##
**Mathematics behind fuzzy logic.**
*(English)*
Zbl 0940.03029

Advances in Soft Computing. Heidelberg: Physica-Verlag. x, 191 p. (1999).

Fuzzy mathematics and contemporary many-valued logic are tightly connected both on the theoretical level and concerning applications. It is therefore not astonishing that there is some common mathematics behind them, as shown by the monograph under review. The book consists of five chapters.

Chapter 1 starts with lattices, equivalence relations, filters and residuated lattices and comes to the concept of BL-algebra, introduced by Hájek as a particular case of residuated lattices. The chapter ends with a thorough study of deductive systems in a BL-algebra.

Chapter 2 begins with Wajsberg algebras, which have many important properties and direct applications to fuzzy logic. In fact, Wajsberg algebras are in the core of fuzzy set theory. Then the chapter studies MV-algebras, by giving first a long definition which in fact points out some basic properties of this algebraic structure. It is proved that there is a one-to-one correspondence between MV-algebras and Wajsberg algebras. Besides, every MV-algebra is a BL-algebra. Complete MV-algebras are studied in some detail. Locally finite MV-algebras coincide with locally finite BL-algebras and are isomorphic to certain subalgebras of Łukasiewicz(-Moisil) algebras. Semi-simple BL-algebras are MV-algebras and they can be represented as algebras of fuzzy sets. Complete MV-algebras can be made into pseudo-Boolean algebras, i.e., pseudo-complemented bounded distributive lattices.

Chapter 3 focuses in detail on Pavelka’s fuzzy sentential logic in the algebraic formulation due to the author. A discussion about the postulates to be satisfied by this logic yields the conclusion that the set \(L\) of truth values should be a complete MV-algebra and if \(L\) is the interval \([0,1]\) then this requirement amounts to \(L\) being an injective MV-algebra. The semantic consequence operator \(C^{\text{sem}}\) is then introduced. The syntax has the peculiarity that a fuzzy rule of inference consists of two components. The first component operates on formulas and is, in fact, a rule of inference in the usual sense; the second component operates on truth values and says how the truth value of the conclusion is to be computed from the truth values of the premises such that the degree of truth be preserved. The most important result is the completeness theorem: if the set of truth functions is an injective MV-algebra then the semantic operator \(C^{\text{sem}}\) coincides with the syntactic consequence operator \(C^{\text{syn}}\). If \(L=[0,1]\) is a residuated lattice but not a complete MV-algebra then there exist fuzzy theories \(T\) that are not axiomatizable, i.e., \(C^{\text{sem}}(T)\neq C^{\text{syn}}(T)\).

Chapter 4 is devoted to fuzzy relations. Necessary and sufficient conditions for the existence of solutions of general fuzzy equations are given. Then the study, within the framework of a complete BL-algebra \(L\), is focused on fuzzy similarity relations and their connections to approximate reasoning. A new method for many-valued reasoning based on fuzzy similarity is introduced which does not require any defuzzification technique.

The book contains many exercises, varying from routine computations to more sophisticated theoretical ones. Chapter 5 provides full solutions. Numerous examples of applications to everyday life, social sciences, medicine etc., are quite convincing.

All in all, this monograph is an excellent contribution to the field. It will certainly stimulate further research.

Reviewer’s remarks. 1. Curiously enough, the monograph by A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy relation equations and their applications to knowledge engineering (Kluwer Academic Publishers, Dordrecht) (1989; Zbl 0694.94025) is not quoted. 2. On page 12, line 16, read \(0\neq 1\).

Chapter 1 starts with lattices, equivalence relations, filters and residuated lattices and comes to the concept of BL-algebra, introduced by Hájek as a particular case of residuated lattices. The chapter ends with a thorough study of deductive systems in a BL-algebra.

Chapter 2 begins with Wajsberg algebras, which have many important properties and direct applications to fuzzy logic. In fact, Wajsberg algebras are in the core of fuzzy set theory. Then the chapter studies MV-algebras, by giving first a long definition which in fact points out some basic properties of this algebraic structure. It is proved that there is a one-to-one correspondence between MV-algebras and Wajsberg algebras. Besides, every MV-algebra is a BL-algebra. Complete MV-algebras are studied in some detail. Locally finite MV-algebras coincide with locally finite BL-algebras and are isomorphic to certain subalgebras of Łukasiewicz(-Moisil) algebras. Semi-simple BL-algebras are MV-algebras and they can be represented as algebras of fuzzy sets. Complete MV-algebras can be made into pseudo-Boolean algebras, i.e., pseudo-complemented bounded distributive lattices.

Chapter 3 focuses in detail on Pavelka’s fuzzy sentential logic in the algebraic formulation due to the author. A discussion about the postulates to be satisfied by this logic yields the conclusion that the set \(L\) of truth values should be a complete MV-algebra and if \(L\) is the interval \([0,1]\) then this requirement amounts to \(L\) being an injective MV-algebra. The semantic consequence operator \(C^{\text{sem}}\) is then introduced. The syntax has the peculiarity that a fuzzy rule of inference consists of two components. The first component operates on formulas and is, in fact, a rule of inference in the usual sense; the second component operates on truth values and says how the truth value of the conclusion is to be computed from the truth values of the premises such that the degree of truth be preserved. The most important result is the completeness theorem: if the set of truth functions is an injective MV-algebra then the semantic operator \(C^{\text{sem}}\) coincides with the syntactic consequence operator \(C^{\text{syn}}\). If \(L=[0,1]\) is a residuated lattice but not a complete MV-algebra then there exist fuzzy theories \(T\) that are not axiomatizable, i.e., \(C^{\text{sem}}(T)\neq C^{\text{syn}}(T)\).

Chapter 4 is devoted to fuzzy relations. Necessary and sufficient conditions for the existence of solutions of general fuzzy equations are given. Then the study, within the framework of a complete BL-algebra \(L\), is focused on fuzzy similarity relations and their connections to approximate reasoning. A new method for many-valued reasoning based on fuzzy similarity is introduced which does not require any defuzzification technique.

The book contains many exercises, varying from routine computations to more sophisticated theoretical ones. Chapter 5 provides full solutions. Numerous examples of applications to everyday life, social sciences, medicine etc., are quite convincing.

All in all, this monograph is an excellent contribution to the field. It will certainly stimulate further research.

Reviewer’s remarks. 1. Curiously enough, the monograph by A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy relation equations and their applications to knowledge engineering (Kluwer Academic Publishers, Dordrecht) (1989; Zbl 0694.94025) is not quoted. 2. On page 12, line 16, read \(0\neq 1\).

Reviewer: S.Rudeanu (Bucureşti)

### MSC:

03B52 | Fuzzy logic; logic of vagueness |

06D35 | MV-algebras |

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

03E72 | Theory of fuzzy sets, etc. |

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