An Interpretation of Aristotle’s Syllogistic and a Certain Fragment of Set Theory in Propositional Calculi

In [1] Chapter IV Lukasiewicz presents a system of syllogistic which is an extension of Aristotle’s ordinary syllogistic. In spite of this difference Lukasiewicz speaks about it, as do we, as the Aristotelian system. One of the well-known interpretation of syllogistic is Leibnitz’s interpretation described in [1] (pp. 126–129). Syllogistic formulas are interpreted there in an arithmetical manner. A second, very natural interpretation, has been given by S lupecki (see below), who interprets syllogistic formulas set theoretically. Although every formula which is not a syllogistic thesis can be rejected by using a finite number of objects (natural numbers in Leibnitz’s interpretation, and sets in S lupecki’s interpretation), there is not any fixed finite number of objects that would falsify every formula not being a syllogistic thesis. Our first interpretation (comp. Theorem 1) has the advantage of interpreting Aristotle’s syllogistic in a finite(four-) valued propositional calculus. We also give an interpretation of Aristotle’s syllogistic and a fragment of set theory in the modal calculus S5. Let Trm be an infinite set of terms of Aristotle’s syllogistic (AS). Elementary formulas of AS are expressions of the following forms: Aab, Iab