Journal of Philosophical Logic

, Volume 38, Issue 2, pp 151–177 | Cite as

A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5

  • Dorota Leszczyńska-JasionEmail author


The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.

Key words

logic of questions loop-free procedures the method of Socratic proofs transitive modal logics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fitting, M. (1983). Proof methods for modal and intuitionistic logic, vol. 169 of Synthese Library. Dordrecht: Reidel.Google Scholar
  2. 2.
    Goré, R. (1999). Tableau methods for modal and temporal logics. In M. D’Agostino, D. M. Gabbay, R. Hähnle & J. Posegga (Eds.), Handbook of tableau methods (pp. 297–396). Dordrecht: Kluwer.Google Scholar
  3. 3.
    Heuerding, A., Seyfried, M., & Zimmermann, H. (1996). Efficient loop-check for backward proof search in some non-classical propositional logics. Proceedings of the 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, LNCS 1071, pp. 210–225.Google Scholar
  4. 4.
    Horrocks, I., & Sattler, U. (1999). A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation 9(No. 3), 385–410.CrossRefGoogle Scholar
  5. 5.
    Horrocks, I., Sattler, U., & Tobies, S. (2000). Practical reasoning for very expressive description logics. Logic Journal of the IGPL 8(No. 3), 239–263.CrossRefGoogle Scholar
  6. 6.
    Hughes, G., & Cresswell, M. (1968). An introduction to modal logic. London: Methuen.Google Scholar
  7. 7.
    Indrzejczak, A. (2006). Hybrydowe systemy dedukcyjne w logikach modalnych, (Hybrid deductive systems for modal logics). Łódź: University of Łódź Press.Google Scholar
  8. 8.
    Leszczyńska, D. (2004). Socratic proofs for some normal modal propositional logics. Logique et Analyse 47(No. 185–188), 259–285.Google Scholar
  9. 9.
    Leszczyńska, D. The method of Socratic proofs for normal modal propositional logics, PhD Thesis, University of Zielona Góra, Zielona Góra 2006; printed by Adam Mickiewicz University Press, Poznań 2007.Google Scholar
  10. 10.
    Leszczyńska-Jasion, D. (2008). The method of Socratic proofs for modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. Studia Logica 89(No. 3), 371–405.Google Scholar
  11. 11.
    Matsumoto, T. (2003). A tableau system for modal logic S4 with an efficient proof-search procedure. Conference paper, Proceeding of the 5th Japan Society for Software Science and Technology Workshop on Programming and Programming Languages, pp. 75–86.Google Scholar
  12. 12.
    Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.Google Scholar
  13. 13.
    Rautenberg, W. (1983). Modal tableau calculi and interpolation. Journal of Philosophical Logic 12(No. 4), 403–423.CrossRefGoogle Scholar
  14. 14.
    Smullyan, R. M. (1968). First-order logic. Berlin: Springer.Google Scholar
  15. 15.
    Wiśniewski, A. (1995). The posing of questions: Logical foundations of erotetic inferences. Dordrecht: Kluwer.Google Scholar
  16. 16.
    Wiśniewski, A. (2004). Socratic proofs. Journal of Philosophical Logic 33(No. 3), 299–326.CrossRefGoogle Scholar
  17. 17.
    Wiśniewski, A., & Shangin, V. (2006). Socratic proofs for quantifiers. Journal of Philosophical Logic 35(No. 2), 147–178.CrossRefGoogle Scholar
  18. 18.
    Wiśniewski, A., Vanackere, G., & Leszczyńska, D. (2005). Socratic proofs and paraconsistency: A case study. Studia Logica 80(No. 2–3), 433–468.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Section of Logic and Cognitive Science, Institute of PsychologyAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations