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**Mathematical principles of fuzzy logic.**
*(English)*
Zbl 0940.03028

The Kluwer International Series in Engineering and Computer Science. 517. Dordrecht: Kluwer Academic Publishers. xiii, 320 p. (1999).

The expression “fuzzy logic” is often used as an umbrella term for different applied techniques which stem from the original idea of fuzzy sets and fuzzy logic, techniques whose successes range from expert systems to control. Such applications as fuzzy control (often called “fuzzy logic control”) are rather far away from logic, but there is a definite logical part in fuzzy methodology, and this part is what the book under review is about.

After a brief introduction to logic, model theory, and main concepts of multi-valued logics, the authors proceed, in Chapter 4, to describe fuzzy logic as a natural generalization of the traditional two-valued logic. This generalization is, in many cases, more adequate in representing human reasoning: e.g., it successfully avoids paradoxes like the well-known heap paradox (One grain of sand is not a heap; if \(n\) grains are not a heap, then adding 1 extra grain does not make it a heap; by induction, for every \(k\), \(k\) grains are not a heap). For this generalized logic, the authors manage to prove analogues of complex theorems known in 2-valued case, such as Gödel’s completeness theorem. In the authors’ exposition, these generalizations sound rather simple and very natural, but they are a great triumph of serious research, because historically the first (seemingly natural) analogues of these classical logical theorems were proved to be false in multi-valued logic.

Chapter 5, which mainly contains the results of I. Perfilieva, describes interesting results about the possibility to represent (or at least approximate) an arbitrary fuzzy logical operation or relation by an appropriate propositional formula (i.e., as a composition of basic fuzzy logic operations). Chapter 6 expands the results of the previous chapter by adding operations such as hedges (“very”, “almost”, etc.) which have no 2-valued analogues. These operations lead to a natural explanation of the formulas used in the original Mamdani approach to fuzzy control. Finally, Chapter 7 analyzes fuzzy logic from the viewpoint of category theory.

The book is a pleasure to read. The only (minor) pedagogical drawback is that although the authors go into great detail to introduce even the most basic notions of propositional and first-order logic, they use terms from category theory (such as pullback) without explicitly defining them. This is OK for algebraist readers, but it may make the book somewhat more difficult to read for readers who are specialists in mathematical logic, readers who, without any doubt, stand to benefit from this book.

After a brief introduction to logic, model theory, and main concepts of multi-valued logics, the authors proceed, in Chapter 4, to describe fuzzy logic as a natural generalization of the traditional two-valued logic. This generalization is, in many cases, more adequate in representing human reasoning: e.g., it successfully avoids paradoxes like the well-known heap paradox (One grain of sand is not a heap; if \(n\) grains are not a heap, then adding 1 extra grain does not make it a heap; by induction, for every \(k\), \(k\) grains are not a heap). For this generalized logic, the authors manage to prove analogues of complex theorems known in 2-valued case, such as Gödel’s completeness theorem. In the authors’ exposition, these generalizations sound rather simple and very natural, but they are a great triumph of serious research, because historically the first (seemingly natural) analogues of these classical logical theorems were proved to be false in multi-valued logic.

Chapter 5, which mainly contains the results of I. Perfilieva, describes interesting results about the possibility to represent (or at least approximate) an arbitrary fuzzy logical operation or relation by an appropriate propositional formula (i.e., as a composition of basic fuzzy logic operations). Chapter 6 expands the results of the previous chapter by adding operations such as hedges (“very”, “almost”, etc.) which have no 2-valued analogues. These operations lead to a natural explanation of the formulas used in the original Mamdani approach to fuzzy control. Finally, Chapter 7 analyzes fuzzy logic from the viewpoint of category theory.

The book is a pleasure to read. The only (minor) pedagogical drawback is that although the authors go into great detail to introduce even the most basic notions of propositional and first-order logic, they use terms from category theory (such as pullback) without explicitly defining them. This is OK for algebraist readers, but it may make the book somewhat more difficult to read for readers who are specialists in mathematical logic, readers who, without any doubt, stand to benefit from this book.

Reviewer: V.Ya.Kreinovich (El Paso)

### MSC:

03B52 | Fuzzy logic; logic of vagueness |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |