Vaguely defined objects. Representations, fuzzy sets and nonclassical cardinality theory
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 descriptions, is very nice. The endeavour to be general led the author to introducing ``two functions'' approximation \((f(A), g(A))\) of membership functions. This enabled him to include the other above-mentioned means for the description of vaguely defined objects. One of the main tools the author presents for further work is Łukasiewicz logic. A this place, it is suprising that he completely disregards fuzzy logic in the narrow sense (of Łukasiewicz style) as presented, e.g., by \textit{P. Hájek} [``Fuzzy logic as logic'', in: G. Coletti et al. (eds.), Mathematical models for handling partial knowledge in artificial intelligence, Sel. Pap. Workshop, Sicily, Italy, 1994, 21-30 (1995)] and the reviewer [Int. J. Gen. Syst. 24, 377-405 (1996; Zbl 0855.03010)], though the latter, providing tools for description of the vagueness phenomenon, is better prepared for this task than pure Łukasiewicz logic. Chapters 5-14 provide a systematic explanation of generalized cardinality theory. The cardinal of a vaguely defined object is, in general, a fuzzy set. The author demonstrates that his concept behaves well with respect to all requirements that cardinal numbers should have (such as equipotency and others). The explanation includes also arithmetic of generalized cardinal numbers (addition, multiplication, generalized operations). Chapter 15 outlines a further generalization to the representation of vaguely defined objects by free couples. The final Chapter 16 shortly discusses other generalizations (the use of t-norms or lattice-valued fuzzy sets). This book offers the first successful (as far as I know) theory of generalized cardinal numbers in fuzzy set theory. Maybe it is somewhat over-formalized, which makes its reading quite difficult. In any case, it is a valuable contribution to the literature on fuzzy sets and fuzzy logic.
scientific article; zbMATH DE number 911351