Bas Kortenbach

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.

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it.

Incurvati and Schlöder (_Journal of Philosophical Logic, 51_(6), 1549–1582, 2022) have recently proposed to define supervaluationist logic in a multilateral framework, and claimed that this defuses well-known objections concerning supervaluationism’s apparent departures from classical logic. However, we note that the unconventional multilateral syntax prevents a straightforward comparison of inference rules of different levels, across multi- and unilateral languages. This leaves it unclear how the supervaluationist multilateral logics actually relate to classical logic, and raises questions about Incurvati and Schlöder’s response to the objections. We overcome the obstacle, by developing a general method for comparisons of strength between multi- and unilateral logics. We apply it to establish precisely on which inferential levels the supervaluationist multilateral logics defined by Incurvati and Schlöder are classical. Furthermore, we prove general limits on how classical a multilateral logic can be while remaining supervaluationistically acceptable. Multilateral supervaluationism leads to sentential logic being classical on the levels of theorems and regular inferences, but necessarily strictly weaker on meta- and higher-levels, while in a first-order language with identity, even some classical theorems and inferences must be forfeited. Moreover, the results allow us to fill in the gaps of Incurvati and Schlöder’s strategy for defusing the relevant objections.