Abstract
This is a companion paper to Braüner (2004b, Journal of Logic and Computation 14, 329–353) where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order hybrid logic.
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References
Areces, C., Blackburn, P., and Marx, M., 2003, “Repairing the interpolation theorem in quantified modal logic,” Annals of Pure and Applied Logic 124, 287–299.
Basin, D., Matthews, S., and Viganò, L., 1997, “Labelled propositional modal logics: Theory and practice,” Journal of Logic and Computation 7, 685–717.
Basin, D., Matthews, S., and Viganò, L., 1998, “Labelled modal logics: Quantifiers,” Journal of Logic, Language and Information 7, 237–263.
Bencivenga, E., 2002, “Free logics,” in Hand-book of Philosophical Logic, 2nd Edition, Vol. 5., D. Gabbay and F. Guenthner, eds., Dordrecht: Kluwer Academic Publishers, pp. 147–196.
Blackburn, P., 2000, “Internalizing labelled deduction,” Journal of Logic and Computation 10, 137–168.
Blackburn, P. and Marx, M., 2002, “Tableaux for quantified hybrid logic,” pp. 38–52 in Automated Reasoning with Analytic Tableaux and Related Methods, TABLEAUX 2002, Vol. 2381, U. Egly and C. Fermüller (eds.), Lecture Notes in Artificial Intelligence, Springer-Verlag.
Blackburn, P., de Rijke, M., and Venema, Y., 2001, Modal Logic, Vol. 53, Cambridge Tracts in Theoretical Computer Science, Cambridge: Cambridge University Press.
Braüner, T., 2002a, “Modal logic, truth, and the master modality,” Journal of Philosophical Logic 31, 359–386. Revised and extended version of paper in Advances in Modal Logic, Vol. 3, World Scientific.
Braüner, T., 2002b, “Natural deduction for first-order hybrid logic,” pp. 37–51 in Workshop Proceedings of Fourth Workshop on Hybrid Logics. Affiliated to IEEE Symposium on Logic in Computer Science, C. Areces, P. Blackburn, M. Marx, and U. Sattler (eds.).
Braüner, T., 2004a, “Axioms for classical and constructive hybrid logic,” Manuscript, Hard-copies available from the author.
Braüner, T., 2004b, “Natural deduction for hybrid logic,” Journal of Logic and Computation 14, 329–353. Revised and extended version of paper in Workshop Proceedings of Methods for Modalities 2.
Braüner, T., 2004c, “Two natural deduction systems for hybrid logic: A comparison,” Journal of Logic, Language and Information 13, 1–23.
Braüner, T. and de Paiva, V., 2003, “Towards constructive hybrid logic (Extended Abstract),” p. 15 in Workshop Proceedings of Methods for Modalities 3. C. Areces and P. Blackburn, eds.
Braüner, T. and Ghilardi, S., 2005, “First-order modal logic,” in Handbook of Modal Logic, P. Blackburn, J. van Benthem, and F. Wolter, eds., Elsevier, in preparation.
Fitting, M. and Mendelsohn, R., 1998, First-Order Modal Logic, Dordrecht: Kluwer Academic Publishers.
Garson, J., 2001, “Quantification in modal logic,” pp. 267–323 in Handbook of Philosophical Logic, 2nd Edition, Vol. 3., D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers.
Gentzen, G., 1969, “Investigations into logical deduction,” pp. 68–131 in The Collected Papers of Gerhard Gentzen. M.E. Szabo, ed., North-Holland Publishing Company.
Hasle, P., Oslash;hrstrOslash;m, P., Braüner, T., and Copeland, J., eds., 2003, Revised and Expanded Edition of Arthur N. Prior: Papers on Time and Tense, Oxford University Press, 342 pp.
Hughes, G. and Cresswell, M., 1996, A New Introduction to Modal Logic, Routledge.
Oslash;hrstrOslash;m, P. and Hasle, P., 1993, “A. N. prior’s rediscovery of tense logic,” Erkenntnis 39, 23–50.
Prawitz, D., 1965, Natural Deduction. A Proof-Theoretical Study, Stockholm: Almqvist and Wiksell.
Prawitz, D., 1971, “Ideas and results in proof theory,” pp. 235–307 in Proceedings of the Second Scandinavian Logic Symposium, Vol. 63 of Studies in Logic and The Foundations of Mathematics, J.E. Fenstad, ed., North-Holland.
Prior, A., 1968, Papers on Time and Tense, Clarendon/Oxford University Press.
Seligman, J., 1997, “The logic of correct description,” pp. 107–135 in Advances in Intensional Logic, M. de Rijke, ed., Applied Logic Series, Dordrecht: Kluwer Academic Publishers.
Simpson, A., 1994, “The proof theory and semantics of intuitionistic modal logic,” Ph.D. Thesis, University of Edinburgh.
Troelstra, A. and Schwichtenberg, H., 1996, Basic Proof Theory, Vol. 43, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.
Vickers, S., 1988, Topology via Logic, Vol. 5, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.
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BraÜner, T. Natural Deduction for First-Order Hybrid Logic. J Logic Lang Inf 14, 173–198 (2005). https://doi.org/10.1007/s10849-005-3927-y
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DOI: https://doi.org/10.1007/s10849-005-3927-y