Abstract
A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel–Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.
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References
Basin, D., Matthews, S. and Viganò, L. (1998) Natural deduction for non-classical logics, Studia Logica 60, 119–160.
Blackburn, P. (2000) Representation, reasoning, and relational structures: A hybrid logic manifesto, Logic J. IGPL 8, 339–365.
Blackburn, P., de Rijke, M. and Venema, Y. (2001) Modal Logic, Cambridge University Press.
Braüner T. (2000) A cut-free Gentzen formulation of the modal logic S5, Logic J. IGPL 8, 629–643.
Bull, R. and Segerberg, K. (1984) Basic modal logic, in D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, Vol. 2, Kluwer, pp. 1–88.
Castellini, C. (2005) Automated reasoning in quantified modal and temporal logic, Ph.D. thesis, School of Informatics, University of Edinburgh.
Castellini, C. and Smaill, A. (2002) A systematic presentation of quantified modal logics, Logic J. IGPL 10, 571–599.
Catach, L. (1991) TABLEAUX: A general theorem prover for modal logic, J. Automated Reasoning 7, 489–510.
Fitting, M. (1983) Proof Methods for Modal and Intuitionistic Logics, Synthese Library, Vol. 169, Kluwer.
Goré, R. (1998) Tableau methods for modal and temporal logis, in M. D'Agostino et al. (eds.), Handbook of Tableau Methods, Kluwer.
Kanger, S. (1957) Provability in Logic, Almqvist & Wiksell, Stockholm.
Kracht, M. (1996) Power and weakness of the modal display calculus, in Wansing (1996).
Kushida, H. and Okada, M. (2003) A proof-theoretic study of the correspondence of classical logic and modal logic, J. Symbolic Logic 68, 1403–1414.
Leivant, D. (1981) On the proof theory of the modal logic for arithmetic provability, J. Symbolic Logic 46, 531–538.
Mints, G. (1968) Cut-free calculi of type S5, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 8, 166–174 (in Russian). Translated in English in A. O. Slisenko (ed.), Studies in Constructive Mathematics and Mathematical Logic, Vol. 8, Part II, Leningrad, 1970.
Mints, G. (1992) A Short Introduction to Modal Logic, CSLI Lecture Notes 30, Stanford.
Mints, G. (1997) Indexed systems of sequents and cut-elimination, J. Philos. Logic 26, 671–696.
Moen, A. (2001) The proposed algorithms for eliminating cuts in the provability calculus GLS do not terminate, Nordic Workshop on Programming Theory.
Negri, S. (2003) Contraction-free sequent calculi for geometric theories, with an application to Barr's theorem, Arch. Math. Logic 42, 389–401.
Negri, S. and von Plato, J. (1998) Cut elimination in the presence of axioms, Bull. Symbolic Logic 4, 418–435.
Negri, S. and von Plato, J. (2001) Structural Proof Theory, Cambridge University Press.
Negri, S. and von Plato, J. (2004) Proof systems for lattice theory, Math. Structures Comput. Sci. 14, 507–526.
Negri, S., von Plato, J. and Coquand, Th. (2004) Proof-theoretical analysis of order relations, Arch. Math. Logic 43, 297–309.
Nerode, A. (1991) Some lectures on modal logic, in F. L. Bauer (ed.), Logic, Algebra, and Computation, NATO ASI Series F79, Springer.
Ohlbach, H. J. (1993) Translation methods for non-classical logics – An overview, Bull. IGPL 1, 69–91.
Popkorn, S. (1994) First Steps in Modal Logic, Cambridge University Press.
Sambin, G. and Valentini, S. (1982) The modal logic of provability. The sequential approach, J. Philos. Logic, 311–342.
Shvarts, G. (1989) Gentzen style systems for K45 and K45D, in Logic at Botik '89 (Pereslavl'-Zalesskiy), Lecture Notes in Comput. Sci. 363, Springer, Berlin, pp. 245–256.
Simpson, A. (1994) Proof theory and semantics of intuitionistic modal logic, Ph.D. thesis, School of Informatics, University of Edinburgh.
Troelstra, A. and Schwichtenberg, H. (1996) Basic Proof Theory, Cambridge University Press, second edition 2000.
Valentini, S. (1983) The modal logic of provability: Cut-elimination, J. Philos. Logic 12, 471–476.
Van Benthem, J. (1984) Correspondence theory, in D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, Vol. 2, Kluwer, pp. 167–247.
Viganò, L. (2000) Labelled Non-Classical Logics, Kluwer.
Wansing, H. (1996) Proof Theory of Modal Logic, Proceedings of the workshop held at the University of Hamburg, Hamburg, November 19–20, 1993, edited by Heinrich Wansing, Applied Logic Series 2, Kluwer Academic Publishers, Dordrecht.
Wansing, H. (2002) Sequent systems for modal logics, in D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, Vol. 8, 2nd edn, Kluwer, pp. 61–145.
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Negri, S. Proof Analysis in Modal Logic. J Philos Logic 34, 507–544 (2005). https://doi.org/10.1007/s10992-005-2267-3
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DOI: https://doi.org/10.1007/s10992-005-2267-3