In a recent paper, Barrio, Pailos and Szmuc (BPS) show that there are logics that have exactly the validities of classical logic up to arbitrarily high levels of inference. They suggest that a logic therefore must be identified by its valid inferences at every inferential level. However, Scambler shows that there are logics with all the validities of classical logic at every inferential level, but with no antivalidities at any inferential level. Scambler concludes that in order to identify a logic, we at least need to look at the validities and the antivalidities of every inferential level. In this paper, I argue that this is still not enough to identify a logic. I apply BPS’s techniques in a super/sub-valuationist setting to construct a logic that has exactly the validities and antivalidities of classical logic at every inferential level. I argue that the resulting logic is nevertheless distinct from classical logic.
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Porter, B. Supervaluations and the Strict-Tolerant Hierarchy. J Philos Logic 51, 1367–1386 (2022). https://doi.org/10.1007/s10992-021-09624-6
- Strict-Tolerant logic
- Substructural Logic
- metainferential hierarchy
- Classification problem