Abstract
C. I. Lewis’ systems were the first axiomatisations of modal logics. However some of those systems are non-normal modal logics, since they do not admit a full rule of necessitation, but only a restricted version thereof. We provide G3-style labelled sequent calculi for Lewis’ non-normal propositional systems. The calculi enjoy good structural properties, namely admissibility of structural rules and admissibility of cut. Furthermore they allow for straightforward proofs of admissibility of the restricted versions of the necessitation rule. We establish completeness of the calculi and we discuss also related systems.
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References
Chagrov, A., and M. Zakharyaschev, Modal Logic, Oxford University Press, Clarendon Press, 1997.
Dyckhoff, R., and S. Negri, Proof analysis in intermediate logics, Archive for Mathematical Logic 51:71–92, 2012.
Gilbert, D. R., and P. Maffezioli, Modular sequent calculi for classical modal logics, Studia Logica 103:175–217, 2015.
Hughes, G. E., and M. J. Cresswell, An Introduction to Modal Logic, University of California, 1968.
Kripke, S., Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi, in J. W. Addison, L. Henkin and A. Tarski, (eds.), The Theory of Models, North Holland, Amsterdam, 1965, pp. 206–220.
Lavendhomme, R., and T. Lucas, Sequent calculi and decision procedures for weak modal systems, Studia Logica 65:121–145, 2000.
Lellmann, B. and E. Pimentel, Proof search in nested sequent calculi, in M. Davis, A. Fehnker, A. McIver, and A. Voronkov, (eds.), LPAR-20. LNCS, vol. 9450, Springer, Berlin, 2015, pp. 558–574.
Lewis, C. I., and C. H. Langford Symbolic Logic, New York Century Company, 1932. Reprinted: New York Dover Publication, 2 edition 1959.
Matsumoto, K., Decision procedure for modal sentential calculus S3, Osaka Mathematical Journal 12:167–175, 1960.
Minari, P., Labelled sequent calculi for modal logic and implicit contractions, Archive for Mathematical Logic 52:881–907, 2013.
Negri, S., Proof Analysis in Modal Logic, Journal of Philosophical Logic 34(4):507–544, 2005.
Negri, S., Proof theory for non-normal modal logics: The neighbourhood formalism and basic results, IfCoLog Journal of Logics and their Applications, Mints’ memorial issue 4(4):1241–1286, 2017.
Negri, S., and J. von Plato, Proof Analysis, Cambridge University Press, Cambridge, 2011.
Ohnishi, M., Gentzen decision procedures for Lewis’ systems S2 and S3, Osaka Mathematical Journal 13:125–137, 1961.
Ohnishi, M., and K. Matsumoto, Gentzen method in modal calculi, Osaka Math. J. 9(2):113–130, 1957.
Priest, G., An Introduction to Non-Classical Logic. From If to Is, Cambridge University Press, Cambridge, 2001.
Segerberg, K., An Essay in Classical Modal Logic, Filosofiska Studier, vol. 13, Uppsala Universitet, Uppsala, 1971.
Troelstra, A., and H. Schwichtenberg, Basic Proof Theory, Cambridge University Press, Cambridge, 2000.
Whitehead, A., and B. Russell, Principia Mathematica, Cambridge University Press, Cambridge, 1910.
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Presented by Heinrich Wansing; Received January 22, 2020
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Tesi, M. Labelled Sequent Calculi for Lewis’ Non-normal Propositional Modal Logics. Stud Logica 109, 725–757 (2021). https://doi.org/10.1007/s11225-020-09924-z
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DOI: https://doi.org/10.1007/s11225-020-09924-z