Aristotle, Boole, and Categories

Abstract

We propose new axiomatizations of the 24 assertoric syllogisms of Aristotle’s syllogistic, and the $2^{2^n}$ n-ary operations of Boole’s algebraic logic. The former organizes the syllogisms as a $6\times 4$ table partitioned into four connected components according to which term if any must be inhabited. We give two natural-deduction style axiomatizations, one with four axioms and four rules, the second with one axiom and six rules. The table provides immediately visualizable proofs of soundness and completeness. We give an elementary category-theoretic semantics for the axioms along with criteria for determining the term if any required to be nonempty in each syllogism. We base the latter on Lawvere’s notion of an algebraic theory as a category with finite products having as models product-preserving set-valued functors. The benefit of this axiomatization is that it avoids the dilemma of whether a Boolean algebra is a numerical ring as envisaged by Boole, a logical lattice as envisaged by Peirce, Jevons, and Schroeder, an intuitionistic Heyting algebra on a middle-excluding diet as envisaged by Heyting, or any of several other candidates for the “true nature” of Boolean logic. Unlike general rings, Boolean rings have only finitely many n-ary operations, permitting a uniform locally finite axiomatization of their theory in terms of a certain associative multiplication of finite 0–1 matrices.