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Semantical Approach to Cut Elimination and Subformula Property in Modal Logic

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Structural Analysis of Non-Classical Logics

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

Abstract

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.

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Notes

  1. 1.

    Quite recently, we proved in our joint work with T. Kowalski that subformula property implies analytic cut property in a certain general setting. Thus, Theorem 7 follows from Theorem 6.

References

  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. Belardinelli, F., Jipsen, P., Ono, H.: Algebraic aspects of cut elimination. Stud. Log. 77, 209–240 (2004)

    Article  Google Scholar 

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science 53, (2001)

    Google Scholar 

  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. Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides, Clarendon Press, vol.35 (1997)

    Google Scholar 

  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)

    Article  Google Scholar 

  7. Curry, H.: The elimination theorem when modality is present. J. Symb. Log. 17, 249–265 (1952)

    Google Scholar 

  8. Fitting, M.: Model existence theorems for modal and intuitionistic logics. J. Symb. Log. 38, 613–627 (1973)

    Article  Google Scholar 

  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. Gentzen, G.: Untersuchungen über das logische Schliessen I. II. Mathematische Zeitschrift 39, (176-210, 405-431) (1934, 1935)

    Google Scholar 

  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. Okada, M., Terui, K.: The finite model property for various fragments of intuitionistic linear logic. J. Symb. Log. 64, 790–802 (1999)

    Article  Google Scholar 

  13. Ono, H.: On some intuitionistic modal logics, Publ. Res. Inst. Math. Sci. Kyoto University, 13, 687–722 (1977)

    Google Scholar 

  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. 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. Schütte, K.: Syntactical and semantical properties of simple type theory. J. Symb. Log. 25, 305–326 (1960)

    Article  Google Scholar 

  17. Schütte, K.: Vollständige Systeme modaler und intuitionistischer Logik. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, vol. 42 (1968)

    Google Scholar 

  18. Takano, M.: Subformula property as a substitute for cut-elimination in modal propositional logics. Math. Jpn. 37, 1145–1192 (1992)

    Google Scholar 

  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. Takano, M.: A modified subformula property for the modal logics K5 and K5D. Bull. Sect. Log. 30, 115–122 (2001)

    Google Scholar 

  21. Takeuti, G.: Proof Theory. Stud. Log. Found. Math. North-Holland, vol. 81 (1975)

    Google Scholar 

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Authors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Hiroakira Ono

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  1. Hiroakira Ono
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Correspondence to Hiroakira Ono .

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Editors and Affiliations

  1. Department of Philosophy, National Taiwan University, Taipei, Taiwan

    Syraya Chin-Mu Yang

  2. Department of Philosophy, National Taiwan University, Taipei, Taiwan

    Duen-Min Deng

  3. Department of Philosophy, University of California, Davis, CA, Australia

    Hanti Lin

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Ono, H. (2016). Semantical Approach to Cut Elimination and Subformula Property in Modal Logic. In: Yang, SM., Deng, DM., Lin, H. (eds) Structural Analysis of Non-Classical Logics. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48357-2_1

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