Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics

  • Rosalie Iemhoff
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5407)

Abstract

The admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a proof-theoretic framework where the basic objects of the systems are sequent rules. Here, the framework is extended to cover derivability of the admissible rules of intermediate logics and a wider class of modal logics, in this case, by taking hypersequent rules as the basic objects.

Keywords

Modal Logic Proof System Basic Object Intuitionistic Logic Logical Rule 
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.
    Avron, A.: A constructive analysis of RM. Journal of Symbolic Logic 52(4), 939–951 (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1996)MATHGoogle Scholar
  3. 3.
    Ciabattoni, A., Ferrari, M.: Hypersequent calculi for some intermediate logics with bounded Kripke models. Journal of Logic and Computation 11(2), 283–294 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ghilardi, S.: Unification in intuitionistic logic. Journal of Symbolic Logic 64(2), 859–880 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ghilardi, S.: Best solving modal equations. Annals of Pure and Applied Logic 102(3), 184–198 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ghilardi, S., Sacchetti, L.: Filtering unification and most general unifiers in modal logic. Journal of Symbolic Logic 69(3), 879–906 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Iemhoff, R.: On the admissible rules of intuitionistic propositional logic. Journal of Symbolic Logic 66(1), 281–294 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Iemhoff, R.: Intermediate logics and Visser’s rules. Notre Dame Journal of Formal Logic 46(1), 65–81 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iemhoff, R., Metcalfe, G.: Proof theory for admissible rules. (submitted), http://www.math.vanderbilt.edu/people/metcalfe/publications
  10. 10.
    Jerábek, E.: Admissible rules of modal logics. Journal of Logic and Computation 15, 411–431 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer Series in Applied Logic, vol. 36. Springer, Heidelberg (to appear, 2009)MATHGoogle Scholar
  12. 12.
    Rozière, P.: Regles Admissibles en calcul propositionnel intuitionniste. PhD thesis, Université Paris VII (1992)Google Scholar
  13. 13.
    Rybakov, V.: Admissibility of Logical Inference Rules. Elsevier, Amsterdam (1997)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Rosalie Iemhoff
    • 1
  • George Metcalfe
    • 2
  1. 1.Department of PhilosophyUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations