Semantical Approach to Cut Elimination and Subformula Property in Modal Logic

Part of the Logic in Asia: Studia Logica Library book series (LIAA)


This is a short survey of semantical study of cut elimination and subformula property in modal logics. Cut elimination is a basic proof-theoretic notion in sequent systems, and subformula property is the most important consequence of cut elimination. A special feature of our presentation is its unified semantical approach to them based on Kripke models. Along the same lines as Takano’s works on subformula property, these properties, together with finite model property, will be discussed as modifications of standard construction of canonical Kripke models. These semantical approaches will be compared with algebraic approaches in modal logics, which often take the forms of various kinds of embedding theorems. In the last part of the paper, an attempt is made to clarify connections between semantical approach to cut elimination and algebraic one.


Cut elimination Subformula property Finite model property Modal logics Embedding theorems 


  1. 1.
    Amano, S.J.: The finite embeddability property for some modal algebras, Master thesis. Japan Advanced Institute of Science and Technology (2006)Google Scholar
  2. 2.
    Belardinelli, F., Jipsen, P., Ono, H.: Algebraic aspects of cut elimination. Stud. Log. 77, 209–240 (2004)CrossRefGoogle Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science 53, (2001)Google Scholar
  4. 4.
    Bull, R.: Some modal calculi based on IC. Formal systems and recursive functions. In: Crossley, J.N., Dummett, M.A.E., 3-7 (1965)Google Scholar
  5. 5.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides, Clarendon Press, vol.35 (1997)Google Scholar
  6. 6.
    Ciabattoni, A., Galatos, N., Terui, K.: Algebraic proof theory for substructural logics: cut-elimination and completions. Ann. Pure Appl. Log. 163, 266–290 (2012)CrossRefGoogle Scholar
  7. 7.
    Curry, H.: The elimination theorem when modality is present. J. Symb. Log. 17, 249–265 (1952)Google Scholar
  8. 8.
    Fitting, M.: Model existence theorems for modal and intuitionistic logics. J. Symb. Log. 38, 613–627 (1973)CrossRefGoogle Scholar
  9. 9.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: an algebraic glimpse at substructural logics. Studies in Logic and the Foundations of Mathematics, Elsevier, vol. 151 (2007)Google Scholar
  10. 10.
    Gentzen, G.: Untersuchungen über das logische Schliessen I. II. Mathematische Zeitschrift 39, (176-210, 405-431) (1934, 1935)Google Scholar
  11. 11.
    Ohnishi, M., Matsumoto, K.: Gentzen method in modal calculi, Osaka Math. J. 9, 113-130 (1957) (Correction ibid. 10 (1958), p.147)Google Scholar
  12. 12.
    Okada, M., Terui, K.: The finite model property for various fragments of intuitionistic linear logic. J. Symb. Log. 64, 790–802 (1999)CrossRefGoogle Scholar
  13. 13.
    Ono, H.: On some intuitionistic modal logics, Publ. Res. Inst. Math. Sci. Kyoto University, 13, 687–722 (1977)Google Scholar
  14. 14.
    Ono, H.: Proof-theoretic methods for nonclassical logic—an introduction. Theories of Types and Proofs (MSJ Memoirs 2). In: Takahashi, M., Okada, M., Dezani-Ciancaglini M. (eds.) Mathematical Society of Japan, 207-254 (1998)Google Scholar
  15. 15.
    Sato, M.: A study of Kripke-type models for some modal logic by Gentzen’s sequential method. Publ. Res. Inst. Math. Sci. Kyoto University, 13, 381-468 (1977)Google Scholar
  16. 16.
    Schütte, K.: Syntactical and semantical properties of simple type theory. J. Symb. Log. 25, 305–326 (1960)CrossRefGoogle Scholar
  17. 17.
    Schütte, K.: Vollständige Systeme modaler und intuitionistischer Logik. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, vol. 42 (1968)Google Scholar
  18. 18.
    Takano, M.: Subformula property as a substitute for cut-elimination in modal propositional logics. Math. Jpn. 37, 1145–1192 (1992)Google Scholar
  19. 19.
    Takano, M.: Semantical proofs of cut elimination and subformula property (in Japanese), abstract of talk at Japan Advanced Institute of Science and Technology (2000)Google Scholar
  20. 20.
    Takano, M.: A modified subformula property for the modal logics K5 and K5D. Bull. Sect. Log. 30, 115–122 (2001)Google Scholar
  21. 21.
    Takeuti, G.: Proof Theory. Stud. Log. Found. Math. North-Holland, vol. 81 (1975)Google Scholar

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Authors and Affiliations

  1. 1.Japan Advanced Institute of Science and TechnologyNomiJapan

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