Constructing Cut Free Sequent Systems with Context Restrictions Based on Classical or Intuitionistic Logic

  • Björn Lellmann
  • Dirk Pattinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7750)

Abstract

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic \(\mathbb{VA}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abadi, M.: Variations in Access Control Logic. In: van der Meyden, R., van der Torre, L. (eds.) DEON 2008. LNCS (LNAI), vol. 5076, pp. 96–109. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Alechina, N., Mendler, M., de Paiva, V., Ritter, E.: Categorical and Kripke Semantics for Constructive S4 Modal Logic. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 292–307. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Avron, A., Lahav, O.: Kripke Semantics for Basic Sequent Systems. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 43–57. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Bellin, G., de Paiva, V., Ritter, E.: Extended Curry-Howard correspondence for a basic contructive modal logic. In: Areces, C., de Rijke, M. (eds.) M4M-2. ILLC Amsterdam (2001)Google Scholar
  5. 5.
    Benton, P., Bierman, G., de Paiva, V.: Computational types from a logical perspective. J. Funct. Programming 8(2), 177–193 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. In: LICS 2008, pp. 229–240. IEEE Computer Society (2008)Google Scholar
  7. 7.
    Fairtlough, M., Mendler, M.: Propositional lax logic. Inform. and Comput. 137(1), 1–33 (1997)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gentzen, G.: Untersuchungen über das logische Schließen. I. Math. Z. 39(2), 176–210 (1934)MathSciNetGoogle Scholar
  9. 9.
    Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compos. Math. 4, 119–136 (1937)MathSciNetGoogle Scholar
  10. 10.
    Lellmann, B., Pattinson, D.: Cut Elimination for Shallow Modal Logics. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 211–225. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Lellmann, B., Pattinson, D.: Sequent Systems for Lewis’ Conditional Logics. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 320–332. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Lewis, D.: Counterfactuals. Blackwell (1973)Google Scholar
  13. 13.
    Mendler, M., Scheele, S.: Cut-free Gentzen calculus for multimodal CK. Inform. and Comput. 209, 1465–1490 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34, 507–544 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Negri, S., von Plato, J.: Structural proof theory. Cambridge University Press (2001)Google Scholar
  16. 16.
    Pattinson, D., Schröder, L.: Generic modal cut elimination applied to conditional logics. Log. Methods Comput. Sci. 7(1) (2011)Google Scholar
  17. 17.
    Pfenning, F., Davies, R.: A judgmental reconstruction of modal logic. Math. Structures Comput. Sci. 11(4), 511–540 (2001)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge Tracts Theoret. Comput. Sci. Cambridge University Press (2000)Google Scholar
  19. 19.
    Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8. Springer (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Björn Lellmann
    • 1
  • Dirk Pattinson
    • 2
  1. 1.Department of ComputingImperial College LondonUK
  2. 2.Research School of Computer ScienceThe Australian National UniversityAustralia

Personalised recommendations