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A treatise on many-valued logics. (English) Zbl 1048.03002
Studies in Logic and Computation 9. Baldock: Research Studies Press (ISBN 0-86380-262-1/hbk). xii, 604 p. (2001).
Many-valued logics have acquired increasing importance in the last two decades because of their applications. Correspondingly, the mathematical theory of these logics has been greatly expanded and various deep results have been obtained. Several books and survey papers have appeared in the last six years covering recent developments. These include P. Hájek’s book, Metamathematics of fuzzy logic [Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)], the book by R. Cignoli, I. D’Ottaviano, and the present reviewer, Algebraic foundations of many-valued reasoning [Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)], the two chapters on many-valued logic in Vol. 2 of the second edition of the Handbook of philosophical logic, edited by D. M. Gabbay et al. [Dordrecht: Kluwer Academic Publishers (2001; Zbl 0996.03002)], respectively by A. Urquhart [“Basic many-valued logic”, pp. 249–295 (2001; Zbl 1003.03523)] and R. Hähnle [“Advanced many-valued logics”, pp. 297–395 (2001; Zbl 1003.03522)]. The scope of the treatise under review is much wider than that of all the above books and surveys. The author has divided his exposition into four parts, covering the theoretical foundations of logical many-valuedness, from the first principles to some of the most advanced results and applications.
This book has more than 600 pages and is an expansion of the 1989 monograph, Mehrwertige Logik, by the same author [Berlin: Akademie-Verlag (1989; Zbl 0714.03022)]. Its main intended readers are logicians, as well as computer scientists interested in applied nonclassical logic. Part I deals with introductory logical semantical and syntactical notions, and ends with a six-pages outline of the history of many-valued logic. Part II deals with connectives and sets of truth degrees, sequent and tableaux calculi, functional completeness, decidability of propositional systems. Part III, the core of the book, deals with t-norm based systems. As shown in Hájek’s book, by a celebrated result of P. S. Mostert and A. L. Shields [Ann. Math. 65, 117–143 (1957; Zbl 0096.01203)] the building blocks of every continuous t-norm are just three special kinds of generalized conjunction operations over the real unit interval: Łukasiewicz conjunction (so called because of its interdefinability with the infinite-valued implication of Łukasiewicz), infimum, and product. One then has three corresponding propositional logics, respectively known as Łukasiewicz calculus, Gödel (-Dummett) logic, and product logic. More than half of Part III is devoted to Łukasiewicz logic and its algebras, MV-algebras. In the late fifties, Chang invented MV-algebras to give a purely algebraic proof of the completeness of the Łukasiewicz axioms, after the syntactic proof of Rose and Rosser. Beginning with the late eighties, MV-algebras have been attracting increasing attention, because they are essentially the same as (more precisely, they are categorically equivalent to) lattice-ordered Abelian groups with strong unit, and hence, via Grothendieck’s group, countable MV-algebras are a complete and invariant classifier for an important class of approximately finite-dimensional \(C^*\)-algebras of operators. This section features complete proofs of the Rosser-Turquette finite-valued completeness theorem, of McNaughton’s representation theorem for infinite-valued truth-functions, and of the completeness of the infinite-valued Łukasiewicz calculus, via Chang’s completeness theorem for MV-algebras. The rest of Part III is devoted to Hájek’s Basic Logic (BL), first-order extensions, graded identity, Post systems. The section on BL culminates in a proof of Hájek’s completeness theorem stating that BL is the logic of all continuous t-norms. Variants of BL-algebras have recently given rise to extensive research on residuated monoidal structures, as an appropriate tool for the algebraic investigation of a host of nonclassical logical systems existing in the literature.
Having to deal in less than 250 pages with the extensive bulk of results on Łukasiewicz logic, MV-algebras, t-norm logics and their extensions, the author has made a careful choice of a few main topics and results, addressing the interested reader to the relevant literature for further details. Thus, e.g., one may consult Hájek’s book for a comprehensive account of BL, and see the book by Cignoli, D’Ottaviano and the present reviewer for a fairly complete account on MV-algebras and Łukasiewicz infinite-valued calculus. Finally, Part IV is a miscellanea of topics related to logical many-valuedness. These include fuzzy sets and their t-norm-based Cartesian products, relations, equivalence relations, partitions, orderings, and model-theoretic logics. Other sections deal with the relationships between many-valued logics and presuppositions, alethic modalities, intuitionism.
The final sections are concerned with Boolean-valued models, independence and consistency results in set theory, also surveying classical results by Skolem, Chang and Fenstad. The author has made a valuable effort to give a coherent presentation of a huge bulk of material in a situation where lack of space obviously made it impossible to cover all important developments of many-valued logic in a single book. Here is, however, a tentative selection of topics that readers of this book, including the present reviewer, may wish to be given more extensive coverage in future editions of this treatise: Complexity-theoretic issues for decision problems of propositional and predicate calculi, game-theoretic semantics of infinite-valued logics and their proof-theoretic counterparts via hypersequent calculi, left-continuous t-norm-based logics and algebras.
The bibliography has 601 entries, and is a useful updating of the extensive bibliographies given by N. Rescher (in his book, Many-valued logic [New York: McGraw-Hill (1969; Zbl 0248.02023)]), and by R. G. Wolf in his survey on many-valued logic [in: J. M. Dunn and G. Epstein (eds.), Modern uses of multiple-valued logic. Dordrecht: Reidel, 167–323 (1977; Zbl 0372.02012)]. This treatise is an indispensable source to all researchers in many-valued logics.

03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
06D35 MV-algebras