On the hierarchy of t-norm based residuated fuzzy logics

  • Authors:

  • Francesc Esteva;Lluís Godo;Àngel García-Cerdaña

  • Affiliations:

  • Institut d'Investigació en Intel.ligència Artificial - CSIC, Campus Univ. Autònoma de Barcelona s/n, 08193 Bellaterra, Spain;Institut d'Investigació en Intel.ligència Artificial - CSIC, Campus Univ. Autònoma de Barcelona s/n, 08193 Bellaterra, Spain;Institut d'Investigació en Intel.ligència Artificial - CSIC, Campus Univ. Autònoma de Barcelona s/n, 08193 Bellaterra, Spain

  • Venue:
  • Beyond two
  • Year:
  • 2003

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Abstract

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.