Every Logic is Unique: Transfinite Metainferences in Classical Set Theory

Abstract What is the proper way to transfinitely extend the usual hierarchy of finite metainferential levels? McAllister ( Journal of Philosophical Logic, 51 , 1345–1365, 2022; Belief Revision About Logic , PhD Thesis, University of Auckland, 2024) has proven that classical logic and numerous other logics are non-unique in classical set theory. On the basis of these results, she argues for a range of philosophical consequences, including problems for logical monism, classical set theory, and the identification of logics. This paper demonstrates that McAllister’s key results are merely artifacts of the level labels which she adds to the transfinite hierarchy. It is proven that, in the absence of labels, every logic is unique in classical set theory. Moreover, I argue that if labels are not innocent additions, but instead influence the results in substantial ways, then they must be left out of the hierarchy. Therefore, in the final analysis, every logic is unique. The various problems which McAllister extracts from non-uniqueness accordingly dissolve.