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**Uncertainty theory. An introduction to its axiomatic foundations.**
*(English)*
Zbl 1072.28012

Studies in Fuzziness and Soft Computing 154. Berlin: Springer (ISBN 3-540-21333-3/hbk). x, 411 p. (2004).

Uncertainty theory is used by the author as generic term of probability theory, fuzzy set theory and rough set theory. The author presents axiomatic foundations of these three approaches (Chapters 2–4) and of all nine possible combinations of two such approaches (Chapters 5–13). For instance random variables with fuzzy values are subject of fuzzy random theory in Chapter 5, random fuzzy theory (Chapter 6) deals with fuzzy sets on a universe of probability distributions, fuzzy rough theory stands for rough sets on a universe of fuzzy sets. To complete this list, we find in a more or less similar presentation: bifuzzy theory (Chapter 7), birandom theory (8), rough random theory (9), rough fuzzy theory (10), random rough theory (11) and birough theory (13).

The typical procedure of the author is the following: Definition of an appropriate variable (e.g., fuzzy variable, fuzzy random variable a.s.o.) and an associated (not necessarily additive) measure (e.g., probability, credibility, trust, chance measure) with its distribution function. So called optimistic and pessimistic values of these measures (some kind of quantiles) and an always real-valued expectation are introduced. Finally, convergence concepts are investigated. The presentation is clear but somewhat tedious because of the structural similarity of all Chapters. The value of these formal approaches should be discussed. Classical fuzzy set theory and rough set theory live from the excluded middle law which leads to essentials like possibility and necessity, upper and lower bounds. Using the arithmetic mean of these essentials, obviously we obtain an at least self dual measure and all the “probability-like” investigations presented in this book are possible. It seems to me, however, e.g. fuzzy theory with credibility measures is a “defuzzified” theory.

The typical procedure of the author is the following: Definition of an appropriate variable (e.g., fuzzy variable, fuzzy random variable a.s.o.) and an associated (not necessarily additive) measure (e.g., probability, credibility, trust, chance measure) with its distribution function. So called optimistic and pessimistic values of these measures (some kind of quantiles) and an always real-valued expectation are introduced. Finally, convergence concepts are investigated. The presentation is clear but somewhat tedious because of the structural similarity of all Chapters. The value of these formal approaches should be discussed. Classical fuzzy set theory and rough set theory live from the excluded middle law which leads to essentials like possibility and necessity, upper and lower bounds. Using the arithmetic mean of these essentials, obviously we obtain an at least self dual measure and all the “probability-like” investigations presented in this book are possible. It seems to me, however, e.g. fuzzy theory with credibility measures is a “defuzzified” theory.

Reviewer: Wolfgang NĂ¤ther (Freiberg)

### MSC:

28E10 | Fuzzy measure theory |

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60A05 | Axioms; other general questions in probability |

03E72 | Theory of fuzzy sets, etc. |