Fuzzy sets and residuated logic
type: Item
instance of: scholarly article
title: Fuzzy sets and residuated logic
zbMATH Open document ID: 0763.03030
DOI: 10.1016/0165-0114(92)90180-C
published in: Fuzzy Sets and Systems
publication date: 1992-09-27T00:00:00Z
review text: Based on a logic which has residuated lattices as its intended truth value structures and using a modalized language, the author develops a formal set theory RZF, which is intended to act as a framework for fuzzy set theory and which has a natural embedding of the classical ``crisp'' sets. Its semantics is provided by a suitably modified ``residuated'' Kripke-Joyal semantics based on a forcing style definition of its basic relations. The presentation is in a clear style; a minor irritation is only caused on p. 212 by a proof starting within an actual proof and without a clear claim. Nevertheless it would have been helpful to see some actual semantic objects in an elementary presentation to give a better idea of the relation to usual sets. In comparison with the prevailing discussions in the field of fuzzy sets it is nice to have this attitude toward residuated lattices which continues work of \textit{J. Pavelka}, which is referred to, and \textit{V. Novak}, who has not been mentioned [cf., e.g., his book ``Fuzzy sets and their applications'' (1989; Zbl 0683.94018)]; in any case, it seems to be an approach which is superior to those ones using Heyting-valued structures or (usual) topos theory. Unfortunately, the author gives only very scarce hints to other approaches toward full systems of set theories/set theoretic universes for fuzzy sets. For approaches not based on residuated lattices or Łukasiewicz logic this may count as a minor deficiency, but in other cases, like the embedding of the crisp sets into the fuzzy sets of \textit{D. Klaua} [Monatsber. Deutsch. Akad. Wiss. Berlin 9, 258-272 (1967; Zbl 0221.02050)] or the reviewer's approach toward a cumulative universe of fuzzy sets based on Łukasiewicz logic and a generalized extensionality [Fuzzy Sets Syst. 2, 125-151 (1979; Zbl 0408.03042); ibid. 3, 49-74 (1980; Zbl 0421.03043)], this is a serious gap.
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 64932
zbMATH Keywords: residuated logic
zbMATH Keywords: modal logic
zbMATH Keywords: residuated lattices
zbMATH Keywords: formal set theory RZF
author: K. Lano
reviewed by: Siegfried J. Gottwald
MaRDI profile type: Publication
cites work: A New Approach to Quantum Logic
cites work: Modality and possibility in some intuitionistic modal logics
cites work: Harvey Friedman's research on the foundations of mathematics
cites work: Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic
cites work: Fuzzy sets and toposes - towards higher order fuzzy logic
cites work: Q3852176
cites work: The logic of inexact concepts
cites work: Set theory. An introduction to independence proofs
cites work: Q4124795
cites work: On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi
cites work: Q3328546
cites work: Q3748967
full work available at URL: 0165-0114(92)90180-c
OpenAlex ID: W1993598011
Recommended article: Fuzzy sets and formal logics
Recommended article: Fuzzy logic and fuzzy set theory
Recommended article: Fuzzy sets, logics and reasoning about knowledge
Recommended article: Q4624017
Recommended article: Fuzzy sets and geometric logic
Recommended article: Many-valued logic and fuzzy set theory
Recommended article: Q4836224
Recommended article: Q4697932
Recommended article: Q3105675
description: scientific article; zbMATH DE number 64932