On the Hierarchy of t-norm Based Residuated Fuzzy Logics
Francesc Esteva, Lluı́s Godo, Àngel García-Cerdaña · Studies in fuzziness and soft computing · 2003
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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.