About the set theory model of categorical syllogisms
Dan Constantin Radulescu · 2019
The Set Theoretical Model (STM) of categorical syllogisms was initially developed by George Boole and Lewis Carroll, who worked with a “universe of discourse”, U, which contains the pairwise complementary sets, or categorical terms, S,S'(non-S),P,P'(non-P),M,M'(non-M), and is thus partitioned into 8 subsets: SPM:=S∩P∩M, S'PM,...,S'P'M'. As George Boole and Lewis Carroll noticed, any logical consequence (LC) of a pair of categorical premises (PCP) pinpoints to just one and only one of the eight subsets partitioning U, and asserts one of the following: one subset remains possibly non empty but the other three subsets are empty out of the four subsets partitioning either one of the sets S,P,S',P' - this type of LC is entailed by the Barbara type (1) PCPs – they contain two universal premises, emptying subsets of both M and M'; one subset remains possibly non empty but the other three subsets are empty out of the four subsets partitioning either M or M' - this type of LC is entailed by the Darapti type (2) PCPs – they contain two universal premises, emptying subsets of either M or M'; one subset out of eight is definitely not empty - this type of LC is entailed by Darii/Datisi subtype (3a) PCPs, or by Disamis/Dimaris subtype (3b) PCPs – they contain one universal plus one particular premise, emptying subsets of, and placing set elements in, either M or M'. If both positive terms, S,P,M, and negative terms, S',P',M', are allowed into PCPs and LCs, then each one of the types (1), (2), (3a), (3b) PCPs contains 8 distinct PCPs and each PCP entails at least one LC, thus generating at least one valid categorical argument (VCA). Nevertheless, all the VCAs of the same (sub)type are equivalent: any one of them can be recast as any other VCA of the same (sub)type. A tree like method, (Carroll's method of subscripts), immediately finds the LC of any VCA. Besides the VCAs and their LCs, one discusses mutual compatibility of various PCPs, sorites, “distribution conservation”, empty set constraints (ESC), and one comments on valid syllogisms (VS), a “contrived” subset of the VCAs, and the main subject of the Classic Categorical Syllogistic (CCS).