A complete many-valued logic with product-conjunction
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.
scientific article; zbMATH DE number 881130
scientific article