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Soft Computing

, Volume 4, Issue 2, pp 106–112 | Cite as

Basic Fuzzy Logic is the logic of continuous t-norms and their residua

  • R. Cignoli
  • F. Esteva
  • L. Godo
  • A. Torrens
Original paper

Abstract

 In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hájek in [4]. In fact, Hájek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Gödel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.

Key words many-valued logic (basic Łukasiewicz product and Gödel logics) equational classes fuzzy logic t-norms and standard completeness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • R. Cignoli
    • 1
  • F. Esteva
    • 2
  • L. Godo
    • 2
  • A. Torrens
    • 3
  1. 1.Departamento de Matemáticas, Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires – CONICET, Pabellón 1, Ciudad Universitaria 1428 Buenos Aires, Argentina E-mail: cignoli@mate.dm.uba.arAR
  2. 2.Institut d'Investigació en Intel·ligència Artificial (IIIA) Consejo Superior de Investigaciones Científicas (CSIC) Campus Universitat Autònoma de Barcelona, s/n 08193 Bellaterra, Spain E-mail: {esteva,godo}@iiia.csic.esES
  3. 3.Departament de Lògica, Facultat de Matemàtiques Universitat de Barcelona, Gran Via de les Corts Catalanes, 585 08007 Barcelona, Spain E-mail: torrens@mat.ub.esES

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