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.

我表明，在经典集合论中工作时，不可能唯一地表征经典逻辑。通过基于 Eduardo Barrio、Federico Pailos 和 Damian Szmuc 最近的工作，我表明对于每个推理级别（有限和超限），要么经典逻辑在该级别上不是唯一的，要么存在该级别的直觉上有效的推论不是可以在现代经典集合论中定义。古典逻辑学家因此面临一个三角困境：放弃唯一性但保留可表征性，放弃可表征性并保留唯一性，或（可能）保留两者但放弃现代经典集合论。在证明了主要结果之后，我通过在次一致集合论中对经典逻辑的解释来简要探讨第三个选项。