented for Storrs McCalΓs 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. 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. Also, a decision procedure for determining the validity and invalidity of Aristotelian nonmodal 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 provide a semantics for L-X-M.) I shall assume that L-X-M's relationship to Aristotle'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 substantial 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 selfcontained. The syntax of LXM is as follows. *I am grateful to David Bostock and a referee of this journal for comments on an earlier draft of this paper. And I extend my thanks to Timothy Smiley, who read a second version of my paper with remarkable care and made numerous significant corrections and improvements in it. Received June 29, 1987