Parallel Dialogue Games and Hypersequents for Intermediate Logics

  • Christian G. Fermüller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2796)

Abstract

A parallel version of Lorenzen’s dialogue theoretic foundation for intuitionistic logic is shown to be adequate for a number of important intermediate logics. The soundness and completeness proofs proceed by relating hypersequent derivations to winning strategies for parallel dialogue games. This also provides a computational interpretation of hypersequents.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramsky, S., Jagadeesan, R.: Games and Full Completeness for Multiplicative Linear Logic. J. Symbolic Logic 59(2), 543–574 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blass, A.: A Game Semantics for Linear Logic. Annals of Pure and Applied Logic 56, 183–220 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4, 225–248 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ciabattoni, A., Ferrari, M.: Hypersequent calculi for some intermediate logics with bounded Kripke models. Journal of Logic and Computation 2(11), 283–294 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ciabattoni, A., Gabbay, D.M., Olivetti, N.: Cut-free proof systems for logics of weak excluded middle. Soft Computing 2(4), 147–156 (1998)Google Scholar
  6. 6.
    Dummett, M.: A propositional calculus with denumerable matrix. J. Symbolic Logic 24, 97–106 (1959)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dunn, J.M., Meyer, R.K.: Algebraic completeness results for Dummett’s LC and its extensions. Z. Math. Logik Grundlagen Math. 17, 225–230 (1971)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Felscher, W.: Dialogues, Strategies, and Intuitionistic Provability. Annals of Pure and Applied Logic 28, 217–254 (1985)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Felscher, W.: Dialogues as Foundation for Intuitionistic Logic. In: Gabbay, D., Günther, F. (eds.) Handbook of Philosophical Logic, vol. III, pp. 341–372. Reidel, Dordrecht (1986)Google Scholar
  10. 10.
    Fermüller, C.G., Ciabattoni, A.: From Intuitionistic Logic to Gödel-Dummett Logic via Parallel Dialogue Games. In: 33rd Intl.Symp.on Multiple-Valued Logic, Tokyo, May 2003, pp. 188–193. IEEE Press, Los Alamitos (2003)CrossRefGoogle Scholar
  11. 11.
    Gödel, K.: Zum intuitionistischen Aussagenkalkül. Anz. Akad. Wiss. Wien 69, 65–66 (1932)Google Scholar
  12. 12.
    Jankov, V.: The calculus of the weak “law of excluded middle”. Mathematics of the USSR 8, 648–658 (1968)Google Scholar
  13. 13.
    Krabbe, E.C.W.: Formal Systems of Dialogue Rules. Synthese 63, 295–328 (1985)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Krabbe, E.C.W.: Dialogue Sequents and Quick Proofs of Completeness. In: Hoepelman, J.P. (ed.) Representation and Reasoning, pp. 135–140. Max Niemeyer Verlag (1988)Google Scholar
  15. 15.
    Lorenzen, P.: Logik und Agon. In: Atti Congr. Internat. di Filosofia, Sansoni, Firenze, vol. 4, pp. 187–194 (1960)Google Scholar
  16. 16.
    Rahman, S.: Über Dialoge, Protologische Kategorien und andere Seltenheiten. Europäische Hochschulschriften, Peter Lang (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Christian G. Fermüller
    • 1
  1. 1.Technische Universität WienAustria

Personalised recommendations