Editing Fuzzy logic (section)

=== Propositional fuzzy logics ===
The most important propositional fuzzy logics are:
* [[MTL (logic)|Monoidal t-norm-based propositional fuzzy logic]] MTL is an [[Axiomatic system#Axiomatization|axiomatization]] of logic where [[Logical conjunction|conjunction]] is defined by a [[left continuous]] [[t-norm]] and implication is defined as the residuum of the t-norm. Its [[structure (mathematical logic)|model]]s correspond to MTL-algebras that are pre-linear commutative bounded integral [[residuated lattice]]s.
* [[BL (logic)|Basic propositional fuzzy logic]] BL is an extension of MTL logic where conjunction is defined by a continuous t-norm, and implication is also defined as the residuum of the t-norm. Its models correspond to BL-algebras.
* [[Lukasiewicz fuzzy logic|Łukasiewicz fuzzy logic]] is the extension of basic fuzzy logic BL where standard conjunction is the Łukasiewicz t-norm. It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to [[MV-algebra]]s.
* Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the [[Gödel]] t-norm (that is, minimum). It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras.
* Product fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the product t-norm. It has the axioms of BL plus another axiom for cancellativity of conjunction, and its models are called product algebras.
* Fuzzy logic with evaluated syntax (sometimes also called Pavelka's logic), denoted by EVŁ, is a further generalization of mathematical fuzzy logic. While the above kinds of fuzzy logic have traditional syntax and many-valued semantics, in EVŁ syntax is also evaluated. This means that each formula has an evaluation. Axiomatization of EVŁ stems from Łukasziewicz fuzzy logic. A generalization of the classical Gödel completeness theorem is provable in EVŁ.<ref name="FuzzyLogic2017"></ref>