Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-16T19:43:43.267Z Has data issue: false hasContentIssue false
  • English
  • Français

CUT-FREE COMPLETENESS FOR MODULAR HYPERSEQUENT CALCULI FOR MODAL LOGICS K, T, AND D

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  21 July 2020

SAMARA BURNS  and
RICHARD ZACH
SAMARA BURNS
Affiliation:
DEPARTMENT OF PHILOSOPHY COLUMBIA UNIVERSITY 1150 AMSTERDAM AVENUE NEW YORK, NY10027, USAE-mail: sb4318@columbia.edu
RICHARD ZACH
Affiliation:
UNIVERSITY OF CALGARY DEPARTMENT OF PHILOSOPHY 2500 UNIVERSITY DRIVE NW CALGARY, ABT2N 1N4, CANADAE-mail: rzach@ucalgary.caURL: https://richardzach.org/
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman’s linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Došen’s principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.

Keywords

sequent calculushypersequentslinear nested sequentsmodal logiccompleteness

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03F05: Cut-elimination and normal-form theorems
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 4 , December 2021 , pp. 910 - 929
Check for updates
Copyright
© Association for Symbolic Logic, 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Avron, A. (1996). The method of hypersequents in the proof theory of propositional non-classical logics. In Hodges, W., Hyland, M., Steinhorn, C., and Truss, J., editors. Logic: From Foundations to Applications. Oxford: Oxford University Press, pp. 1–36.Google Scholar
Baelde, D., Lick, A., & Schmitz, S. (2018). A hypersequent calculus with clusters for linear frames. In D’Agostino, G., Bezhanishvili, G., Metcalfe, G., and Studer, T., editors. Advances in Modal Logic, Vol. 12. London: College Publications, pp. 36–55.Google Scholar
Braüner, T. (2000). A cut-free Gentzen formulation of the modal logic S5. Logic Journal of the IGPL, 8(5), 629–643.CrossRefGoogle Scholar
Brünnler, K. (2009). Deep sequent systems for modal logic. Archive for Mathematical Logic, 48(6), 551–577.CrossRefGoogle Scholar
Burns, S. (2018). Hypersequent Calculi for Modal Logics. MA Thesis, University of Calgary.Google Scholar
Goré, R., & Lellmann, B. (2019). Syntactic cut-elimination and backward proof-search for tense logic via linear nested sequents. In Cerrito, S., and Popescu, A., editors. Automated Reasoning with Analytic Tableaux and Related Methods. Lecture Notes in Computer Science, no. 11714. Cham: Springer, pp. 185–202.CrossRefGoogle Scholar
Indrzejczak, A. (2012). Cut-free hypersequent calculus for S4.3. Bulletin of the Section of Logic, 41(1/2), 89–104.Google Scholar
Indrzejczak, A. (2016). Linear time in hypersequent framework. The Bulletin of Symbolic Logic, 22(1), 121–144.CrossRefGoogle Scholar
Indrzejczak, A. (2018). Cut elimination theorem for non-commutative hypersequent calculus. Bulletin of the Section of Logic, 46(1/2), 135–149.Google Scholar
Indrzejczak, A. (2019). Cut elimination in hypersequent calculus for some logics of linear time. The Review of Symbolic Logic, 12(4), 806–822.CrossRefGoogle Scholar
Kuznets, R., & Lellmann, B. (2018). Interpolation for intermediate logics via hyper- and linear nested sequents. In D’Agostino, G., Bezhanishvili, G., Metcalfe, G., and Studer, T., editors. Advances in Modal Logic, Vol. 12. London: College Publications, pp. 473–492.Google Scholar
Lahav, O. (2013). From frame properties to hypersequent rules in modal logics. Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science. Los Alamitos, CA: IEEE Computer Society Press, pp. 408–417.CrossRefGoogle Scholar
Lellmann, B. (2015). Linear nested sequents, 2-sequents and hypersequents. In De Nivelle, H., editor. Automated Reasoning with Analytic Tableaux and Related Methods. 24 th International Conference TABLEAUX 2015. Lecture Notes in Computer Science, no. 9323. Cham: Springer, pp. 135–150.Google Scholar
Lellmann, B., & Pimentel, E. (2019). Modularisation of sequent calculi for normal and non-normal modalities. ACM Transactions on Computational Logic, 20(2), 7:1–7:46.CrossRefGoogle Scholar
Masini, A. (1992). 2-Sequent calculus: A proof theory of modalities. Annals of Pure and Applied Logic, 58(3), 229–246.CrossRefGoogle Scholar
Mints, G. E. (1971). On some calculi of modal logic. Proceedings of the Steklov Institute of Mathematics, 98, 97–124.Google Scholar
Mints, G. E. (1974). Sistemy Lyuisa i sistema T (1965–1973). In Feys, R., editor. Modal’naya Logika. Moscow: Nauka, pp. 422–509.Google Scholar
Ohnishi, M. (1982). A new version to Gentzen decision procedure for modal sentential calculus S5. Mathematical Seminar Notes, 10, 161–170.Google Scholar
Ohnishi, M., & Matsumoto, K. (1957). Gentzen method in modal calculi. Osaka Mathematical Journal, 9(2), 113–130.Google Scholar
Parisi, A. (2017). Second-Order Modal Logic. PhD Dissertation, University of Connecticut, Storrs, CT.Google Scholar
Parisi, A. (2020). A hypersequent solution to the inferentialist problem of modality. Erkenntnis, forthcoming.CrossRefGoogle Scholar
Poggiolesi, F. (2008). A cut-free simple sequent calculus for modal logic S5. The Review of Symbolic Logic, 1(1), 3–15.CrossRefGoogle Scholar
Poggiolesi, F. (2011). Gentzen Calculi for Modal Propositional Logic. Trends in Logic, Vol. 32. Dordrecht, Netherlands: Springer.CrossRefGoogle Scholar
Pottinger, G. (1983). Uniform cut-free formulations of T, S4 and S5. The Journal of Symbolic Logic, 48(3), 900.Google Scholar
Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2), 241–264.CrossRefGoogle Scholar