Quantifier Elimination in Fuzzy Logic

  • Matthias Baaz
  • Helmut Veith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


We investigate quantifier elimination of first order logic over fuzzy algebras. Fuzzy algebras are defined from continuous t-norms over the unit interval, and subsume Łukasiewicz [28, 29], Gödel [16, 12] and Product [19] Logic as most prominent examples.

We show that a fuzzy algebra has quantifier elimination iff it is one of the abovementioned logics. Moreover, we show quantifier elimination for various extensions of these logics, and observe other model-theoretic properties of fuzzy algebras.

Further considerations are devoted to approximation of fuzzy logics by finite-valued logics.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baaz, M.: Infinite-Valued Gödel Logics with 0-1-Projections and Relativizations. In: Proceedings of Gödel 96 - Kurt Gödel’s Legacy. LNL, vol. 6, pp. 23–33. Springer, Heidelberg (1996)Google Scholar
  2. 2.
    Baaz, M., Veith, H.: Interpolation in Fuzzy Logic. Archive for Mathematical Logic (accepted for publication)Google Scholar
  3. 3.
    Baaz, M., Zach, R.: Approximating propositional calculi by finite-valued logics. In: Proc. 24th Int. Symp. on Multiple-Valued Logic, May 25-27, pp. 257–263. IEEE Press, Los Alamitos (1994)Google Scholar
  4. 4.
    Ben-Or, M., Kozen, D.: The Complexity of Elementary Algebra and Geometry. Journal of Computer and System Sciences 32, 251–264 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Canny, J.: Some Algebraic and Geometric Computations in PSPACE. In: Proc. STOC 1988, pp. 460–467 (1988)Google Scholar
  6. 6.
    Chang, C.C.: Algebraic Analysis of Many-Valued Logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  8. 8.
    Cignoli, R., Mundici, D.: An Invitation to Chang’s MV Algebras. In: Goebel, R., Droste, M. (eds.) Advances in algebra and model theory, pp. 171–197. Gordon and Breach Publishers, New York (1997)Google Scholar
  9. 9.
    Collins, E.G.: Quantifier Elimination for Real Closed Fields by Cylindrical Decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  10. 10.
    van den Dries, L.: Tarski’s Elimination Theory for Real Closed Fields. The Journal of Symbolic Logic 53/1, 7–19 (1988)CrossRefGoogle Scholar
  11. 11.
    van den Dries, L.: Algebraic Theories with Definable Skolem Functions. The Journal of Symbolic Logic 49(2), 625–629 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dummett, M.: A Propositional Logic with Denumerable Matrix. Journal of Symbolic Logic 24, 96–107 (1959)MathSciNetGoogle Scholar
  13. 13.
    Fraïssé, R.: Sur l’extension aux relations de quelques propriétés des ordres. Ann. Sci. École Norm. Sup. 71, 363–388 (1954)zbMATHGoogle Scholar
  14. 14.
    Glass, A.M.W., Pierce, K.R.: Existentially complete abelian lattice-ordered groups. Trans. Amer. Math. Society 261, 255–270 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Glass, A.M.W., Pierce, K.R.: Equations and inequations in lattice-ordered groups. In: Smith, J.E., Kenny, G.O., Ball, R.N. (eds.) Ordered Groups, pp. 141–171. Dekker, New York (1980)Google Scholar
  16. 16.
    Gödel, K.: Zum Intuitionistischen Aussagenkalkäl. Ergebnisse eines mathematischen Kolloquiums 4, 34–38 (1933)Google Scholar
  17. 17.
    Grigoriev, D.Y.: Complexity of Deciding Tarski Algebra. J.Symbolic Computation 5, 65–108 (1988)CrossRefGoogle Scholar
  18. 18.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)zbMATHGoogle Scholar
  19. 19.
    Hájek, P., Godo, L., Esteva, F.: A complete many-valued logic with productconjunction. Archive for Math. Logic 35, 191–208 (1996)zbMATHGoogle Scholar
  20. 20.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar
  21. 21.
    Höhle, U.: Commutative residuated l-monoids. In: [22], pp. 53–106.Google Scholar
  22. 22.
    Hóhle, U., Klement, P. (eds.): Non-classical logics and their applications to fuzzy subsets (A handbook of the mathematical foundations of the fuzzy set theory). Kluwer, Dordrecht (1995)Google Scholar
  23. 23.
    Knight, J., Pillay, A., Steinhorn, C.: Definable Sets in Ordered Structures 2. Transactions of the American Mathematical Society 295/2, 593–605Google Scholar
  24. 24.
    Kreiser, L., Gottwald, S., Stelzner, W. (eds.): Nichtklassische Logik. Akademie, Berlin (1988)Google Scholar
  25. 25.
    Lacava, F., Saeli, D.: Sul model-completamento della teoria delle L-catene. Bolletino Unione Matematica Italiana 14-A(5), 107–110 (1977)MathSciNetGoogle Scholar
  26. 26.
    Lacava, F.: Alcune prorietá delle Ł-algebre e delle L-algebre esistenzialmente chiuse. Bolletino Unione Matematica Italiana 16-A(5), 360–366 (1979)MathSciNetGoogle Scholar
  27. 27.
    Ling, H.C.: Representation of associative functions. Publ. Math. Debrecen 12, 182–212 (1965)Google Scholar
  28. 28.
    Lukasiewicz, J.: Zagadnienia prawdy (The problems of truth). Ksiȩga pamia̧tkowa XI zjazdu lekarzy i przyrodników polskich 87, 84–85 (1922)Google Scholar
  29. 29.
    Lukasiewicz, J.: Philosophische Bemerkungen zu mehrwertigen Systemen der Aussagenlogik. Comptes Rendus de la Societe des Science et de Lettres de Varsovie, cl.iii 23, 51–77 (1930)Google Scholar
  30. 30.
    Marker, D.: Introduction to the Model Theory of Fiels. In: Model Theory of Fields, ch. I. LNL, vol. 5. Springer, HeidelbergGoogle Scholar
  31. 31.
    McNaughton, R.: A theorem about infinite-valued sentential logic. Journal of Symbolic Logic 16, 1–13 (1951)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Mundici, D.: Satisfiability in Many-Valued Sentential Logic is NP-complete. Theoretical Computer Science 52, 145–153 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Mundici, D.: Personal CommunicationGoogle Scholar
  34. 34.
    Pillay, A., Steinhorn, C.: Definable Sets in Ordered Structures 1. Transactions of the American Mathematical Society 295/2, 565–592Google Scholar
  35. 35.
    Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology. Springer, Heidelberg (1972)zbMATHGoogle Scholar
  36. 36.
    Schweizer, B., Sklar, A.: Statistical Metric Spaces. Pacific Journal of Mathematics 10, 313–334 (1960)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Publicationes Mathematicae Debrecen 8, 169–186 (1961)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Tarski, A.: A Decision Method of Elementary Algebra and Geometry. University of California Press, Berkeley (1951)Google Scholar
  39. 39.
    Yemelichev, V.A., Kovalev, M.M., Kravtsov, M.K.: Polytopes, Graphs and Optmimisation. Cambridge University Press, Cambridge (1984)Google Scholar
  40. 40.
    Zadeh, L.A.: Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems. In: Zadeh, L.A., Klir, G.J., Yuan, B. (eds.). World Scientific Publishing, Singapore (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Matthias Baaz
    • 1
  • Helmut Veith
    • 2
  1. 1.Institut fär Algebra und Diskrete MathematikTechnische Universität WienAustria
  2. 2.Institut fär InformationssystemeTechnische Universität WienAustria

Personalised recommendations