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Standard completeness theorem for $$\Pi$$MTL. (English) Zbl 1071.03013
$$\Pi$$MTL is a schematic extension of the monoidal t-norm-based logic (MTL) by a new connective that is in $$[0, 1]$$ interpreted by the ordinary product of reals. It is proved that $$\Pi$$MTL satisfies the standard completeness theorem, namely, that a formula of $$\Pi$$MTL is provable iff it is true in a degree 1 in all $$\Pi$$MTL-chains in $$[0, 1]$$ with finitely many Archimedean classes. From the algebraic point of view this means that the class of $$\Pi$$MTL-algebras in $$[0, 1]$$ generates the variety of all $$\Pi$$MTL-algebras.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03B50 Many-valued logic 03G25 Other algebras related to logic
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