Metamathematics of fuzzy logic.

*(English)*Zbl 0937.03030Fuzzy logic in the sense of this book is infinite-valued logic, partly enriched with a graded notion of entailment. The author starts from the observation that t-norms play a crucial role in fuzzy set theory as well as in many-valued logic: the Lukasiewicz systems, the Gödel systems, and also the product logic may be considered as being based in a uniform way upon some particular t-norm. Accordingly, the author first develops a core logic for t-norm based systems of [0,1]-valued logic via some suitable algebraic semantics. The fundamental algebraic structures prove to be commutative residuated lattices, which at the same time are divisible monoids, and satisfy some kind of pre-linearity condition. These “basic algebras” are common generalizations, e.g., of MV-algebras and Heyting algebras. For the particular semigroups formed by a t-norm over the real unit interval, divisibility is the algebraic equivalent of the continuity of the t-norm. The author gives adequate axiomatizations for the propositional as well as the first-order logics of these structures, which also axiomatize the corresponding entailment relations for finite sets of premisses, and which can be extended to adequate axiomatizations, e.g., of the infinite-valued Lukasiewicz and Gödel logics. For the Lukasiewicz case also the Pavelka approach toward a graded notion of entailment and inference is discussed, and coded inside standard Lukasiewicz logic for the case that one adds truth degree constants for all rationals of the unit interval. The investigations in these logical systems are enriched by complexity and undecidability results for propositional as well as for first-order systems. And there are also discussions of applicational aspects: e.g., of approximate inferences, fuzzy rules, fuzzified quantifications and modalities, or of the liar paradox. The last chapter surveys the historical development of many-valued and fuzzy logic. The book contains a wealth of original and important results which have been found by the author in the last couple of years. It is the first monograph by an outstanding logician which shows that the field of fuzzy logics, taken in the narrow sense of this word, should be considered as a serious part of mathematical logic at all.

Reviewer: Siegfried Johannes Gottwald (Leipzig)

##### MSC:

03B52 | Fuzzy logic; logic of vagueness |

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

03B50 | Many-valued logic |

03G10 | Logical aspects of lattices and related structures |

03D35 | Undecidability and degrees of sets of sentences |