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On completeness results for predicate Łukasiewicz, product, Gödel and nilpotent minimum logics expanded with truth-constants. (English) Zbl 1213.03035
Summary: In this paper we deal with generic expansions of first-order predicate logics of some left-continuous t-norms with a countable set of truth-constants. Besides already known results for the case of Łukasiewicz logic, we obtain new conservativeness and completeness results for some other expansions. Namely, we prove that the expansions of predicate product, Gödel and nilpotent minimum logics with truth-constants are conservative, which already implies the failure of standard completeness for the case of product logic. In contrast, the expansions of predicate Gödel and nilpotent minimum logics are proved to be strong standard complete, but, when the semantics is restricted to the canonical algebra, they are proved to be complete only for tautologies. Moreover, when the language is restricted to evaluated formulae, we prove canonical completeness for deductions from finite sets of premises.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
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