Strict Canonical Constructive Systems

  • Arnon Avron
  • Ori Lahav
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6300)

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 strict 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 for proving a strong form of the cut-elimination theorem for such systems, and for providing a decision procedure for them. These results identify a large family of basic constructive connectives, including the standard intuitionistic connectives, together with many other independent connectives.

Keywords

sequent calculus cut-elimination non-classical logics non-deterministic semantics Kripke semantics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avron, A.: Simple Consequence Relations. Information and Computation 92, 105–139 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Avron, A.: Gentzen-Type Systems, Resolution and Tableaux. Journal of Automated Reasoning 10, 265–281 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Avron, A.: Nondeterministic View on Nonclassical Negations. Studia Logica 80, 159–194 (2005)MathSciNetCrossRefMATHGoogle 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., Lahav, O.: Canonical constructive systems. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS, vol. 5607, pp. 62–76. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Avron, A., Lev, I.: Canonical Propositional Gentzen-Type Systems. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 529–544. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Avron, A., Lev, I.: Non-deterministic Multiple-valued Structures. Journal of Logic and Computation 15, 24–261 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Belnap, N.D.: Tonk, Plonk and Plink. Analysis 22, 130–134 (1962)CrossRefGoogle Scholar
  9. 9.
    Bowen, K.A.: An extension of the intuitionistic propositional calculus. Indagationes Mathematicae 33, 287–294 (1971)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ciabattoni, A., Terui, K.: Towards a Semantic Characterization of Cut-Elimination. Studia Logica 82, 95–119 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fernandez, D.: Non-deterministic Semantics for Dynamic Topological Logic. Annals of Pure and Applied Logic 157, 110–121 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Gurevich, Y., Neeman, I.: The Logic of Infons, Microsoft Research Tech. Report MSR-TR-2009-10 (January 2009)Google Scholar
  14. 14.
    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
  15. 15.
    Prior, A.N.: The Runabout Inference Ticket. Analysis 21, 38–39 (1960)CrossRefGoogle Scholar
  16. 16.
    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 2010

Authors and Affiliations

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

Personalised recommendations