Logica Universalis

, Volume 10, Issue 2–3, pp 191–213

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


DOI: 10.1007/s11787-016-0147-y

Cite this article as:
Jacquette, D. Log. Univers. (2016) 10: 191. doi:10.1007/s11787-016-0147-y


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.


Categorical universal and existential generalizations (A-E-I-O sentence types) Contradiction Existence presupposition Frege Gottlob Logical contrariety Square of opposition Subalternation Subcontrariety Syllogistic logic 

Mathematics Subject Classification

03A05 03B05 03B10 03B20 03B65 

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

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