Fuzzy logic and fuzzy set theory
On the semantic side, the authors consider the truth degree set \([0,1]\) with a constant for the degree 0.5, with min, max, Gödel implication, the negation defined by it in the intuitionistic style, but also with Łukasiewicz's (1-..)-negation, bounded sum, and product as connectives, and with sup, inf as quantifiers. Truth degree 1 is the only designated one. On the syntactic side they start from Gentzen's sequent calculus LJ for intuitionistic first-order logic and add two rules (one of them infinitary) and 46 axioms. Their main result concerning this fuzzy logic is the completeness theorem. Turning to fuzzy set theory FZF means to consider a ZF-like axiomatic theory, based on this fuzzy logic, with double complement and Zorn's lemma as additional axioms. For FZF there is given a version of Powell's inner model \(S\) of hereditary stable sets and proven that FZF \(\models\) ``\(S\) is a model of ZFC'', and inside \(S\) Grayson's sheaf model \(S^ l\) over \(l=[0,1]^ S\) is constructed and ``\(S^ l\) is a model of FZF'' proven in \(S\). Finally the authors prove that there exists a functional relation describing a bijection (via a suitable identification of elements of \(S^ l\)!) between \(S^ l\) and the universe of FZF. \{Reviewer's remark: The paper is an interesting contribution to the theoretical foundation of fuzzy set theory. But it is more than astonishing that the authors cite besides their own papers on what they call ``intuitionistic fuzzy logic'' -- and what is not fuzzy logic at all -- no other papers concerned with foundational issues of fuzzy sets\}.
scientific article; zbMATH DE number 484790