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Abstract

We define the notions of a canonical inference rule and a canonical constructive system in the framework of strict single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and develop a corresponding general non-deterministic Kripke-style semantics. We show that every constructive canonical system induces a class of non-deterministic Kripke-style frames, for which it is strongly sound and complete. This non-deterministic semantics is used to show that such a system always admits a strong form of the cut-elimination theorem, and for providing a decision procedure for such systems.

Keywords

Atomic Formula Intuitionistic Logic Natural Deduction Horn Clause Canonical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Avron, A.: Simple Consequence Relations. Information and Computation 92, 105–139 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Avron, A.: Gentzen-Type Systems, Resolution and Tableaux. Journal of Automated Reasoning 10, 265–281 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avron, A.: A Nondeterministic View on Nonclassical Negations. Studia Logica 80, 159–194 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Avron, A.: Non-deterministic Semantics for Families of Paraconsistent Logics. In: Beziau, J.-Y., Carnielli, W., Gabbay, D.M. (eds.) Handbook of Paraconsistency. Studies in Logic, vol. 9, pp. 285–320. College Publications (2007)Google Scholar
  5. 5.
    Avron, A., Lev, I.: Canonical Propositional Gentzen-Type Systems. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 529–544. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Avron, A., Lev, I.: Non-deterministic Multiple-valued Structures. Journal of Logic and Computation 15, 24–261 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Belnap, N.D.: Tonk, Plonk and Plink. Analysis 22, 130–134 (1962)CrossRefGoogle Scholar
  8. 8.
    Ciabattoni, A., Terui, K.: Towards a Semantic Characterization of Cut-Elimination. Studia Logica 82, 95–119 (2006)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fernandez, D.: Non-deterministic Semantics for Dynamic Topological Logic. Annals of Pure and Applied Logic 157, 110–121 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gentzen, G.: Investigations into Logical Deduction. In: Szabo, M.E. (ed.) The Collected Works of Gerhard Gentzen, pp. 68–131. North Holland, Amsterdam (1969)Google Scholar
  11. 11.
    Gurevich, Y., Neeman, I.: The Logic of Infons, Microsoft Research Tech Report MSR-TR-2009-10 (January 2009)Google Scholar
  12. 12.
    Kripke, S.: Semantical Analysis of Intuitionistic Logic I. In: Crossly, J., Dummett, M. (eds.) Formal Systems and Recursive Functions, pp. 92–129. North-Holland, Amsterdam (1965)CrossRefGoogle Scholar
  13. 13.
    Prior, A.N.: The Runabout Inference Ticket. Analysis 21, 38–39 (1960)CrossRefGoogle Scholar
  14. 14.
    Sundholm, G.: Proof theory and Meaning. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 9, pp. 165–198 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

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