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A complete many-valued logic with product-conjunction. (English) Zbl 0848.03005
Studies in the field of fuzzy sets related to triangular norms have shown that the infinitely many-valued logic over the real unit interval with 1 as the only designated truth degree and a product-based conjunction connective has not been discussed much till now, despite some representation theorems of $$t$$-norms which refer to the product as a kind of basic $$t$$-norm.
Choosing the product conjunction, a corresponding implication connective via residuation (i.e., both of them as an adjoint pair), and defining negation formally from implication and the truth degree constant 0 like in intuitionistic logic, the authors constitute the system they investigate.
The completeness proof is given via algebraic studies essentially like the corresponding proof for Łukasiewicz’s infinite-valued logic, but now introducing and investigating product algebras for this product logic instead of the MV-algebras for the Łukasiewicz case.

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