A Matricial Vue of Classical Syllogistic and an Extension of the Rules of Valid Syllogism to Rules of Conclusive Syllogisms with Indefinite Terms

One lists the distinct pairs of categorical premises (PCPs) formulable via only the positive terms, S,P,M, by constructing a six by six matrix obtained by pairing the six categorical P-premises, A(P,M), O(P,M), A(M,P*), O(M,P*), where P* ∈ {P,P′}, with the six, similar, categorical S-premises. One shows how five rules of valid syllogism (RofVS), select only 15 distinct PCPs that entail logical consequences (LCs) belonging to the set L + : = {A(P,S), O(P,S), A(S,P), E(S,P), O(S,P), I(S,P)}. The choice of admissible LCs can be regarded as a condition separated from the conditions (or axioms) contained in the RofVS: the usual eight (Boolean) PCPs that generate valid syllogisms are obtained when the only admissible LCs belong to the set L: = {A(S,P), E(S,P), O(S,P), I(S,P)} and no existential imports are addressed. A 64 PCP-matrix obtains when both PCPs and LCs may contain indefinite terms—the positive, S,P,M, terms, and their complementary sets, S′,P′, M′, in the universe of discourse, U, called the negative terms. Now one can accept eight LCs: A(S*,P*), I(S*,P*), where P* ∈ {P,P′}, S* ∈ {S,S′}, and there are 32 conclusive PCPs, entailing precise, “one partitioning subset of U” LCs. The four rules of conclusive syllogisms (RofCS) predict the less precise LCs, left after eliminating the middle term from the exact LCs. The RofCS also predict that the other 32 PCPs of the 64 PCP-matrix are non-conclusive. The RofVS and the RofCS are generalized, and arguments are given, for also accepting as valid syllogisms the conclusive syllogisms formulable via positive terms which entail the LCs A(P,S) and O(P,S).