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Abstract Algebraic Logic Abstract Algebraic Logic An Introductory Textbook Josep Maria Font College Publications London Josep Maria Font Departament de Matemàtiques i Informàtica Universitat de Barcelona (UB) c Individual author and College Publications 2016 All rights reserved ISBN 978-1-84890-207-7 College Publications Scientific Director: Dov Gabbay Managing Director: Jane Spurr Department of Informatics King’s College London, Strand, London WC2R 2LS, UK www.collegepublications.co.uk Original cover design by Orchid Creative www.orchidcreative.co.uk This cover produced by Laraine Welch Printed by Lightning Source, Milton Keynes, UK All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form, or by any means, electronic, mechanical, photocopyng, recording or otherwise without prior permission, in writing, from the publisher. Short contents Short contents vii Detailed contents ix A letter to the reader xv Introduction and Reading Guide xix 1 Mathematical and logical preliminaries 1 1.1 Sets, languages, algebras . 1 1.2 Sentential logics . 12 1.3 Closure operators and closure systems: the basics . 32 1.4 Finitarity and structurality . 44 1.5 More on closure operators and closure systems . 54 1.6 Consequences associated with a class of algebras . 59 2 The first steps in the algebraic study of a logic 71 2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic . 71 2.2 Implicative logics . 76 2.3 Filters . 88 2.4 Extensions of the Lindenbaum-Tarski process . 98 2.5 Two digressions on first-order logic . 102 3 The semantics of algebras 107 3.1 Transformers, algebraic semantics, and assertional logics . 108 3.2 Algebraizable logics . 115 3.3 A syntactic characterization, and the Lindenbaum-Tarski process again . 123 3.4 More examples, and special kinds of algebraizable logics . 129 3.5 The Isomorphism Theorems . 145 3.6 Bridge theorems and transfer theorems . 159 3.7 Generalizations and abstractions of algebraizability . 176 viii short contents 4 The semantics of matrices 183 4.1 Logical matrices: basic concepts . 183 4.2 The Leibniz operator . 194 4.3 Reduced models and Leibniz-reduced algebras . 203 4.4 Applications to algebraizable logics . 214 4.5 Matrices as relational structures . 220 5 The semantics of generalized matrices 235 5.1 Generalized matrices: basic concepts . 236 5.2 Basic full generalized models, Tarski-style conditions and transfer theorems . 244 5.3 The Tarski operator and the Suszko operator . 254 5.4 The algebraic counterpart of a logic . 270 5.5 Full generalized models . 285 5.6 Generalized matrices as models of Gentzen systems . 304 6 Introduction to the Leibniz hierarchy 317 6.1 Overview . 317 6.2 Protoalgebraic logics . 322 6.3 Definability of equivalence (protoalgebraic and equivalential logics) . 352 6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) . 371 6.5 Algebraizable logics revisited . 400 7 Introduction to the Frege hierarchy 413 7.1 Overview . 413 7.2 Selfextensional and fully selfextensional logics . 419 7.3 Fregean and fully Fregean logics . 449 Summary of properties of particular logics 471 Bibliography 485 Indices 497 Detailed contents Short contents vii Detailed contents ix A letter to the reader xv Introduction and Reading Guide xix Overview of the contents . xxii Numbers, words, and symbols . xxvi Further reading . xxvii 1 Mathematical and logical preliminaries 1 1.1 Sets, languages, algebras . 1 The algebra of formulas . 3 Evaluating the language into algebras . 6 Equations and order relations . 7 Sequents, and other wilder creatures . 8 On variables and substitutions . 9 Exercises for Section 1.1 ....................... 11 1.2 Sentential logics . 12 Examples: Syntactically defined logics . 16 Examples: Semantically defined logics . 18 What is a semantics? . 22 What is an algebra-based semantics? . 24 Soundness, adequacy, completeness . 26 Extensions, fragments, expansions, reducts . 27 Sentential-like notions of a logic on extended formulas . 28 Exercises for Section 1.2 ....................... 30 1.3 Closure operators and closure systems: the basics . 32 Closure systems as ordered sets . 37 Bases of a closure system . 38 The family of all closure operators on a set . 39 The Frege operator . 41 Exercises for Section 1.3 ....................... 42 x detailed contents 1.4 Finitarity and structurality . 44 Finitarity . 45 Structurality . 48 Exercises for Section 1.4 ....................... 52 1.5 More on closure operators and closure systems . 54 Lattices of closure operators and lattices of logics . 54 Irreducible sets and saturated sets . 55 Finitarity and compactness . 57 Exercises for Section 1.5 ....................... 58 1.6 Consequences associated with a class of algebras . 59 The equational consequence, and varieties . 60 The relative equational consequence . 62 Quasivarieties and generalized quasivarieties . 64 Relative congruences . 65 The operator U ............................ 66 Exercises for Section 1.6 ....................... 67 2 The first steps in the algebraic study of a logic 71 2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic . 71 Exercises for Section 2.1 ....................... 76 2.2 Implicative logics . 76 Exercises for Section 2.2 ....................... 86 2.3 Filters . 88 The general case . 88 The implicative case . 91 Exercises for Section 2.3 ....................... 96 2.4 Extensions of the Lindenbaum-Tarski process . 98 Implication-based extensions . 98 Equivalence-based extensions . 100 Conclusion . 101 2.5 Two digressions on first-order logic . 102 The logic of the sentential connectives of first-order logic . 102 The algebraic study of first-order logics . 104 3 The semantics of algebras 107 3.1 Transformers, algebraic semantics, and assertional logics . 108 Exercises for Section 3.1 ....................... 114 3.2 Algebraizable logics . 115 Uniqueness of the algebraization: the equivalent algebraic semantics . 118 Exercises for Section 3.2 ....................... 122 3.3 A syntactic characterization, and the Lindenbaum-Tarski process again . 123 Exercises for Section 3.3 ....................... 128 3.4 More examples, and special kinds of algebraizable logics . 129 detailed contents xi Finitarity issues . 132 Axiomatization . 137 Regularly algebraizable logics . 140 Exercises for Section 3.4 ....................... 143 3.5 The Isomorphism Theorems . 145 The evaluated transformers and their residuals . 146 The theorems, in many versions . 147 Regularity . 155 Exercises for Section 3.5 ....................... 157 3.6 Bridge theorems and transfer theorems . 159 The classical Deduction Theorem . 163 The general Deduction Theorem and its transfer . 166 The Deduction Theorem in algebraizable logics and its applications . 168 Weak versions of the Deduction Theorem . 173 Exercises for Section 3.6 ....................... 175 3.7 Generalizations and abstractions of algebraizability . 176 Step 1: Algebraization of other sentential-like logical systems . 176 Step 2: The notion of deductive equivalence . 178 Step 3: Equivalence of structural closure operators . 180 Step 4: Getting rid of points . 181 4 The semantics of matrices 183 4.1 Logical matrices: basic concepts . 183 Logics defined by matrices . 184 Matrices as models of a logic . 190 Exercises for Section 4.1 ....................... 192 4.2 The Leibniz operator . 194 Strict homomorphisms and the reduction process . 199 Exercises for Section 4.2 ....................... 202 4.3 Reduced models and Leibniz-reduced algebras . 203 Exercises for Section 4.3 ....................... 212 4.4 Applications to algebraizable logics . 214 Exercises for Section 4.4 ....................... 220 4.5 Matrices as relational structures . 220 Model-theoretic characterizations . 225 Exercises for Section 4.5 ....................... 233 5 The semantics of generalized matrices 235 5.1 Generalized matrices: basic concepts . 236 Logics defined by generalized matrices . 237 Generalized matrices as models of logics . 240 Generalized matrices as models of Gentzen-style rules . 242 Exercises for Section 5.1 ....................... 243 5.2 Basic full generalized models, Tarski-style conditions and transfer theorems . 244 xii detailed contents Exercises for Section 5.2 ....................... 251 5.3 The Tarski operator and the Suszko operator . 254 Congruences in generalized matrices . 254 Strict homomorphisms . 259 Quotients . 265 The process of reduction . 266 Exercises for Section 5.3 ....................... 268 5.4 The algebraic counterpart of a logic . 270 The L-algebras and the intrinsic variety of a logic . 273 Exercises for Section 5.4 ....................... 284 5.5 Full generalized models . 285 The main concept . 285 Three case studies . 288 The Isomorphism Theorem . ..
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