Free-Variable Tableaux for Constant-Domain Quantified Modal Logics with Rigid and Non-rigid Designation
This paper presents a sound and complete free-variable tableau calculus for constant-domain quantified modal logics, with a propositional analytical basis, i.e. one of the systems K, D, T, K4, S4. The calculus is obtained by addition of the classical free-variable γ-rule and the “liberalized” δ+-rule  to a standard set of propositional rules. Thus, the proposed system characterizes proof-theoretically the constant-domain semantics, which cannot be captured by “standard” (non-prefixed, non-annotated) ground tableau calculi. The calculi are extended so as to deal also with non-rigid designation, by means of a simple numerical annotation on functional symbols, conveying some semantical information about the worlds where they are meant to be interpreted.
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- 5.S. Cerrito. Herbrand methods for sequent calculi; unification in LL. In K. Apt, editor, Proc. of the Joint International Conference and Symposium on Logic Programming, pages 607–622, 1992.Google Scholar
- 7.M. Cialdea Mayer and S. Cerrito. Variants of first-order modal logics. In R. Dyckhoff, editor, Proc. of Tableaux 2000, LNCS. Springer Verlag, 2000.Google Scholar
- 8.K. Fine. Failures of the interpolation lemma in quantified modal logic. Journal of Symbolic Logic, 44(2), 1979.Google Scholar
- 9.M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel Publishing Company, 1983.Google Scholar
- 12.J. W. Garson. Quantification in modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume II, pages 249–307. D. Reidel Publ. Co., 1984.Google Scholar
- 13.R. Goré. Tableau methods for modal and temporal logics. In M. D’Agostino, G. Gabbay, R. Hähnle, and J. Posegga, editors, Handbook of tableau methods. Kluwer, 1999.Google Scholar
- 15.P. Jackson and H. Reichgelt. A general proof method for first-order modal logic. In Proc. of the 10th Joint Conf. on Artificial Intelligence (IJCAI’ 87), pages 942–944. Morgan Kaufmann, 1987.Google Scholar
- 18.P. D. Lincoln and N. Shankar. Proof search in fit order linear logic and other cut-free sequent calculi. In S. Abramsky, editor, Proc. of the Ninth Annual IEEE Symposium on Logic In Computer Science (LICS), pages 282-91. IEE Computer Society Press, 1994.Google Scholar
- 20.N. Shankar. Proof search in the intuitionist sequent calculus. In D. Kapur, editor, Proc. of the Eleventh International Conference on Automated Deduction (CADE), number 607 in LNCS, pages 522-36. Springer Verlag, 1992.Google Scholar
- 21.L. A. Wallen. Automated Deduction in Nonclassical Logics: Efficient Matrix Proof Methods for Modal and Intuitionistic Logics. MIT Press, 1990.Google Scholar