Abstract
The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for ∀x and ∃x are analogous to the display introduction rules for the modal operators □ and ♦ and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal ‘modal’ predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.
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References
Alechina, N., and M. van Lambalgen, Generalized Quantification as Substructural Logic, Technical Report, ILLC, University of Amsterdam, 1995.
AndrÉka, H., J. van Benthem and I. NÉmeti, ‘Back and forth between modal logic and classical logic’, Journal of the Interest Group in Pure and Applied Logic 3, 685–720, 1995.
Belnap, N., ‘Display Logic’, Journal of Philosophical Logic 11, 375–417, 1982.
Belnap, N., ‘Linear Logic Displayed’ Notre Dame Journal of Formal Logic 31, 14–25, 1990.
Belnap, N., ‘The Display problem’, in H. Wansing (ed.), Proof Theory of Modal Logic, 79–93, Dordrecht, Kluwer Academic Publishers, 1996.
van Benthem, J., Modal Foundations of Predicate Logic, Center for the Study of Language and Information, Report No. CSLI-94-191, Stanford University, 1994.
van Benthem, J., Exploring Logical Dynamics, CSLI Publications, Stanford, 1996.
Dunn, J. M., ‘Gaggle theory: an abstraction of Galois connections and residuation with applications to negation and various logical operations’, in J. van Eijk (ed.), Logics in AI, Proceedings European Workshop JELIA 1990, Lecture Notes in Computer Science 478, 31–51, Berlin, Springer-Verlag, 1990.
Enderton, H., A Mathematical Introduction to Logic, Academic Press, New York, Reidel, Dordrecht, 1972.
Fitting, M., ‘Basic modal logic’, in: D. M. Gabbay et al. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming. Vol. 1, Logical Foundation, Oxford UP, Oxford, 365–448, 1993.
Gabbay D., ‘A general theory of structured consequence relations’, in K. Došen and P. Schroeder-Heister (eds.), Substructural Logics, 109–151, Oxford, Clarendon Press, 1994.
Goldblatt, R., Topoi. The Categorical Analysis of Logic, Amsterdam, North-Holland, 1979.
GorŔ, R., ‘Intuitionistic logic redisplayed’ (manuscript), Department of Computer Science, University of Manchester, 1994.
GorÉ, R., A Uniform Display System for Intuitionistic and Dual Intuitionistic Logic, Technical Report TR-SRS-2-95, Australian National University, 1995.
GorÉ, R., ‘On the completeness of classical modal display lolgic’, in H. Wansing (ed.), Proof Theory of Modal Logic, 137–140, Dordrecht, Kluwer Academic Publishers, 1996.
GorÉ, R., Cut-free Display Calculi for Relation Algebras, Technical Report TR-ARP-19-95, Australian National University, Canberra, 1995.
Grishin, V.N., ‘A nonstandard logic and its application to set theory’ (in Russian), in: Studies in Formalized Languages and Nonclassical Logics (in Russian), Moscow, Nauka, 135–171, 1974.
Hughes, G. E., and M. J. Cresswell, An Introduction to Modal Logic, Methuen, London, 1968.
Kracht, M., ‘Power and weakness of the modal display calculus’, in H. Wansing (ed.), Proof Theory of Modal Logic, 95–122, Dordrecht, Kluwer Academic Publishers, 1996.
Kuhn, S., ‘Quantifiers as modal operators’, Studia Logica 39, 145–158, 1980.
van Lambalgen, M., ‘Natural deduction for generalized quantifiers’, in J. van der Does and J. van Eijck (eds.), Generalized Quantifier Theory and Applications, 143–154, Dutch Network for Language, Logic and Information, Amsterdam, 1991.
Marx, M., and Y. Venema, A Modal Logic of Relations, Technical Report IR-396, Vrije Universiteit Amsterdam, 1995 (to appear in: Studia Logica).
Meyer-Viol, W., Instantial Logic, PhD thesis, University of Amsterdam, Institute of Logic, Language and Computationj, 1995.
Montague, R., ‘Logical necessity, physical necessity, ethics and quantifiers’, Inquiry, 4, 259–269, 1960.
NÉmeti, I., ‘Algebraizations of quantifier logics: an introductory overview’, Studia Logica 50, 485–570, 1991.
Restall, G., Displaying and Deciding Substructural Logics 1: Logics with Contraposition, Technical Report TR-ARP-11-94, Australian National University, Canberra, 1994 (to appear in: Journal of Philosophical Logic).
Restall, G., ‘Display logic and gaggle theory’, Reports on Mathematical Logic 29, 133–146, 1995 (published 1996).
Roorda, D., ‘Dyadic modalities and Lambek calculus’, in: M. de Rijke (ed.), Diamonds and Defaults, 215–253, Dordrecht, Kluwer Academic Publishers, 1993.
de Rijke, M., Extending Modal Logic, PhD thesis, University of Amsterdam, Institute of Logic, Language and Computationj, 1993.
Ryan, M., P.-Y. Schobbens and O. Rodrigues, ‘Counterfactuals and updates as inverse modalities’, in: Y. Shoam (ed.), TARK '96. Proceedings Theoretical Aspects of Rationality and Knowledge, 163–173, San Francisco, Morgan Kaufmann, 1996.
Venema, Y., Many-Dimensional Modal Logic, PhD thesis, University of Amsterdam, Math. Institutue, 1991.
Venema, Y., ‘Cylindric modal logic’, Journal of Symbolic Logic 60, 591–623, 1995.
Venema, Y., ‘A modal logic of quantification and substitution’, in L. Czirmaz, D. Gabbay and M. de Rijke (eds.), Logic Colloquium '92, 293–309, CSLI Publications, 1995.
Wang, H., A Survey of Mathematical Logic, Peking, Science Press, 1962.
Wansing, H., ‘Sequent calculi for normal modal propositional logics’, Journal of Logic and Computation, 4, 125–142, 1994.
Wansing, H., ‘A proof-theoretic proof of functional completeness for many modal and tense logics’, in H. Wansing (ed.), Proof Theory of Modal Logic, 123–136, Dordrecht, Kluwer Academic Publishers, 1996.
Wansing, H., ‘A new axiomatization of K t’, Bulletin of the Section of Logic 25, 1996, 60–62.
Wansing, H., ‘Strong cut-elimination in Display Logic’, Reports on Mathematical Logic 29, 117–131, 1995 (published 1996).
Wansing, H., ‘A full-circle theorem for simple tense logic’, in M. de Rijke (ed.), Advances in Intensional Logic, Kluwer Academic Publishers, Dordrecht, 171–190, 1997.
Wansing, H., ‘Displaying as temporalizing. Sequent systems for subintuitionistic logics’, in S. Akama (ed.), Logic, Language and Computation, Kluwer Academic Publishers, Dordrecht, 159–178, 1997.
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Wansing, H. Predicate Logics on Display. Studia Logica 62, 49–75 (1999). https://doi.org/10.1023/A:1005125717300
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DOI: https://doi.org/10.1023/A:1005125717300