Review of Leonard Bolc

In recent years, interest in many-valued logics for computer science has increased considerably, and there is certainly a need for good and thorough textbooks on the subject geared towards that potential area of application, namely, computer science. The authors of the volume Many-valued Logics 1 , Leonard Bolc and Piotr Borowik, have set out to do just that. The volume under review here is the rst volume of a two-volume set. Volume 1 is devoted to the theoretical foundations of many-valued logics, the second volume to follow shall be devoted to applications of many-valued logics in computer science, in particular automated deduction in many-valued logics. A review of the rst volume should of course take into account the second volume, so the present review can only be partial and tentative. So what should be expected of a volume on the theoretical foundations of many-valued logic? Certainly it should deal with the basics of many-valued logics (matrices and axiomatizations), survey (at least the most important) many-valued systems considered in the literature, and present the most important mathematical results. The major textbook available on the subject so far, Rescher's Many-valued Logic 1969] lacks in the third aspect, and also does not place any emphasis on computer science related aspects. Another, more recent, book by Gottwald 1989] is an excellent work on the foundations of many-valued logics; it is unfortunately only available in German. Chapter 1 (\Preliminaries") introduces and makes precise the formal notions used in the book. In the last two pages of this chapter the central concept of many-valued logic is hidden: the deenition of a logical matrix. These two pages also contain the statement of Lindenbaum's theorem that every propositional logic closed under substitution can be characterized by an innnite matrix, and the product construction for matrices, attributed to Ja skowski (both without proof). Chapter 2 (\Many-valued Propositional Calculi") then takes up the topic of calculi for many-valued logic. Seemingly, the term \calculus" is used in a diierent sense as usual, namely, not only denoting an axiomatic system, but as a synonym for \logical system." The authors use what they call the algebraic approach to deening a propositional calculus: A propositional calculus is a quadruple X = (M; L; v; Pr), where M is a minimal nite-valued matrix, L 1 Hereafter referred to as B.B.