A cylindrical Venn diagram model for categorical syllogisms
One denotes A(M,*) by A1 and A(*,M) by A2, where * stands for either S or P, and the same for the O categorical operator. This allows to dispense with the four syllogistic figures and reduces the number of the 24 classically valid syllogisms (CVS) to only 8 (not 15) distinct CVS plus 6 (not 9) distinct existential import (ei) CVS. Out of the 36 (not 64!) distinct pairs of categorical premises (PCP or just pairs), 19 pairs entail at least one logical conclusion and thus generate valid syllogistic arguments (in short, valid syllogisms or VS) split into two subsets: the CVS and the VS\CVS. The latter ones have A(P,S), O(P,S) and I(S',P') as conclusions, not the “(S,P)-type” conclusions required of the CVS. (S',P',M' are the complementary sets of S,P,M in a universal set U.) The other results are: (i) one can also dispense with most of the rules of valid syllogisms (such as “the middle term has to be distributed in at least one premise” and “two negative premises are not allowed”), (ii) any pair of categorical premises generates at least one VS unless 1. both premises are particular, or, 2. one premise is universal and one particular and they act one on M, the middle term, and the other premise acts on M' – its complement in U, (iii) the VS set is the (disjoint) union of three equivalence classes generated respectively by (a) two universal premises acting on both M and M' , (b) two universal premises both acting on either M or M', (c) one universal premise and one particular premise both acting on either M or M', (iv) inside each equivalence class, via a relabeling transformation of the sets S,P,M, S',P',M', any of the VS can be recast (or reformulated) as any other VS from the same class. This “naming invariance” suggests that, from a set theoretical point of view, the (S,P) conclusion restriction is not meaningful. The VS\CVS subset contains 6 VS and 7 ei VS.