A new presentation of categorical syllogisms

Dan Constantin Radulescu

Published 2018-05-21Version 1

One proposes two simple schemes - a graphical and a tree like one - for finding logical conclusions (LCs) from any pair of categorical premises (PCP), via using a set model for the syllogistic terms S,P,M, non-S (S'), non-P (P'), non-M (M'). In this model, any universal, (resp. particular), premise empties, (resp. lays set elements into), two subsets out of the 8-subset partition of the universal set U which models the categorical statements. (S',P',M' are interpreted as the complementary sets in U of the S,P,M terms respectively.) A cylindrical Venn diagram replaces the 8 irregularly shaped subsets from the usual 3-circle Venn diagram of U by rectangular shapes drawn on a cylinder and provides a very clear presentation of the two LCs finding schemes. One shows, (without using syllogistic moods and figures, syllogistic axioms and inference rules, nor valid syllogism rules and syllogism reduction), that any LC refers to just one of the 8 subsets of U and that any PCP entailing an LC, (called valid syllogistic argument (VCA), may be recast as, e.g., one of the Barbara, Darapti or Darii valid syllogisms (VSs). The recasting is done via set relabelings instead of the syllogism reduction (used, e.g., by Aristotle and J. Lukasiewicz). One may say that this paper approaches categorical syllogisms in the same spirit as George Boole approached them in The Mathematical Analysis of Logic, but instead of Boole's equations one uses graphic and tree like methods for finding the LCs.