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Abstract

A basic propositional modal fuzzy logic \({\sf {GK}}_{\Box}\) is defined by combining the Kripke semantics of the modal logic K with the many-valued semantics of Gödel logic G. A sequent of relations calculus is introduced for \({\sf {GK}}_{\Box}\) and a constructive counter-model completeness proof is given. This calculus is used to establish completeness for a Hilbert-style axiomatization and Gentzen-style hypersequent calculus admitting cut-elimination, and to show that the logic is PSPACE-complete.

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References

  1. 1.
    Avron, A.: A constructive analysis of RM. Journal of Symbolic Logic 52(4), 939–951 (1987)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4(3–4), 225–248 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baaz, M., Ciabattoni, A., Fermüller, C.G.: Hypersequent calculi for Gödel logics: a survey. Journal of Logic and Computation 13, 1–27 (2003)CrossRefGoogle Scholar
  4. 4.
    Baaz, M., Fermüller, C.G.: Analytic calculi for projective logics. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 36–50. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Bou, F., Esteva, F., Godo, L., Rodríguez, R.: On the minimum many-valued logic over a finite residuated lattice (manuscript)Google Scholar
  6. 6.
    Caicedo, X., Rodríguez, R.: A Gödel modal logic (manuscript)Google Scholar
  7. 7.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1996)Google Scholar
  8. 8.
    Ciabattoni, A., Metcalfe, G., Montagna, F.: Adding modalities to MTL and its extensions. In: Proceedings of the 26th Linz Symposium (to appear)Google Scholar
  9. 9.
    Dummett, M.: A propositional calculus with denumerable matrix. Journal of Symbolic Logic 24, 97–106 (1959)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dyckhoff, R.: A deterministic terminating sequent calculus for Gödel-Dummett logic. Logic Journal of the IGPL 7(3), 319–326 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fitting, M.C.: Many-valued modal logics. Fundamenta Informaticae 15(3-4), 235–254 (1991)MATHMathSciNetGoogle Scholar
  12. 12.
    Fitting, M.C.: Many-valued modal logics II. Fundamenta Informaticae 17, 55–73 (1992)MATHMathSciNetGoogle Scholar
  13. 13.
    Gödel, K.: Zum intuitionisticschen Aussagenkalkül. Anzeiger Akademie der Wissenschaften Wien, mathematisch-naturwiss. Klasse 32, 65–66 (1932)Google Scholar
  14. 14.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)MATHGoogle Scholar
  15. 15.
    Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Applied Logic, vol. 36. Springer, Heidelberg (2009)MATHGoogle Scholar
  17. 17.
    Priest, G.: Many-valued modal logics: a simple approach. Review of Symbolic Logic 1, 190–203 (2008)Google 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.
    Straccia, U.: Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research 14, 137–166 (2001)MATHMathSciNetGoogle Scholar
  20. 20.
    Wolter, F.: Superintuitionistic companions of classical modal logics. Studia Logica 58(2), 229–259 (1997)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Zhang, Z., Sui, Y., Cao, C., Wu, G.: A formal fuzzy reasoning system and reasoning mechanism based on propositional modal logic. Theoretical Computer Science 368(1-2), 149–160 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • George Metcalfe
    • 1
  • Nicola Olivetti
    • 2
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2.LSIS-UMR CNRS 6168Université Paul CézanneMarseille Cedex 20France

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