On the interpretation of Aristotelian syllogistic
John C. Shepherdson · Journal of Symbolic Logic · 1956
The main purpose of this note is to prove (theorem 11, § 5) that, in any interpretation of the formalisation of Aristotelian syllogistic given by Łukasiewicz [4], it is always possible to associate with each element a a non-null sub-class φ ( a ) of some ‘universal’ class V in such a way that ‘ Aab’ (all a are b ), ‘ Iab ’ (some a are b ) are equivalent respectively to ‘ φ ( a ) is contained in φ ( b )’, ‘ φ ( a ) has a non-null intersection with φ ( b )’. Similarly (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 → φ ( a ) as above such that ‘ Aab ’ is equivalent to ‘ φ ( a ) is contained in φ( b )’ and φ( a ‘ is equal to φ( a )’, the complement of φ( 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 [11], [12] has proved that Łukasiewicz'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.