On the Hierarchy of t-norm Based Residuated Fuzzy Logics

  • Francesc Esteva
  • Lluís Godo
  • Àngel García-Cerdaña
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 114)


In this paper we overview recent results, both logical and algebraic, about [0, 1]-valued logical systems having a t-norm and its residuum as truth functions for conjunction and implication. We describe their axiomatic systems and algebraic varieties and show they can be suitably placed in a hierarchy of logics depending on their characteristic axioms. We stress that the most general variety generated by residuated structures in [0, 1], which are defined by left-continuous t-norms, is not the variety of residuated lattices but the variety of pre-linear residuated lattices, also known as MTL-algebras. Finally, we also relate t-norm based logics to substructural logics, in particular to Ono’s hierarchy of extensions of the Full Lambek Calculus.


Residuated Lattice Axiomatic System Substructural Logic Axiomatic Extension Standard Completeness 
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  1. 1.
    Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic Fuzzy Logic is the logic of continuous t-norms and their residua. Soft Computing 4 (2000) 106–112CrossRefGoogle Scholar
  2. 2.
    Cintula, P.: About axiomatic system of the Product Fuzzy Logic. Soft Computing 5 (2001) 243–244MATHCrossRefGoogle Scholar
  3. 3.
    Di Nola, A., Esteva, F., Garcia, P., Godo, L. and Sessa, S.: Subvarieties of BL-algebras generated by single-component chains. Archive for Mathematical Logic, to appearGoogle Scholar
  4. 4.
    Dummett, M.: A propositional calculus with denumerable matrix. Journal of Symbolic Logic 24 (1959) 97–106MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Esteva, F. and Godo, L.: Monoidal t-norm based Logic: Towards a logic for left continuous t-norms. Fuzzy Sets and Systems vol. 124, 3 (2001) 271–288MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Esteva, F., Gispert, J., Godo, L. and Montagna, F.: On the standard and Rational Completeness of some Axiomatic extensions of Monoidal t-norm Based Logic. To appear in Studia LogicaGoogle Scholar
  7. 7.
    Esteva, F., Godo, L., Hâjek, P. and Navara, M.: Residuated fuzzy logics with an involutive negation. Archive for Mathematical Logic 39 (2000) 103–124MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Esteva, F., Trillas, E. and Domingo, X.: Weak and strong negation functions for Fuzzy Set Theory. Proc. 11th Int. Symposium on Multiple-valued Logic ISMVL’81 (1981) 23–26Google Scholar
  9. 9.
    Ferreirim, I.M.: On varieties and quasivarieties of Hoops and their reducts. Phd. Thesis. University of Illinois at Chicago (1992)Google Scholar
  10. 10.
    Fodor, J.: Nilpotent minimum and related connectives for fuzzy logic. Proc. of FUZZ—IEEE’95 (1995) 2077–2082Google Scholar
  11. 11.
    Girard, J.Y.: Linear Logic. Theoretical Computer Science, 50 (1987) 1–102MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gottwald, S.: A Treatise on Many-valued Logics. Studies in Logic and Computation. Research Studies Press, Baldock (2001)Google Scholar
  13. 13.
    Gottwald, S., Garcia-Cerdaiia, A., Bou, F.: Axiomatizing Monoidal Logic — a Correction. SubmmitedGoogle Scholar
  14. 14.
    Gottwald, S. and Jenei, S.: On a new axiomatization for Involutive Monoidal t-norm based Logic. Fuzzy Sets and Systems, vol. 124, 3 (2001) 303–308MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hâjek, P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4 Kluwer (1998)Google Scholar
  16. 16.
    Hâjek, P.: Observations on the Monoidal t-norm Logic, Fuzzy Sets and Systems (to appear)Google Scholar
  17. 17.
    Hâjek, P., Godo, L., Esteva, F.: A complete many-valued logic with product conjunction, Archive for Mathematical Logic 35 (1996) 191–208MathSciNetMATHGoogle Scholar
  18. 18.
    Höhle, U.: Commutative, residuated 1-monoids. In: Höhle, U. and Klement, E.P. eds., Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Acad. Publ., Dordrecht (1995) 53–106Google Scholar
  19. 19.
    Jenei, S. and Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica, to appearGoogle Scholar
  20. 20.
    Jipsen, P.: Some lists of finite structures at http://atlas.math.vanderbilt.edu/Npjipsen/gap/ Google Scholar
  21. 21.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms, Kluwer (2000)Google Scholar
  22. 22.
    Lambek, J.: Logic without structural rules (Another look at cut elimination). In: Schroeder-Heister, P. and Dosen, K. eds., Substructural Logics, Studies in Logic and Computation 2, Oxford Sciences Pub. (1993) 179–206Google Scholar
  23. 23.
    Okada, M.O. and Terui, K.: The finite model property for various fragments of intuitionistic linear logic Journal of Symbolic Logic 64 (2) (1999) 790–802MathSciNetMATHGoogle Scholar
  24. 24.
    Ono, H.: Logic without contraction rule and residuated lattices I. To appear in the book for the Festschrift of Prof. R.K. Meyer, edited by E. MaresGoogle Scholar
  25. 25.
    Pavelka, J.: On Fuzzy Logic I, II, III. Z.Math. Logik Grundl. Math. 25 (1979) 45–52, 119–134, 447–464Google Scholar
  26. 26.
    Rose, P.A. and Rosser, J.B.: Fragments of many-valued statement calculi, Trans. A.M.S. 87 (1958) 1–53Google Scholar
  27. 27.
    Schweizer, B. and Sklar, A.: Probabilistic metric spaces, North Holland, Amsterdam (1983)Google Scholar
  28. 28.
    Troelstra, A.S.: Lectures in Linear Logic, CSLI Lecture Notes, vol. 29, Center for the Study of Language and Information, Stanford (1991)Google Scholar
  29. 29.
    Zadeh, L.A.: Fuzzy Logic, IEEE Comput. 1, 83 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Francesc Esteva
    • 1
  • Lluís Godo
    • 1
  • Àngel García-Cerdaña
    • 1
  1. 1.Institut d’Investigació en Intel.ligència ArtificialCSIC Campus Univ. Autònoma de Barcelona s/nBellaterraSpain

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