Vaguely defined objects. Representations, fuzzy sets and nonclassical cardinality theory
type: Item
instance of: scholarly article
title: Vaguely defined objects. Representations, fuzzy sets and nonclassical cardinality theory
zbMATH Open document ID: 0856.04002
author: Maciej Wygralak
published in: Theory and Decision Library. Series B: Mathematical and Statistical Methods
publication date: 1996-07-29T00:00:00Z
review text: In comparison with the abundant number of books on various aspects of fuzzy logic and fuzzy sets which mostly present applications and whose theory is brief, this book is highly theoretical where only few applications are briefly outlined. I consider this good and refreshing, especially because the author expended a lot of effort to be mathematically precise and sufficiently general. The main topic of the book is a systematic treatise of the theory of cardinal numbers in fuzzy set theory. The book begins with a non-formal exposition of vague objects which have the role of something ungraspable given by a vague property, and which are mathematically described using various approaches. This is in accordance with the approach taken in the Alternative Set Theory [cf. \textit{P. Vopěnka}, Mathematics in the alternative set theory (1979; Zbl 0499.03042); the reviewer, The alternative mathematical model of linguistic semantics and pragmatics (1992; Zbl 0837.03029)]. The basic mathematical tool for the author are fuzzy sets. A vague object is thus a fuzzy set \(\text{FS}(A)\) described by the membership function \(A: M\to [0, 1]\). Hence, unlike usual mathematical understanding, the author makes a distinction between a fuzzy set and its membership function. I consider this unnecessary and only making the notation more complicated. After an exposition of the basic concepts and operations, also references to other means for grasping vague objects are presented, including ultrafuzzy sets, fuzzy sets of type 2, twofold fuzzy sets, flou sets and partial sets. The author also mentions semisets, but unfortunately only the basic result stating that semisets can consistently be introduced into Gödel-Bernays set theory is considered, and thus the reference to the Alternative Set Theory, which is a radical departure from some principles of classical set theory, is misleading and incorrect. However, the general scheme on p. 32, presenting vague objects and their mathematical descriptio…
reviewed by: Vilém Novák
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 911351
zbMATH Keywords: cardinal arithmetic
zbMATH Keywords: alternative set theory
zbMATH Keywords: cardinal numbers in fuzzy set theory
zbMATH Keywords: vague objects
zbMATH Keywords: membership function
zbMATH Keywords: Łukasiewicz logic
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 911351