T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction.
In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the law of prelinearity, (A → B) ∨ (B → A).
Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied.
Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well.
As members of the family of fuzzy logics, t-norm fuzzy logics primarily aim at generalizing classical two-valued logic by admitting intermediary truth values between 1 (truth) and 0 (falsity) representing degrees of truth of propositions.
The degrees are assumed to be real numbers from the unit interval [0, 1].
T-norm fuzzy logics impose certain natural constraints on the truth function of conjunction.
of conjunction is assumed to satisfy the following conditions: These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (t-norm based).
such that for all x, y, and z in [0, 1], The residuum of a left-continuous t-norm can explicitly be defined as This ensures that the residuum is the pointwise largest function such that for all x and y, The latter can be interpreted as a fuzzy version of the modus ponens rule of inference.
The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic.
Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation
Formulae that always evaluate to 1 are called tautologies with respect to the given left-continuous t-norm
Besides the standard real-valued semantics on [0, 1], the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices.
Since then, a plethora of t-norm fuzzy logics have been introduced and their metamathematical properties have been investigated.
Some of the most important t-norm fuzzy logics were introduced in 2001, by Esteva and Godo (MTL, IMTL, SMTL, NM, WNM),[1] Esteva, Godo, and Montagna (propositional ŁΠ),[6] and Cintula (first-order ŁΠ).
Algebraic semantics is predominantly used for propositional t-norm fuzzy logics, with three main classes of algebras with respect to which a t-norm fuzzy logic