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Algebraizable logics. (English) Zbl 0664.03042
Mem. Am. Math. Soc. 396, 78 p. (1989).
This clear and readable paper is an excellent introduction to the ideas of algebraic semantics for logical systems. Although most of the familiar logical systems are known to have an algebraic counterpart, no general and precise notion of an algebraizable logic exists upon which a systematic investigation of the process of algebraization can be based. The authors propose and begin such an investigation. Their main result is an intrinsic characterization of algebraizability in terms of the Leibniz operator \(\Omega\), which associates to each theory T of a given deductive system S a congruence relation \(\Omega\) T on the formula algebra. \(\Omega\) T identifies all formulas that cannot be distinguished from one another, on the basis of T, by any property expressible in the language of S. The characterization theorem states that a deductive system S is algebraizable if and only if \(\Omega\) is one-to-one and order-preserving on the lattice of S-theories and also preserves directed unions. The authors illustrate these results with a large number of examples from modal and intuitionistic logic, relevance logic, and classical predicate logic.
The paper is organized as follows: Introduction, Ch. 1 - Deductive systems and matrix semantics, Ch. 2 - Equational consequence and algebraic semantics, Ch. 3 - The lattice of theories, Ch. 4 - Two intrinsic characterizations, Ch. 5 - Matrix semantics and algebraizability, Appendix A - Elementary definitional equivalence, Appendix B - An example, Appendix C - Predicate logic, Bibliography - 49 items.
Reviewer: L.Esakia

03G99 Algebraic logic
03B45 Modal logic (including the logic of norms)
03B55 Intermediate logics
03B60 Other nonclassical logic
03C05 Equational classes, universal algebra in model theory
08C15 Quasivarieties
03B20 Subsystems of classical logic (including intuitionistic logic)
03B10 Classical first-order logic
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