Analytic Sequent Calculi for Abelian and Łukasiewicz Logics

  • George Metcalfe
  • Nicola Olivetti
  • Dov Gabbay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2381)

Abstract

In this paper we present the first labelled and unlabelled analytic sequent calculi for abelian logic A, the logic of lattice-ordered abelian groups with characteristic model ℤ, motivated in [10] as a logic of relevance and in [3] as a logic of comparison. We also show that the so-called material fragment of A coincides with Łukasiewicz’s infinite-valued logic Ł, hence giving us as a significant by-product, labelled and unlabelled analytic sequent calculi for Ł.

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References

  1. 1.
    S. Aguzzoli and A. Ciabattoni. Finiteness in infinite-valued Łukasiewicz logic. Journal of Logic, Language and Information, 9(1), 2000.Google Scholar
  2. 2.
    A. R. Anderson and N. D. Belnap Jr. Entailment: The Logic of Relevance and Necessity, Volume 1. Princeton, University Press, 1975.MATHGoogle Scholar
  3. 3.
    E. Casari. Comparative Logics and Abelian l-Groups. In Logic Colloquium’ 88, Ferro et al. Ed, Elsevier, 1989.Google Scholar
  4. 4.
    A. Ciabattoni and D. Luchi. Two connections between linear logic and Łukasiewicz logics. In Proceedings of Computational Logic and Proof Theory, G. Gottlöb, A. Leitsch and D. Mundici Eds, Lecture Notes in Computer Science, Vol. 1289, Berlin: Springer-Verlag, 1997.Google Scholar
  5. 5.
    R. Cignoli, I. M. L. D’Octaviano and D. Mundici. Algebraic foundations of many-valued reasoning. Kluwer, 2000.Google Scholar
  6. 6.
    R. Giles. Łukasiewicz logic and fuzzy set theory. In Fuzzy Reasoning and its Applications, E. H. Mamdani and B. R. Gaines Eds, London: Academic Press, 1981.Google Scholar
  7. 7.
    Reiner Hähnle. Automated Deduction in Many-Valued Logics. Oxford University Press, 1994.Google Scholar
  8. 8.
    P. Hájek. Metamathematics of Fuzzy Logic. Kluwer, 1998.Google Scholar
  9. 9.
    G. Malinowski. Many-Valued Logics. Oxford, 1993.Google Scholar
  10. 10.
    R. K. Meyer and K. Slaney. Abelian Logic (from A to Z). In Paraconsistent Logic Essays on the Inconsistent, G. Priest et al. Ed, Philosophia Verlag, 1989.Google Scholar
  11. 11.
    R. K. Meyer. Intuitionism, entailment, negation. In Truth, Syntax, Modality, H. Leblanc Ed, Amsterdam, 1973.Google Scholar
  12. 12.
    D. Mundici and N. Olivetti. Resolution and model building in the infinite-valued calculus of Łukasiewicz. Theoretical Computer Science, 200(1–2), 1998.Google Scholar
  13. 13.
    N. Olivetti. Tableaux for infinite-valued Łukasiewicz logic. To appear in Studia Logica, 2002.Google Scholar
  14. 14.
    A. Prijatelj. Bounded Contraction and Gentzen style formulation of Łukasiewicz Logics. Studia Logica, 57, 437–456, 1996.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Urquhart. Many-valued logic. In Handbook of Philosophical Logic Volume 3, Dov Gabbay and F. Guenthner Eds, Kluwer, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • George Metcalfe
    • 1
  • Nicola Olivetti
    • 2
  • Dov Gabbay
    • 1
  1. 1.Department of Computer ScienceKing’s College LondonLondonUK
  2. 2.Department of Computer ScienceUniversity of TurinTurinItaly

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