Abstract
Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the non-transitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The non-transitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and more metainferential levels. All the logics in this hierarchy can accommodate a truth predicate, including the logic at the top of the hierarchy—known as CMω—which presumably maintains classicality at all levels. CMω has so far been accounted for exclusively in model-theoretic terms. Therefore, there remains an open question: how do we account for this logic in proof-theoretic terms? Can there be found a proof system that admits each and every classical principle—at all inferential levels—but nevertheless blocks the derivation of the liar? In the present paper, I solve this problem by providing such a proof system and establishing soundness and completeness results. Yet, I also argue that the outcome is philosophically unsatisfactory. In fact, I’m afraid that in light of my results this metainferential solution to the paradoxes can hardly be called a “solution,” let alone a good one.
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This paper was supported by a Minerva fellowship at Freie Universiterlin. The paper benefited from fruitful discussions with Robert Brandom, Ulf Hlobil, Dan Kaplan, Shuhei Shimamura, and Ryan Simonelli. Some of the ideas in the paper were presented at the 10th workshop on philosophical logic organized by the Buenos Aires logic group. My thanks to the organizers of this event, as well as to the audience. I would also like to thank two anonymous referees for this journal, for helpful comments and suggestions.
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Golan, R. On the Metainferential Solution to the Semantic Paradoxes. J Philos Logic 52, 797–820 (2023). https://doi.org/10.1007/s10992-022-09688-y
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DOI: https://doi.org/10.1007/s10992-022-09688-y