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Uncertainty theory. 4th ed. (English) Zbl 1309.28001
Springer Uncertainty Research. Berlin: Springer (ISBN 978-3-662-44353-8/hbk; 978-3-662-44354-5/ebook). xv, 487 p. (2015).
The author of this book is a very active Chinese mathematician and chief of the Uncertainty Theory Laboratory at the Tsinghua University in Beijing. Since 1996 he works on the problem of modelling uncertainty, starting with uncertain programming. The first edition of Uncertainty theory was published in 2004 (see [Zbl 1072.28012]), followed by the second edition in 2007 (see [Zbl 1141.28001]), the third edition in 2010 and now the fourth edition in 2015. However, all the four editions are four very different books. There are only few similarities between the first two editions and the present one. The author’s conviction is that probability theory is only applicable when samples are available and that belief degrees of experts cannot be treated as (subjective) probabilities. For evaluating belief degrees, he introduces more general measures.
In this edition the author starts with uncertain measures, which are somewhat more general than credibility measures used in the first two editions. An uncertain measure is a set function that satisfies the three axioms of normality, duality and subadditivity. Whereas the first two chapters introduce the essentials for uncertain measures und uncertain variables, all the remaining 14 chapters deal with uncertainty in special mathematical disciplines: uncertain programming, uncertain statistics, uncertain risk analysis, uncertain reliability analysis, uncertain propositional logic, uncertain entailment, uncertain set, uncertain logic, uncertain inference, uncertain process, uncertain renewal process, uncertain calculus, uncertain differential equation, uncertain finance. An appendix of 100 pages presents essentials of probability theory and chance theory. Chance theory is the attempt to consider uncertainty and probability simultaneously, where a chance space is the product space of an uncertainty space and a probability space. The last part of the appendix discusses 12 frequently asked questions, e.g., the questions: “Why is fuzzy set unable to model unsharp concept?” and “What goes wrong with Cox’s theorem?”. Cox’s assertion is that any kind of belief measure is “isomorphic” to a probabilty measure.
Due to many included remarks and discussion points, this edition is much more readable than the first two ones. Nevertheless, the textbook is suitable especially for mathematicians.
Reviewer’s remark: The author is very self-confident. To be mentioned in the same breath with Kolmogorov and Zadeh, this is presumably his dream. The first sentence of The Bibliographic Notes at the end of each chapter is always: “The concept of …was first proposed by Liu …”. Preliminary ideas of other researchers are not mentioned there. On the website http://orsc.edu.cn/online of the Uncertainty Theory Laboratory the reader can find already the 5th edition of this book.

28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
28E10 Fuzzy measure theory
60A05 Axioms; other general questions in probability
68T37 Reasoning under uncertainty in the context of artificial intelligence
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