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A Duality for Algebras of Lattice-Valued Modal Logic

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Logic, Language, Information and Computation (WoLLIC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5514))

Abstract

In this paper, we consider some versions of Fitting’s L-valued logic and L-valued modal logic for a finite distributive lattice L. Using the theory of natural dualities, we first obtain a natural duality for algebras of L-valued logic (i.e., L -VL-algebras), which extends Stone duality for Boolean algebras to the L-valued case. Then, based on this duality, we develop a Jónsson-Tarski-style duality for algebras of L-valued modal logic (i.e., L -ML-algebras), which extends Jónsson-Tarski duality for modal algebras to the L-valued case. By applying these dualities, we obtain compactness theorems for L-valued logic and for L-valued modal logic, and the classification of equivalence classes of categories of L -VL-algebras for finite distributive lattices L.

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References

  1. Abramsky, S.: Domain theory in logical form. Ann. Pure Appl. Logic 51, 1–77 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Awodey, S.: Category theory. OUP (2006)

    Google Scholar 

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. CUP (2001)

    Google Scholar 

  4. Bonsangue, M.M.: Topological duality in semantics. Electr. Notes Theor. Comput. Sci. 8 (1998)

    Google Scholar 

  5. Brink, C., Rewitzky, I.M.: A paradigm for program semantics: power structures and duality. CSLI Publications, Stanford (2001)

    Google Scholar 

  6. Burris, S., Sankappanavar, H.P.: A course in universal algebra. Springer, Heidelberg (1981)

    Book  MATH  Google Scholar 

  7. Clark, D.M., Davey, B.A.: Natural dualities for the working algebraist. CUP (1998)

    Google Scholar 

  8. Connes, A.: Noncommutative geometry. Academic Press, London (1994)

    MATH  Google Scholar 

  9. Chagrov, A., Zakharyaschev, M.: Modal logic. OUP (1997)

    Google Scholar 

  10. Doran, R.S., Belfi, V.A.: Characterizations of C*-algebras; The Gelfand-Naimark theorems. Marcel Dekker Inc., New York (1986)

    MATH  Google Scholar 

  11. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique: I. Le langage des schémas. Publications Mathématiques de l’IHÉS 4, 225–228 (1960)

    Google Scholar 

  12. Hansoul, G.: A duality for Boolean algebras with operators. Algebra Universalis 17, 34–49 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eleftheriou, P.E., Koutras, C.D.: Frame constructions, truth invariance and validity preservation in many-valued modal logic. J. Appl. Non-Classical Logics 15, 367–388 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 239, 345–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fitting, M.C.: Many-valued modal logics. Fund. Inform. 15, 235–254 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Fitting, M.C.: Many-valued modal logics II. Fund. Inform. 17, 55–73 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Fitting, M.C.: Many-valued non-monotonic modal logics. In: Nerode, A., Taitslin, M.A. (eds.) LFCS 1992. LNCS, vol. 620, pp. 139–150. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  19. Fitting, M.C.: Tableaus for many-valued modal logic. Studia Logica 55, 63–87 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Koutras, C.D., Zachos, S.: Many-valued reflexive autoepistemic logic. Logic Journal of the IGPL 8, 33–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koutras, C.D., Peppas, P.: Weaker axioms, more ranges. Fund. Inform. 51, 297–310 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Koutras, C.D.: A catalog of weak many-valued modal axioms and their corresponding frame classes. J. Appl. Non-Classical Logics 13, 47–72 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Maruyama, Y.: Algebraic study of lattice-valued logic and lattice-valued modal logic. In: Ramanujam, R., Sarukkai, S. (eds.) ICLA 2009. LNCS (LNAI), vol. 5378, pp. 172–186. Springer, Heidelberg (2009)

    Google Scholar 

  24. Maruyama, Y.: The logic of common belief, revisited (in preparation)

    Google Scholar 

  25. Stone, M.H.: The representation of Boolean algebras. Bull. Amer. Math. Soc. 44, 807–816 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  26. Straßburger, L.: What is a logic, and what is a proof? Logica Universalis 2nd edn., 135–152 (2007)

    Google Scholar 

  27. Teheux, B.: A duality for the algebras of a Łukasiewicz n + 1-valued modal system. Studia Logica 87, 13–36 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Humanistic Informatics, Graduate School of Letters, Kyoto University, Japan

    Yoshihiro Maruyama

Authors
  1. Yoshihiro Maruyama
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Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Ishikawa, Japan

    Hiroakira Ono

  2. National Institute of Informatics, 2–1–2 Hitotsubashi, Chiyoda-ku, 101–8430, Tokyo, Japan

    Makoto Kanazawa

  3. Centro de Informática, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Ruy de Queiroz

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Maruyama, Y. (2009). A Duality for Algebras of Lattice-Valued Modal Logic. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2009. Lecture Notes in Computer Science(), vol 5514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02261-6_23

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  • DOI: https://doi.org/10.1007/978-3-642-02261-6_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02260-9

  • Online ISBN: 978-3-642-02261-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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