Abstract
In his Formalization of Logic (1943) Carnap pointed out that there are non-normal interpretations of classical logic: non-standard interpretations of the connectives and quantifiers that are consistent with the classical consequence relation of a language. Different ways around the problem have been proposed. In a recent paper, Bonnay and Westerståhl argue that the key to a solution is imposing restrictions on the type of interpretation we take into account. More precisely, they claim that if we restrict attention to interpretations that are (a) compositional, (b) non-trivial and (c) in the case of the quantifiers, invariant under permutations of the domain, Carnap’s Problem is avoided. This paper has two goals. The first is to show that Bonnay and Westerståhl’s solution to Carnap’s Problem doesn’t work. The second is to argue that something similar to their proposal seems to do the job. The problems with Bonnay and Westerståhl’s approach trace back to issues concerning the (un)definability of subsets of the domain of first-order structures, as well as to the compositionality of first-order languages. After expanding on these problems, I’ll propose a way to modify Bonnay and Westerståhl’s account and solve Carnap’s Problem.
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Acknowledgements
I would like to thank two anonymous reviewers from this journal, as well as Bogdan Dicher, Bruno Jacinto, Julian J. Schlöder, Julien Murzi and Bas Kortenbach for their very helpful comments on this material. Earlier versions of this paper were presented at the LanCog Seminar in the University of Lisbon, the Logic Seminar of the Scuola Normale Superiore, and at Logica 2023 and PhD’s in Logic 2023. I’m also grateful to the audiences of these events for their valuable feedback.
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del Valle-Inclán, P. Carnap’s Problem, Definability and Compositionality. J Philos Logic (2024). https://doi.org/10.1007/s10992-024-09767-2
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DOI: https://doi.org/10.1007/s10992-024-09767-2