Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fitting, M. (1983). Proof methods for modal and intuitionistic logic, vol. 169 of Synthese Library. Dordrecht: Reidel.
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.
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.
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.
Horrocks, I., Sattler, U., & Tobies, S. (2000). Practical reasoning for very expressive description logics. Logic Journal of the IGPL 8(No. 3), 239–263.
Hughes, G., & Cresswell, M. (1968). An introduction to modal logic. London: Methuen.
Indrzejczak, A. (2006). Hybrydowe systemy dedukcyjne w logikach modalnych, (Hybrid deductive systems for modal logics). Łódź: University of Łódź Press.
Leszczyńska, D. (2004). Socratic proofs for some normal modal propositional logics. Logique et Analyse 47(No. 185–188), 259–285.
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.
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.
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.
Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.
Rautenberg, W. (1983). Modal tableau calculi and interpolation. Journal of Philosophical Logic 12(No. 4), 403–423.
Smullyan, R. M. (1968). First-order logic. Berlin: Springer.
Wiśniewski, A. (1995). The posing of questions: Logical foundations of erotetic inferences. Dordrecht: Kluwer.
Wiśniewski, A. (2004). Socratic proofs. Journal of Philosophical Logic 33(No. 3), 299–326.
Wiśniewski, A., & Shangin, V. (2006). Socratic proofs for quantifiers. Journal of Philosophical Logic 35(No. 2), 147–178.
Wiśniewski, A., Vanackere, G., & Leszczyńska, D. (2005). Socratic proofs and paraconsistency: A case study. Studia Logica 80(No. 2–3), 433–468.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leszczyńska-Jasion, D. A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5. J Philos Logic 38, 151–177 (2009). https://doi.org/10.1007/s10992-008-9089-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-008-9089-z