Abstract

I show that it is not possible to uniquely characterize classical logic when working within classical set theory. By building on recent work by Eduardo Barrio, Federico Pailos, and Damian Szmuc, I show that for every inferential level (finite and transfinite), either classical logic is not unique at that level or there exist intuitively valid inferences of that level that are not definable in modern classical set theory. The classical logician is thereby faced with a three-horned dilemma: Give up uniqueness but preserve characterizability, give up characterizability and preserve uniqueness, or (potentially) preserve both but give up modern classical set theory. After proving the main result, I briefly explore this third option by developing an account of classical logic within a paraconsistent set theory. This account of classical logic ensures unique characterizability in some sense, but the non-classical set theory also produces highly non-classical meta-results about classical logic.