Algebraic Study of Lattice-Valued Logic and Lattice-Valued Modal Logic

  • Yoshihiro Maruyama
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5378)

Abstract

In this paper, we study lattice-valued logic and lattice-valued modal logic from an algebraic viewpoint. First, we give an algebraic axiomatization of L-valued logic for a finite distributive lattice L. Then we define the notion of prime L-filters and prove an L-valued version of prime filter theorem for Boolean algebras, by which we show a Stone-style representation theorem for algebras of L-valued logic and the completeness with respect to L-valued semantics. By the representation theorem, we can show that a strong duality holds for algebras of L-valued logic and that the variety generated by L coincides with the quasi-variety generated by L. Second, we give an algebraic axiomatization of L-valued modal logic and establish the completeness with respect to L-valued Kripke semantics. Moreover, it is shown that L-valued modal logic enjoys finite model property and that L-valued intuitionistic logic is embedded into L-valued modal logic of S4-type via Gödel-style translation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal logic, CUP (2001)Google Scholar
  2. 2.
    Burris, S., Sankappanavar, H.P.: A course in universal algebra. Springer, Heidelberg (1981)CrossRefMATHGoogle Scholar
  3. 3.
    Clark, D.M., Davey, B.A.: Natural dualities for the working algebraist, CUP (1998)Google Scholar
  4. 4.
    Grätzer, G.A.: Universal algebra. Springer, Heidelberg (1979)MATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Epstein, G.: The lattice theory of Post algebras. Trans. Amer. Math. Soc. 95, 300–317 (1960)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 239, 345–371 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand 94, 13–45 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fitting, M.C.: Many-valued modal logics. Fund. Inform. 15, 235–254 (1991)MathSciNetMATHGoogle Scholar
  11. 11.
    Fitting, M.C.: Many-valued modal logics II. Fund. Inform. 17, 55–73 (1992)MathSciNetMATHGoogle Scholar
  12. 12.
    Fitting, M.C.: Tableaus for many-valued modal logic. Studia Logica 55, 63–87 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Koutras, C.D., Nomikos, C., Peppas, P.: Canonicity and completeness results for many-valued modal logics. J. Appl. Non-Classical Logics 12, 7–41 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Koutras, C.D., Peppas, P.: Weaker axioms, more ranges. Fund. Inform. 51, 297–310 (2002)MathSciNetMATHGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yoshihiro Maruyama
    • 1
  1. 1.Faculty of Integrated Human StudiesKyoto UniversityJapan

Personalised recommendations