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A systematic study of particular t-norm fuzzy logics and their classes began with Hájek's (1998) monograph Metamathematics of Fuzzy Logic, which presented the notion of the logic of a continuous t-norm, the logics of the three basic continuous t-norms (Łukasiewicz, Gödel, and product), and the 'basic' fuzzy logic BL of all continuous t-norms ...
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic.
Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic.
Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. There is in fact more than one possible generalization. The most widely used operations are called standard fuzzy set operations ; they comprise: fuzzy complements , fuzzy intersections , and fuzzy unions .
Instead of min and max one can use t-norm and t-conorm, respectively; [4] for example, min(a, b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.
In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the t-norm fuzzy logics.It belongs to the broader class of substructural logics, or logics of residuated lattices; [1] it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FL ew, or intuitionistic logic without ...
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-algebras. Gödel fuzzy logic is the extension of basic fuzzy logic BL where conjunction is the Gödel t-norm (that is ...
In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; [1] it extends the logic MTL of all left-continuous t-norms.