A Multiple-Conclusion Calculus for First-Order Gödel Logic

  • Arnon Avron
  • Ori Lahav
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6651)

Abstract

We present a multiple-conclusion hypersequent system for the standard first-order Gödel logic. We provide a constructive, direct, and simple proof of the completeness of the cut-free part of this system, thereby proving both completeness for its standard semantics, and the admissibility of the cut rule in the full system. The results also apply to derivations from assumptions (or “non-logical axioms”), showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions. Finally, the results about the multiple-conclusion system are used to show that the usual single-conclusion system for the standard first-order Gödel logic also admits (strong) cut-admissibility.

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References

  1. 1.
    Avellone, A., Ferrari, M., Miglioli, P.: Duplication-free Tableaux Calculi Together with Cut-free and Contraction-free Sequent Calculi for the Interpolable Propositional Intermediate Logics. Logic J. IGPL 7, 447–480 (1999)CrossRefMATHGoogle Scholar
  2. 2.
    Avron, A.: Using Hypersequents in Proof Systems for Non-classical Logics. Annals of Mathematics and Artificial Intelligence 4, 225–248 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Avron, A.: Gentzen-Type Systems, Resolution and Tableaux. Journal of Automated Reasoning 10, 265–281 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Avron, A.: A Simple Proof of Completeness and Cut-admissibility for Propositional Gödel Logic. Journal of Logic and Computation (2009), doi:10.1093/logcom/exp055Google Scholar
  5. 5.
    Avron, A., Konikowska, B.: Decomposition Proof Systems for Gödel Logics. Studia Logica 69, 197–219 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Baaz, M., Ciabattoni, A., Fermüller, C.G.: Hypersequent Calculi for Gödel Logics - a Survey. Journal of Logic and Computation 13, 835–861 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baaz, M., Preining, N., Zach, R.: First-order Gödel Logics. Annals of Pure and Applied Logic 147, 23–47 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Baaz, M., Zach, R.: Hypersequents and the Proof Theory of Intuitionistic Fuzzy Logic. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 187–201. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Ciabattoni, A., Galatos, N., Terui, K.: From Axioms to Analytic Rules in Nonclassical Logics. In: Proceedings of LICS, pp. 229–240 (2008)Google Scholar
  10. 10.
    Corsi, G.: Semantic Trees for Dummett’s Logic LC. Studia Logica 45, 199–206 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dyckhoff, D.: A Deterministic Terminating Sequent Calculus for Gödel-Dummett Logic. Logic J. IGPL 7, 319–326 (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Dyckhoff, D., Negri, S.: Decision Methods for Linearly Ordered Heyting Algebras. Archive for Mathematical Logic 45, 411–422 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dummett, M.: A Propositional Calculus with a Denumerable matrix. Journal of Symbolic Logic 24, 96–107 (1959)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gabbay, D.: Semantical Investigations in Heyting’s Intuitionistic Logic. Reidel, Dordrechtz (1983)MATHGoogle Scholar
  15. 15.
    Gödel, K.: On the Intuitionistic Propositional Calculus. In: Feferman, S., et al. (eds.) Collected Work, vol. 1, Oxford University Press, Oxford (1986)Google Scholar
  16. 16.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)CrossRefMATHGoogle Scholar
  17. 17.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer, Heidelberg (2009)MATHGoogle Scholar
  18. 18.
    Sonobe, O.: A Gentzen-type Formulation of Some Intermediate Propositional Logics. Journal of Tsuda College 7, 7–14 (1975)MathSciNetGoogle Scholar
  19. 19.
    Takano, M.: P Another proof of the strong completeness of the intuitionistic fuzzy logic. Tsukuba J. Math. 11, 851–866 (1984)MathSciNetGoogle Scholar
  20. 20.
    Takeuti, G.: Proof Theory. North-Holland, Amsterdam (1975)MATHGoogle Scholar
  21. 21.
    Takeuti, G., Titani, T.: Intuitionistic Fuzzy Logic and Intuitionistic Fuzzy Set Theory. Journal of Symbolic Logic 49, 851–866 (1984)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Arnon Avron
    • 1
  • Ori Lahav
    • 1
  1. 1.School of Computer ScienceTel-Aviv UniversityIsrael

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