Subalternation and existence presuppositions in an unconventionally formalized canonical square of opposition

Abstract

An unconventional formalization of the canonical (Aristotelian-Boethian) square of opposition in the notation of classical symbolic logic secures all but one of the canonical square’s grid of logical interrelations between four A-E-I-O categorical sentence types. The canonical square is first formalized in the functional calculus in Frege’s Begriffsschrift, from which it can be directly transcribed into the syntax of contemporary symbolic logic. Difficulties in received formalizations of the canonical square motivate translating I categoricals, ‘Some S is P’, into symbolic logical notation, not conjunctively as \({\exists x[Sx\wedge Px]}\), but unconventionally instead in an ontically neutral conditional logical symbolization, as \({\exists x[Sx\rightarrow Px]}\). The virtues and drawbacks of the proposal are compared at length on twelve grounds with the explicit existence expansion of A and E categoricals as the default strategy for symbolizing the canonical square preserving all original logical interrelations.