On the Interpretation of Aristotelian Syllogistic

(theorem 6, ? 4) we show that in Wedberg's system [14] with primitives 'Aab', 'a" (not a), it is possible to find a mapping a -T(a) as above such that 'Aab' is equivalent to '92(a) is contained in 9p(b)' and 9p(a') is equal to p(a)', the complement of p(a) with respect to V. Thus, if we make the preliminary step of identifying elements a, b such that Aab and Aba both hold (i.e. taking equivalence classes with respect to the relation Aab & Aba), we are left with essentially only one kind of interpretation for these systems, namely the 'normal' interpretation by classes. Slupecki [1 1], [12] has proved that Lukasiewicz's system is a complete and decidable theory of the relations of inclusion and intersection of non-null classes, and Wedberg [14] has proved that his system is a complete and decidable theory of the relation of inclusion and the operation of complementation for nonnull, non-universal classes. Using the above-mentioned embedding theorem, we are able to obtain (theorems 9, 6, ?? 5, 4) very simple proofs of these results. For the sake of completeness and simplicity of presentation we deal first with the theory of inclusion and complementation for general (i.e. not necessarily non-null, non-universal) classes. In this way all the results follow with very little further effort from our theorem 1 which says, essentially, that a sufficient (and obviously necessary) condition for a partially ordered set S with a 'complement' operation "' to be embeddable, with preservation of complements, in a Boolean algebra is that, for all a, b of S, a" = a, a ? b .). b' ? a', and a ? a' .D. a ? b.