 Besides MTL (the logic of all left-continuous t-norms) and Łukasiewicz and Gödel–Dummett logics (each induced by a particular t-norm), one can consider logics induced by sets of t-norms or, in general, arbitrary axiomatic extensions of MTL. In particular, the logic of all continuous t-norms (Hájek’s basic fuzzy logic BL) is obtained by adding the axiom   \[\varphi\mathbin{\&amp;}(\varphi\to{{\psi}}) \to \psi\mathbin{\&amp;}(\psi\to\varphi)\]   to those of MTL. Actually, for any set of continuous t-norms there is a finite axiomatization of the corresponding logic (Esteva, Godo, &amp; Montagna 2003; Haniková 2014); in most cases the axiomatization captures semantic consequence from finite sets of premises. In particular, the logic of the third prominent continuous t-norm (algebraic product), known as Product logic, is the extension of BL by the axiom   \[\neg\varphi \vee ((\varphi\to\varphi\mathbin{\&amp;}{{\psi}})\to{{\psi}}).\]   On the other hand, not all axiomatic extensions of MTL can be given a t-norm based semantics. For example, classical logic can be axiomatized as MTL \(+\) \(\varphi\vee\neg \varphi\), but the axiom of excluded middle is not a tautology under any t-norm based interpretation.