The Display Problem

  • Nuel Belnap
Part of the Applied Logic Series book series (APLS, volume 2)

Abstract

A Gentzen calculus has the ‘display property’ if every antecedent [consequent] constituent can be displayed as the antecedent [consequent] standing alone. It is explained why this property is interesting. The ‘display problem’ is the problem of designing a calculus with the display property. A perspective is suggested from which the solution of Wansing [19] can easily be seen to be incomparable with that of [4]. The perspective suggests some other solutions, which are briefly surveyed. Additional questions are raised.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Allwein, C. Hartonas. Duality for bounded lattices. Unpublished manuscript, Indiana University Logic Group Preprint Series, 1993.Google Scholar
  2. [2]
    A. R. Anderson, N. D. Belnap. Entailment: the logic of relevance and necessity. Vol. I, Princeton University Press, Princeton, 1975.Google Scholar
  3. [3]
    A. R. Anderson, N. D. Belnap, J. M. Dunn. Entailment: the logic of relevance and necessity. Vol. II, Princeton University Press, Princeton, 1992.Google Scholar
  4. [4]
    N. Belnap. Display logic. Journal of Philosophical Logic,11 375–417, 1982. Republished with minor changes as §62 of [3].Google Scholar
  5. [5]
    N. Belnap. Linear logic displayed. Notre Dame Journal of Formal Logic, 31, 14–25, 1990.CrossRefGoogle Scholar
  6. [6]
    H. B. Curry. A theory of formal deducibility. Notre Dame mathematical lectures, vol. no. 6, Notre Dame University Press, Notre Dame, 1950.Google Scholar
  7. [7]
    J. M. Dunn. Gaggle theory: an abstraction of Galois connections and residuation with applications to negation and various logical operations. In: J. van Eijck (ed.), Logics in Al: Proceedings European Workshop JELJA 1990, pp. 31–51, Lecture Notes in Computer Sci. 478, Springer-Verlag, Berlin, 1990.Google Scholar
  8. [8]
    J. M. Dunn. Partial-gaggles applied to sub-structural logics. In: P. Schroeder-Heister, K. Dosen (eds.), Substructural logics, pp. 63–108, Oxford University Press, Oxford, 1993.Google Scholar
  9. [9]
    J. M. Dunn. A representation of relation algebras using Routley-Meyer frames. Unpublished manuscript, Indiana University Logic Group Preprint Series, 1993.Google Scholar
  10. [10]
    J. M. Dunn. Perp and Star: two treatments of negation. In: J. Tomberlin (ed.), Philosophical Perspectives, vol. 7, Language and Logic, pp. 331–157, 1993.Google Scholar
  11. [11] G. Gentzen. Untersuchungen über das logische Schliessen. Mathenuuische Zeitschrift, 39, 176-210, 405-431,1934. English translation (Investigations into logical deduction)
    by M. E. Szabo in: The Collected Papers of Gerhard Gentzen, M. E. Szabo (ed.), pp. 69–131, North-Holland, Amsterdam, 1969.Google Scholar
  12. [12]
    R. Goré. Displayed intuitionistic modal logics. Technical report in preparation, Australian National University, April, 1994.Google Scholar
  13. [13]
    R. Goré. Displaying sub-intuitionistic logics. Technical report in preparation, Australian National University, April 1994b.Google Scholar
  14. [14]
    R. Goré. Intuitionistic logic redisplayed. Technical report, TR-ARP-1–1995, Australian National University, January 1995.Google Scholar
  15. [15]
    C. Hartonas. Lattices with operators: a unified semantics for substructural logics. Unpublished manuscript, Indiana University Logic Group Preprint Series, 1993.Google Scholar
  16. [16]
    C. Hartonas, J. M. Dunn. Duality theorems for partial orders, semilattices, and Galois connections and lattices. Unpublished manuscript, Indiana University Logic Group Preprint Series, 1993.Google Scholar
  17. [17]
    M. Kracht. Power and weakness of the modal display calculus. Technical report, Free University of Berlin, Mathematics Institute II, August, 1993.Google Scholar
  18. [18]
    R. K. Meyer. A general Gentzen system for implicational calculi. Relevance Logic Newsletter, 1, 198–201, 1976.Google Scholar
  19. [19]
    H. Wansing. Sequent calculi for normal modal propositional logics. Technical report, n. LP-92–12, University of Amsterdam, 1992, Journal of Logic and Computation, 4, 124–142, 1994.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Nuel Belnap
    • 1
  1. 1.University of PittsburghUSA

Personalised recommendations