Fuzzy logic and fuzzy set theory
type: Item
instance of: scholarly article
title: Fuzzy logic and fuzzy set theory
zbMATH Open document ID: 0786.03039
DOI: 10.1007/BF01270392
author: Gaisi Takeuti
author: Satoko Titani
published in: Archive for Mathematical Logic
publication date: 1994-01-13T00:00:00Z
review text: 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\}.
reviewed by: Siegfried J. Gottwald
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B52
zbMATH DE Number: 484790
zbMATH Keywords: truth degree set
zbMATH Keywords: Gentzen's sequent calculus
zbMATH Keywords: fuzzy logic
zbMATH Keywords: completeness theorem
zbMATH Keywords: fuzzy set theory FZF
zbMATH Keywords: inner model
zbMATH Keywords: hereditary stable sets
zbMATH Keywords: sheaf model
MaRDI profile type: Publication
cites work: Extending Gödel's negative interpretation to ZF
cites work: Q4143279
cites work: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory
cites work: Globalization of intuitionistic set theory
cites work: Q3722491
description: scientific article; zbMATH DE number 484790