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Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. (English) Zbl 1168.03052
This is an algebraic study of t-norm based logics. Hájek’s Basic Logic is the logic of all continuous t-norms and their residual implications (see [P. Hájek, Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]). Esteva and Godo observed that left-continuity (rather than continuity) is sufficient, and necessary, for a t-norm to have a residuum, and proposed a weaker logic, called MTL, conjecturing that this is the logic of all left-continuous t-norms and their residual implications. This conjecture was proved by Jenei and Montagna. Interestingly, MTL can be characterized as Full Lambek calculus plus exchange, weakening and prelinearity. The authors consider three completeness properties with respect to the semantics given by linearly ordered MTL-algebras. Five different kinds of semantical generators are considered, namely the real, the rational, the hyperreal unit interval, the strict hyperreals (i.e., those ultrapowers which are a proper extension of the real unit interval) and finite chains. In a final section all these completeness properties and semantics are discussed for first-order logics, and a number of results are proved using a variety of techniques, including formal grammars and languages.

MSC:
03G25 Other algebras related to logic
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
06F05 Ordered semigroups and monoids
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