Abstract
We give complete sequent-like tableau systems for the modal logics KB, KDB, K5, and KD5. Analytic cut rules are used to obtain the completeness. Our systems have the analytic superformula property and can thus give a decision procedure. Using the systems, we prove the Craig interpolation lemma for the mentioned logics.
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References
Chellas, B., 1980, Modal Logic: An Introduction, Cambridge Univ. Press.
Craig, W., 1957, 'Linear reasoning. A new form of the Herbrand-Gentzen theorem', pp. 250-268. Three uses of the Hebrand-Gentzen theorem in relating model theory to proof theory, pp. 269-285', Journal of Symbolic Logic 22.
Fitting, M., 1983, Proof Methods for Modal and Intuitionistic Logics, volume 169 of Synthese Library, D. Reidel, Dordrecht, Holland.
Gabbay, D.: 1972, 'Craig's interpolation theorem for modal logics', in: Conference in Mathematical Logic — London'70, Lecture Notes in Mathematics vol 255, pp. 111-127.
GorÉ, R.: 1999, 'Tableau methods for modal and temporal logics', in: D'Agostino, Gabbay, Hähnle, Posegga (eds.), Handbook of Tableau Methods, Kluwer Academic Publishers, pp. 297-396.
Hintikka, K., 1955, 'Form and content in quantification theory', Acta Philosophica Fennica 8, 3-55.
Hudelmaier, J., 1996, 'Improved decision procedures for the modal logics K, T, S4', in: H. Kleine Büuning, editor, Proceedings of CSL'95, LNCS 1092, pp. 320-334.
Hughes, G. and M. Cresswell, 1968, An Introduction to Modal Logic, Methuen.
Kripke, S., 1963, 'Semantical analysis of modal logic, I. Normal modal propositional calculus', Z. Math. Logic Grundlag 9, 67-96.
Maksimova, L., 1991, 'Amalgamation and interpolation in normal modal logics', Studia Logica 50, 458-471.
Massacci, F., 1994, 'Strongly analytic tableaux for normal modal logics', in: A. Bundy, editor, Proceedings of CADE-12, LNAI 814, pp. 723-737.
Nguyen, L., 2000a, 'Clausal tableau systems and space bounds for the modal logics K, KD, T, KB, KDB, and B', Technical Report TR 00-01(261), Warsaw University.
Nguyen, L., 2000b, 'Sequent-Like tableau systems with the analytic superformula property for the modal logics KB, KDB, K5, KD5', in: R. Dyckhoff, editor, Proceedings of TABLEAUX 2000, LNAI 1847, pp. 341-351.
Rautenberg, W., 1983, 'Modal tableau calculi and interpolation', Journal of Philosophical Logic 12, 403-423.
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Nguyen, L.A. Analytic Tableau Systems and Interpolation for the Modal Logics KB, KDB, K5, KD5. Studia Logica 69, 41–57 (2001). https://doi.org/10.1023/A:1013834410884
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DOI: https://doi.org/10.1023/A:1013834410884