Lattice-valued logic. An alternative approach to treat fuzziness and incomparability
The book is a result of work of several people over more than 10 years done with a very limited admission to the literature. Working in such isolation, a lot of results were rediscovered. However, the book is written on a good mathematical level and is worth reading. The text contains mostly references to Chinese works only. However, in the year of publishing, the authors added an introductory chapter in which they put the necessary references and clarifications of their work with respect to the results in the subject obtained elsewhere. The key concept is that of a lattice implication algebra, which is isomorphic to that of a Wajsberg algebra. Moreover, each lattice implication algebra can be made an MV-algebra and vice-versa. Therefore, the book deals, in fact, with fuzzy logic with evaluated syntax (known also as Pavelka logic), which is proved to be complete for the case when the truth values form either a finite MV-algebra, or the Łukasiewicz MV-algebra (standard MV-algebra on \([0, 1]\)). This logic has been introduced by \textit{J. Pavelka} [``On fuzzy logic. I, II, III'', Z. Math. Logik Grundlagen Math. 25, 45--52 (1979; Zbl 0435.03020), 119--134 (1979; Zbl 0446.03015), 447--464 (1979; Zbl 0446.03016)] and further elaborated by V. Novák [for complete reference see \textit{V. Novák, I. Perfilieva} and \textit{J. Močkoř}: Mathematical principles of fuzzy logic. Kluwer, Boston/Dordrecht (1999; Zbl 0940.03028)]. The book is divided into three parts. The first part consists of Chapter 1 only, in which the authors explain the position of their work in the world of fuzzy logic. Part~II introduces the logic algebras, namely the mentioned lattice implication algebras and studies their properties. Part~III introduces the fuzzy logic with evaluated syntax based on lattice implication algebras. First, they introduce and analyse propositional logic and its slight generalisation -- gradational lattice-valued logic -- where the authors have further generalized the concept of consequence relation. This logic is then extended to the predicate first-order logic (including its gradational variant). Finally, the authors focus on models of reasoning under uncertainty as well as on the \(\alpha\)-resolution principle, which is a generalisation of the resolution principle given a threshold \(\alpha\). The mathematical level of the book is high and it is written with necessary precision and care. Therefore, it may serve as a good source of knowledge about fuzzy logic with evaluated syntax.
scientific article; zbMATH DE number 1980913