Advertisement

Sequent Calculi for Normal Update Logics

  • Katsuhiko SanoEmail author
  • Minghui Ma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11600)

Abstract

Normal update logic is the temporalization of normal conditional logic. Sequent calculi for the least normal update logic \(\mathbf {UCK}\) by Andreas Herzig (1998) and some of its extensions are developed. The subformula property of these sequent calculi is shown by Takano’s semantic method. Consequently we prove the finite model property and decidability of these sequent calculi.

References

  1. 1.
    Alenda, R., Olivetti, N., Pozzato, G.L.: Nested sequent calculi for normal conditional logics. J. Logic Comput. 26(1), 7–50 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burgess, J.P.: Basic tense logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, Reidel, Dordrecht, vol. II, pp. 89–133 (1984)Google Scholar
  3. 3.
    Chellas, B.F.: Basic conditional logic. J. Philosophcal Logic 4(2), 133–153 (1975)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chellas, B.F.: Modal Logic. Cambridge University Press, Cambridge (1980)CrossRefGoogle Scholar
  5. 5.
    de Swart, H.C.: A Gentzen-or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS. J. Symbolic Logic 48(1), 1–20 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gent, P.: A sequent- or tableau-style system for Lewis’s counterfactual logic. Notre Dame J. Formal Logic 33(3), 369–382 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gentzen, G.: Untersuchungen über das logische Schließen, Mathematische Zeitschrift, 39(1), pp. 176–210 (1935)Google Scholar
  8. 8.
    Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L.: Standard sequent calculi for Lewis’ logics of counterfactuals. In: Michael, L., Kakas, A. (eds.) JELIA 2016. LNCS (LNAI), vol. 10021, pp. 272–287. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-48758-8_18CrossRefzbMATHGoogle Scholar
  9. 9.
    Herzig, A.: Logics for belief base updating. In: Dubois, D., Przde, H. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management System, vol. 3, pp. 189–231. Springer, Dordrecht (1998).  https://doi.org/10.1007/978-94-011-5054-5_5CrossRefGoogle Scholar
  10. 10.
    Kowalski, T., Ono, H.: Analytic cut and interpolation for bi-intuisionistic logic. The Review of Symbolic Logic, 10(2), pp. 259–283 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lellmann, B., Pattinson, D.: Sequent systems for Lewis’ conditional logics. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS (LNAI), vol. 7519, pp. 320–332. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-33353-8_25CrossRefzbMATHGoogle Scholar
  12. 12.
    Maruyama, A.: Towards combined system of modal logics - a syntactic and semantic study, Ph.D. thesis, School of Information Science, Japan Advanced Institute of Science and Technology (2003)Google Scholar
  13. 13.
    Maruyama, A., Tojo, S., Ono, H.: Decidability of temporal epistemic logics for multi-agent models. In: Proceedings of the ICLP’01 Workshop on Computational Logic in Multi-Agent Systems (CLIMA-01), pp. 31–40 (2001)Google Scholar
  14. 14.
    Negri, S., Sabrdolini, G.: Proof analysis for Lewis counterfactuals. Rev. Symbolic Logic 9(1), 44–75 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Negri, S., Von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  16. 16.
    Nishimura, H.: A study of some tense logics by Gentzen’s sequential method. Publ. Res. Inst. Math. Sci. 16, 343–353 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ohnishi, M., Matsumoto, K.: Gentzen method in modal calculi II. Osaka J. Math. 11(2), 115–120 (1959)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ono, H.: Semantical approach to cut elimination and subformula property in modal logic. In: Yang, S.C.-M., Deng, D.-M., Lin, H. (eds.) Structural Analysis of Non-Classical Logics. LASLL, pp. 1–15. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-48357-2_1CrossRefGoogle Scholar
  19. 19.
    Pattinson, D., Schröder, L.: Generic modal cut elimination applied to conditional logic. Logical Methods Comput. Sci. 7(1:4), 1–28 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Takano, M.: Subformula property as a substitute for cut-elimination in modal propositional logics. Math. Jpn. 37(6), 1129–1145 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Takano, M.: A modified subformula property for the modal logics K5 and K5D. Bull. Sect. Logic 30(2), 115–122 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Takano, M.: A semantical analysis of cut-free calculi for modal logics. Rep. Math. Logic 53, 43–65 (2018)Google Scholar
  23. 23.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  24. 24.
    Zach, R.: Non-analytic tableaux for Chellas’s conditional logic CK and Lewis’s logic of counterfactuals VC. Australas. J. Logic 15(3), 609–628 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of LettersHokkaido UniversitySapporoJapan
  2. 2.Institute of Logic and CognitionSun Yat-Sen UniversityGuangzhouChina

Personalised recommendations