Abstract
I provide a tableau system and completeness proof for a revised version of Carnap’s semantics for quantified modal logic. For Carnap, a sentence is possible if it is true in some first order model. However, in a similar fashion to second order logic, no sound and complete proof theory can be provided for this semantics. This factor contributed to the ultimate disappearance of Carnapian modal logic from contemporary philosophical discussion. The proof theory I discuss comes close to Carnap’s semantic vision and provides an interesting counterpoint to mainstream approaches to modal logic. Despite its historical origins, my intention is to demonstrate that this approach to modal logic is worthy of contemporary attention and that current debate is the poorer for its absence.
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References
Bernays, Paul : Review of modalities and quantification by Rudolf Carnap. The Journal of Symbolic Logic 13(4), 218–219 (1948)
Burgess, John P.: Which modal logic is the right one?. Notre Dame Journal of Formal Logic 40(1), 81–93 (1999)
Carnap, Rudolf : Modalities and quantification. Journal of Symbolic Logic 11(2), 33–64 (1946)
Carnap, Rudolf, Meaning and Necessity. Chicago, 1956.
Cocchiarella, Cocchiarella : On the primary and secondary semantics of logic necessity. The Journal of Philosophical Logic 4(1), 13–27 (1975)
Fine, Kit : Model theory for modal logic. part i - the de re/de dicto distinction. Journal of Symbolic Logic 7, 125–156 (1978)
Hendry, Herbert E., Pokriefka M. L.: Carnapian extensions of s5. The Journal of Philosophical Logic 14, 111–128 (1985)
Henkin, Leon : The completeness of the first-order functional calculus. Journal of Symbolic Logic 14, 159–166 (1949)
Kripke, Saul : A completeness theorem in modal logic. Journal of Symbolic Logic 24(1), 1–14 (1959)
Kripke, Saul : Semantical considerations on modal logic. Acta Philosophica Fennica – Modal and Many-valued Logics 16, 83–94 (1963)
Leitgeb, Hannes, Towards a logic of type-free modality and truth. In C. Dimitracopoulos et al. (eds.), Logic Colloquium 05, Lecture Notes in Logic, Cambridge University Press, Cambridge 2008.
Lewis C. I., Langford C. H.: Symbolic Logic. Dover Publications, New York (1959)
Prawitz, Dag : Completeness and hauptsatz for second order logic. Theoria 33(3), 246–258 (1967)
Schurz, Gerhard, Carnap’s modal logic. In Zwischen traditioneller und moderner Logic. Nichtklassische Ansatze, Mentis, Paderborn, 2001.
Schweizer, Schweizer : Quantified Quinnean S5. Journal of Philosophical Logic 22, 589–605 (1993)
Shapiro, Stewart : Foundations without Foundationalism: a case for second order logic. OUP, Oxford (1991)
Skyrms, Brian : An imaculate conception of modality or how to confuse use and mention. Journal of Philosophy 75, 368–387 (1978)
Smullyan, Raymond M.: First-Order Logic. Dover, New York (1968)
Wittgenstein, Ludwig : Tractatus Logico-Philosophicus. Routledge, New York (2001)
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Meadows, T. Revising Carnap’s Semantic Conception of Modality. Stud Logica 100, 497–515 (2012). https://doi.org/10.1007/s11225-012-9412-0
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DOI: https://doi.org/10.1007/s11225-012-9412-0