Advertisement

Sound and Complete Tree-Sequent Calculus for Inquisitive Logic

  • Katsuhiko Sano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5514)

Abstract

We introduce a tree-sequent calculus for inquisitive logic (Groenendijk 2008) as a special form of labelled deductive system (Gabbay 1996). In particular, we establish that (i) our tree-sequent calculus is sound and complete with respect to Groenendijk’s inquisitive semantics and that (ii) our tree-sequent calculus is decidable and enjoys cut-elimination theorem. (ii) is semantically revealed by our argument for (i). The key idea which allows us to obtain these results is as follows: In Groenendijk’s inquisitive semantics, a formula of propositional logic is evaluated against a pair of worlds on a model. Given the appropriate pre-order on the set of such pairs, any inquisitive model can be regarded as a Kripke model for intuitionistic logic. This representation enables us to connect inquisitive semantics with the tree-sequent technique for non-classical logics (Kashima 1999).

Keywords

Propositional Logic Intuitionistic Logic Kripke Model Sequent Calculus Hybrid Logic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, pp. 821–868. Elsevier, Amsterdam (2007)CrossRefGoogle Scholar
  2. 2.
    Blackburn, P., ten Cate, B.: Pure extensions, proof rules, and hybrid axiomatics. Studia Logica 84, 277–322 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ten Cate, B., Shan, C.: Axiomatizing Groenendijk’s logic of interrogation. In: Aloni, M., Butler, A., Dekker, P. (eds.) Questions in Dynamic Semantics, pp. 63–82. Elsevier, Oxford (2007)Google Scholar
  4. 4.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides, vol. 35. Oxford Science Publications, Oxford (1997)MATHGoogle Scholar
  5. 5.
    Ciardelli, I., Roelofsen, F.: Generalized inquisitive logic (submitted for publication, 2009)Google Scholar
  6. 6.
    Gabbay, D.: Labelled Deductive Systems. Clarendon Press, Oxford (1996)MATHGoogle Scholar
  7. 7.
    Groenendijk, J.: The logic of interrogation. In: Matthews, T., Strolovitch, D. (eds.) The Proceedings of the Ninth Conference on Semantics and Linguistic Theory, pp. 109–126. CLC Publications, NY (1999)Google Scholar
  8. 8.
    Groenendijk, J.: Inquisitive semantics: Two possibilities for disjunction. Prepublication Series PP-2008-26, ILLC (2008)Google Scholar
  9. 9.
    Hasuo, I., Kashima, R.: Kripke completeness of first-order constructive logics with strong negation. Logic Journal of the IGPL 11(6), 615–646 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ishigaki, R., Kikuchi, K.: A tree-sequent calculus for a natural predicate extension of Visser’s propositional logic. Logic Journal of the IGPL 15(2), 149–164 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ishigaki, R., Kashima, R.: Sequent calculi for some strict implication logics. Logic Journal of the IGPL 16(2), 155–174 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kashima, R.: Sequent calculi of non-classical logics - Proofs of completeness theorems by sequent calculi (in Japanese). In: Proceedings of Mathematical Society of Japan Annual Colloquium of Foundations of Mathematics, pp. 49–67 (1999)Google Scholar
  13. 13.
    Mascarenhas, S.: Inquisitive semantics and logic. The Institute for Logic, Language and Computation, the University of Amsterdam, Forthcoming Master ThesisGoogle Scholar
  14. 14.
    Mascarenhas, S.: Inquisitive semantics and logic. ILLC, Amsterdam (2008) (manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Katsuhiko Sano
    • 1
  1. 1.Department of Humanistic Informatics Graduate School of LettersKyoto University / JSPSJapan

Personalised recommendations