Monoidal t-norm logic
Monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity.
T-norms are binary functions on the real unit interval [0, 1] which are often used to represent a conjunction connective in fuzzy logic. Every left-continuous t-norm has a unique residuum, that is, a function such that for all x, y, and z,
The residuum of a left-continuous t-norm can explicitly be defined as
This ensures that the residuum is the 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. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.
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