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.


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