Algebraizable logics

The main result of the paper is an intrinsic characterization of algebraizability in terms of the Leibniz operator [capital Greek]Omega, which associates with each theory [italic]T of a given deductive system [script]S a congruence relation [capital Greek]Omega[italic]T on the formula algebra. [capital Greek]Omega[italic]T identifies all formulas that cannot be distinguished from one another, on the basis of [italic]T, by any property expressible in the language of [script]S. The characterization theorem states that a deductive system [script]S is algebraizable if and only if [capital Greek]Omega is one-to-one and order-preserving on the lattice of [script]S-theories, and in addition preserves directed unions. Several other characteristics are given. The results and concepts are illustrated by a large number of examples from modal and intuitionistic logic, relevance logic, and classical predicate logic. Title Page......Page 1 Abstract......Page 3 Contents......Page 4 Introduction......Page 7 1 Deductive Systems and Matrix Semantics......Page 11 1.1 The Lattice of Theories......Page 12 1.2 Matrix Semantics......Page 14 1.3 Deductive Systems as Elementary Theories......Page 15 1.4 The Elementary Leibniz Equivalence Relation......Page 16 1.4.1 Protoalgebraic Logics......Page 18 2 Equational Consequence and Algebraic Semantics......Page 19 2.1 Algebraic Semantics......Page 20 2.2 Equivalent Algebraic Semantics......Page 25 2.2.1 Uniqueness......Page 28 2.2.2 Axiomatization......Page 30 3 The Lattice of Theories......Page 33 4.1 The Leibniz Operator......Page 40 4.2 A Second Intrinsic Characterization......Page 45 5.1 Matrix Semantics and Algebraic Semantics......Page 48 5.2.1 Modal Logics......Page 52 5.2.2 Entailment and Relevance Logics......Page 54 5.2.3 Pure Implicational Logics......Page 55 5.2.4 Two Logics with the Same Algebraization......Page 60 5.2.5 Intuitionistic Propositional Logic without Implication......Page 62 5.2.6 Equivalential Logic......Page 63 A Elementary Definitional Equivalence......Page 66 B An Example......Page 69 C Predicate Logic......Page 73 Bibliography......Page 79 Index......Page 83 W. J. Blok and Don Pigozzi set out to try to answer the question of what it means for a logic to have algebraic semantics. In this seminal book they transformed the study of algebraic logic by giving a general framework for the study of logics by algebraic means. The Dutch mathematician W. J. Blok (1947-2003) received his doctorate from the University of Amsterdam in 1979 and was Professor of Mathematics at the University of Illinois, Chicago until his death in an automobile accident. Don Pigozzi (1935- ) grew up in Oakland, California, received his doctorate from the University of California, Berkeley in 1970, and was Professor of Mathematics at Iowa State University until his retirement in 2002. The Advanced Reasoning Forum is pleased to make available in its Classic Reprints series this exact reproduction of the 1989 text, with a new errata sheet prepared by Don Pigozzi. W.j. Blok And Don Pigozzi. Volume 77, Number 396 (third Of 4 Numbers). Includes Index. Bibliography: P. 73-76.