Uncertainty and information: Foundations of generalized information theory

The book is truly an outstanding work giving a comprehensive exposition of a “generalized information theory” (GIT) having been developed in the early 1990s by Klir. The theory addresses the issue of information-based uncertainty and uncertainty-based information in their various manifestations. The book is organized in ten chapters. Chapter 1 provides the roots, motivations, framework, relevant terminology, and notation. Chapters 2 and 3 overview the two classical uncertainty theories. Chapter 2 presents the theory based on possibility and the associated notion necessity. Chapter 3 deals with probability-based uncertainty theory. Chapters 4–6 generalize the classical probability-based uncertainty theory in the frame of classical set theory. The generalizations are obtained by replacing the additivity assumption characteristic of probability measures with the weaker assumption of monotonicity with respect to the subsethood ordering. Chapter 4 focuses on the unifying features of the theories of imprecise probabilities whereas Chapter 5 explores the great diversity of these theories. Chapter 6 deals with the methods of measuring the amount of uncertainty and the associated information—especially with the generalizations of the Hartley and Shannon measures in the various theories of imprecise probabilities (introduced in Chapters 4 and 5). In the rest of the chapters, further generalization of the uncertainty theories is given by extending the classical sets to different types of fuzzy sets. Chapter 7 presents briefly the basic concepts of fuzzy set theory. Chapter 8 is on several fuzzification methods of uncertainty theories. Chapter 9 examines four methodological principles of uncertainty, i.e., principle of minimum uncertainty, principle of maximum uncertainty, principle of requisite generalization, and principle of uncertainty invariance. Chapter 10 summarizes GIT in the sense of both solved and still open problems. Seven appendices complement the book. The bibliography is really impressive—it occupies 29 pages. There is also subject index and name index. Each chapter is followed by “Notes” with many relevant historic and bibliographic remarks. The “Notes” deserve special attention—they are a sort of fingerprints typical for Klir’s books—providing a reader with not only additional common knowledge but also with the interesting personal points of view and corrections given by the author to a wide range of the problems from the propriety of some names being used in the literature to the comparison of GIT to “generalized theory of uncertainty” (GTU proposed by Zadeh).