Categorical syllogisms – a set theory do-over
Authors: Radulescu Dan Constantin 
Published: July 2018

There are 36 distinct pairs of categorical premises (PCP) expressed via only the S,P,M terms and the categorical operators (or quantifiers) A,O,E,I applied to pairs of these 3 terms. Modeling such premises on a cylindrical Venn diagram or Karnaugh map with n=3, where the middle term M occupies one row and M', its complementary set, occupies a 2nd row, one sees that the PCPs do not act symmetrically on the M and M' rows: four P-premises (resp. S-premises) out of six act on M and only two act on M'. Only by adding premises explicitly containing the M' term, namely A(M',*) and O(M',*) where the * stands for either S or P, one can get to 8 P-premises and 8 S-premises, with 4 P-premises, (resp. 4 S-premises) acting on the M set and 4 P-premises, (resp. 4 S-premises) acting in similar ways on the M' set. Only with the “M' premises addition” one gets to a set of 64 distinct PCPs which remains unchanged under the relabelings M↔M', S↔S' and P↔P', (where S',P',M' are the complementary sets of S,P,M in a universal set U). One models the usual S,P,M terms which appear in the 3 statements of a categorical syllogism as sets. This way the two premises and the conclusion of a categorical syllogism become statements about sets. Since sets are not “distributed”, do not appear in “figures”, do not have “moods”, etc., such concepts will not be used in this do-over of the categorical syllogisms. The 36 distinct pairs of categorical premises (PCP) which are worded using only the S,P,M terms, as well as the “extended” 64 PCPs are naturally partitioned into 5 subsets: two subsets whose PCPs do not entail any logical conclusion (LC), and three subsets whose PCPs do each entail at least one LC and thus generate a valid categorical argument (VCA). This way one obtains three classes of VCAs, each class containing some of the valid syllogisms (VS). By definition a VS is a VCA whose LC is one of the categorical operators (or quantifiers) A,O,E,I applied to the ordered pair (S,P). The LCs of the VCA\VS set have one of the formats A(P,S), O(P,S) or I(S',P'). Using a P↔S relabeling one transforms a VCA\VS PCP which entails an A(P,S) or O(P,S) LC, into a PCP which has an A(S,P) or O(S,P) LC; if the latter PCP is a VS PCP, then one may argue that such VCA\VS are of no interest since they are, up to a P↔S relabeling, identical to usual VSs. One may also argue that LCs of the format I(S',P') are of no practical interest, (maybe because an I(S',P') LC introduces the S', P' terms which do not appear in the (English) wording of the 64 PCPs). Such arguments, (outside of set theory), would play an equivalent role to the rules of valid syllogisms – whose main role is to eliminate, (under the “guise” of distribution, of “two negative premises are not allowed”, etc.), the VCA\VS set as containing only invalid syllogisms, even if all of them are valid categorical arguments. Using conversions, contrapositions and obversions, or, equivalently, set relabelings involving all the six sets S,P,M,S',P',M', one can show that the VCAs inside each of the three VCA classes are transformed into each other, i.e., are equivalent to one another. Existential import (ei) means that one adds, to a PCP made of two universal premises, the supposition that one of the sets S,P,M,S',P',M' is not empty.