On the Hierarchy of t-norm Based Residuated Fuzzy Logics

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 to substructural logics, in particular to Ono's hierarchy of extensions of the Full Lambek Calculus.