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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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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