Studia Logica

, Volume 62, Issue 1, pp 49–75 | Cite as

Predicate Logics on Display

  • Heinrich Wansing

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.

predicate logic quantifiers display logic modal logic 

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References

  1. [1]
    Alechina, N., and M. van Lambalgen, Generalized Quantification as Substructural Logic, Technical Report, ILLC, University of Amsterdam, 1995.Google Scholar
  2. [2]
    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.Google Scholar
  3. [3]
    Belnap, N., ‘Display Logic’, Journal of Philosophical Logic 11, 375–417, 1982.Google Scholar
  4. [4]
    Belnap, N., ‘Linear Logic Displayed’ Notre Dame Journal of Formal Logic 31, 14–25, 1990.Google Scholar
  5. [5]
    Belnap, N., ‘The Display problem’, in H. Wansing (ed.), Proof Theory of Modal Logic, 79–93, Dordrecht, Kluwer Academic Publishers, 1996.Google Scholar
  6. [6]
    van Benthem, J., Modal Foundations of Predicate Logic, Center for the Study of Language and Information, Report No. CSLI-94-191, Stanford University, 1994.Google Scholar
  7. [7]
    van Benthem, J., Exploring Logical Dynamics, CSLI Publications, Stanford, 1996.Google Scholar
  8. [8]
    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.Google Scholar
  9. [9]
    Enderton, H., A Mathematical Introduction to Logic, Academic Press, New York, Reidel, Dordrecht, 1972.Google Scholar
  10. [10]
    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.Google Scholar
  11. [11]
    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.Google Scholar
  12. [12]
    Goldblatt, R., Topoi. The Categorical Analysis of Logic, Amsterdam, North-Holland, 1979.Google Scholar
  13. [13]
    GorŔ, R., ‘Intuitionistic logic redisplayed’ (manuscript), Department of Computer Science, University of Manchester, 1994.Google Scholar
  14. [14]
    GorÉ, R., A Uniform Display System for Intuitionistic and Dual Intuitionistic Logic, Technical Report TR-SRS-2-95, Australian National University, 1995.Google Scholar
  15. [15]
    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.Google Scholar
  16. [16]
    GorÉ, R., Cut-free Display Calculi for Relation Algebras, Technical Report TR-ARP-19-95, Australian National University, Canberra, 1995.Google Scholar
  17. [17]
    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.Google Scholar
  18. [18]
    Hughes, G. E., and M. J. Cresswell, An Introduction to Modal Logic, Methuen, London, 1968.Google Scholar
  19. [19]
    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.Google Scholar
  20. [20]
    Kuhn, S., ‘Quantifiers as modal operators’, Studia Logica 39, 145–158, 1980.Google Scholar
  21. [21]
    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.Google Scholar
  22. [22]
    Marx, M., and Y. Venema, A Modal Logic of Relations, Technical Report IR-396, Vrije Universiteit Amsterdam, 1995 (to appear in: Studia Logica).Google Scholar
  23. [23]
    Meyer-Viol, W., Instantial Logic, PhD thesis, University of Amsterdam, Institute of Logic, Language and Computationj, 1995.Google Scholar
  24. [24]
    Montague, R., ‘Logical necessity, physical necessity, ethics and quantifiers’, Inquiry, 4, 259–269, 1960.Google Scholar
  25. [25]
    NÉmeti, I., ‘Algebraizations of quantifier logics: an introductory overview’, Studia Logica 50, 485–570, 1991.Google Scholar
  26. [26]
    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).Google Scholar
  27. [27]
    Restall, G., ‘Display logic and gaggle theory’, Reports on Mathematical Logic 29, 133–146, 1995 (published 1996).Google Scholar
  28. [28]
    Roorda, D., ‘Dyadic modalities and Lambek calculus’, in: M. de Rijke (ed.), Diamonds and Defaults, 215–253, Dordrecht, Kluwer Academic Publishers, 1993.Google Scholar
  29. [29]
    de Rijke, M., Extending Modal Logic, PhD thesis, University of Amsterdam, Institute of Logic, Language and Computationj, 1993.Google Scholar
  30. [30]
    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.Google Scholar
  31. [31]
    Venema, Y., Many-Dimensional Modal Logic, PhD thesis, University of Amsterdam, Math. Institutue, 1991.Google Scholar
  32. [32]
    Venema, Y., ‘Cylindric modal logic’, Journal of Symbolic Logic 60, 591–623, 1995.Google Scholar
  33. [33]
    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.Google Scholar
  34. [34]
    Wang, H., A Survey of Mathematical Logic, Peking, Science Press, 1962.Google Scholar
  35. [35]
    Wansing, H., ‘Sequent calculi for normal modal propositional logics’, Journal of Logic and Computation, 4, 125–142, 1994.Google Scholar
  36. [36]
    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.Google Scholar
  37. [37]
    Wansing, H., ‘A new axiomatization of K t’, Bulletin of the Section of Logic 25, 1996, 60–62.Google Scholar
  38. [38]
    Wansing, H., ‘Strong cut-elimination in Display Logic’, Reports on Mathematical Logic 29, 117–131, 1995 (published 1996).Google Scholar
  39. [39]
    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.Google Scholar
  40. [40]
    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.Google Scholar

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© Kluwer Academic Publishers 1999

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  • Heinrich Wansing

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