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## Abstract algebraic logic : an introductory textbook / Josep Maria Font.

Editor: London : College Publications, c2016Descripción: xxix, 521 p. ; 24 cmISBN: 9781848902077Otra clasificación: 03-01 (03G27)
Contenidos:
```Detailed contents
Short contents vii
Detailed contents ix
A letter to the reader xv
Overview of the contents xxii
Numbers, words, and symbols xxvi
i Mathematical and logical preliminaries 
1.1 Sets, languages, algebras 
The algebra of formulas 
Evaluating the language into algebras 
Equations and order relations 
Sequents, and other wilder creatures 
On variables and substitutions 
Exercises for Section 1.1 
1.2 Sentential logics 
Examples: Syntactically defined logics 
Examples: Semantically defined logics 
What is a semantics? 
What is an algebra-based semantics? 
Extensions, fragments, expansions, reducts 
Sentential-like notions of a logic on extended formulas 
Exercises for Section 1.2 
1.3 Closure operators and closure systems: the basics 
Closure systems as ordered sets 
Bases of a closure system 
The family of all closure operators on a set 
The Frege operator 
Exercises for Section 1.3 
1.4 Finitarity and structurality 
Finitarity 
Structurality 
Exercises for Section 1.4 
1.5 More on closure operators and closure systems 
Lattices of closure operators and lattices of logics 
Irreducible sets and saturated sets 
Finitarity and compactness 
Exercises for Section 1.5 
1.6 Consequences associated with a class of algebras 
The equational consequence, and varieties 
The relative equational consequence 
Quasivarieties and generalized quasivarieties 64 Relative congruences 
The operator U 
Exercises for SecHon 1.6 
2 The first steps in the algebraic study of a logic 
2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic 
Exercises for Section 2.1 
2.2 Implicative logics 
Exercises for Section 2.2 
2.3 Filters 
Hie general case 
The implicative case 
Exercises for Section 2.3 
24 Extensions of the Lindenbaum-Tarski process 
Implication-based extensions 
Equivalence-based extensions 
Conclusion 
2.5 Two digressions on first-order logic 
The logic of the sentential connectives of first-order logic 
The algebraic study of first-order logics 
3 The semantics of algebras 
3.1 Transformers, algebraic semantics, and assertional logics 
Exercises for Section 3.1 
3.2 Algebraizable logics 
Uniqueness of the algebraization: the equivalent algebraic semantics 
Exercises for Section 
3.3 A syntactic characterization, and the Lindenbaum-Tarski process again 
Exercises for Section 3.3 
3.4 More examples, and special kinds of algebraizable logics 
Finitarity issues 
Axiomatization 
Regularly algebraizable logics 
Exercises for Section 3.4 
3.5 The Isomorphism Theorems 
The evaluated transformers and their residuals 
The theorems, in many versions 
Regularity 
Exercises for Section 3.5 
3.6 Bridge theorems and transfer theorems 
The classical Deduction Theorem 
The general Deduction Theorem and its transfer 
The Deduction Theorem in algebraizable logics and its applications 
Weak versions of the Deduction Theorem 
Exercises for Section 3.6 
3.7 Generalizations and abstractions of algebraizability 
Step 1: Algebraization of other sentential-like logical systems 
Step 2: The notion of deductive equivalence 
Step 3: Equivalence of structural closure operators 
Step 4: Getting rid of points 
4 The semantics of matrices 
4.1 Logical matrices: basic concepts 
Logics defined by matrices 
Matrices as models of a logic 
Exercises for Section 4.1 
4.2 The Leibniz operator 
Strict homomorphisms and the reduction process 
Exercises for Section 4.2 
4.3 Reduced models and Leibniz-reduced algebras 
Exercises for Section 4.3 
4.4 Applications to algebraizable logics 
Exercises for Section 4.4 
4.5 Matrices as relational structures 
Model-theoretic characterizations 
Exercises for Section 4.5 
5 The semantics of generalized matrices 
5.1 Generalized matrices: basic concepts 
Logics defined by generalized matrices 
Generalized matrices as models of logics 
Generalized matrices as models of Gentzen-style rules 
Exercises for Section 5.1 
5.2 Basic full generalized models, Tarski-style conditions and transfer theorems 
Exercises for Section 
5.3 The Tarski operator and the Susako operator 
Congruences in generalized matrices 
Strict homomorphisms 
Quotients 
The process of reduction 266 Exercises for Section 5.3 
5.4 The algebraic counterpart of a logic 
The L-algebras and the intrinsic variety of a logic 
Exercises for Section 5.4 
5.5 Full generalized models 
The main concept 
Three case studies 
The Isomorphism Theorem 
The Galois adjunction of the compatibility relation 
Exercises for Section 5.5 
5.6 Generalized matrices as models of Gentzen systems 
The notion of full adequacy 

6 Introduction to the Leibniz hierarchy
6.1 Overview 
6.2 Protoalgebraic logics 
Basic concepts 
The fundamental set and the syntactic characterization 
Monotonicity, and its applications 
A model-theoretic characterization 
The Correspondence Theorem 
Protoalgebraic logics and the Deduction Theorem 
Full generalized models of protoalgebraic logics and Leibniz filters 
Exercises for Section 6.2 
6.3 Definability of equivalence (protoalgebraic and equivalential logics) 
Definability of the Leibniz congruence, with or without parameters 
Equivalential logics: Definition and general properties 
Equivalentiality and properties of the Leibniz operator 
Model-theoretic characterizations 
Relation with relative equational consequences 
Exercises for Section 6.3 
6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) 
Implicit and explicit (equational) definitions of truth 
Truth-equational logics 
Assertional logics 
Weakly algebraizable logics 
Regularly weakly algebraizable logics 
Exercises for Section 6.4 
6.5 Algebraizable logics revisited 
Full generalized models of algebraizable logics 
On the bridge theorem concerning the Deduction Theorem 
Regularly algebraizable logics revisited 
Exercises for Section 6.5 
7 Introduction to the Frege hierarchy 
7.1 Overview 
7.2 Selfextensional and fully selfextensional logics 
Selfextensional logics with conjunction 
Semilattice-based logics with an algebraizable assertional companion 
Selfextensional logics with the Uniterm Deduction Theorem 
Exercises for Sections 7.1 and 7.2 
7.3 Fregean and fully Fregean logics 
Fregean logics and truth-equational logics 
Fregean logics and protoalgebraic logics 
Exercises for Section 7.3 
Summary of properties of particular logics 
Classical logic and its fragments 
Intuitionistic logic, its fragments and extensions, and related logics 
Other logics of implication 
Modal logics 
Many-valued logics 
Substructural logics 
Other logics 
Bibliography 
Indices 
Author index 
Index of logics 
Index of classes of algebras 
General index 
Index of acronyms and labels 
Symbol index ``` Average rating: 0.0 (0 votes)
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Incluye referencias bibliográficas (p. -496) e índices.

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 --
i Mathematical and logical preliminaries  --
1.1 Sets, languages, algebras  --
The algebra of formulas  --
Evaluating the language into algebras  --
Equations and order relations  --
Sequents, and other wilder creatures  --
On variables and substitutions  --
Exercises for Section 1.1  --
1.2 Sentential logics  --
Examples: Syntactically defined logics  --
Examples: Semantically defined logics  --
What is a semantics?  --
What is an algebra-based semantics?  --
Extensions, fragments, expansions, reducts  --
Sentential-like notions of a logic on extended formulas  --
Exercises for Section 1.2  --
1.3 Closure operators and closure systems: the basics  --
Closure systems as ordered sets  --
Bases of a closure system  --
The family of all closure operators on a set  --
The Frege operator  --
Exercises for Section 1.3  --
1.4 Finitarity and structurality  --
Finitarity  --
Structurality  --
Exercises for Section 1.4  --
1.5 More on closure operators and closure systems  --
Lattices of closure operators and lattices of logics  --
Irreducible sets and saturated sets  --
Finitarity and compactness  --
Exercises for Section 1.5  --
1.6 Consequences associated with a class of algebras  --
The equational consequence, and varieties  --
The relative equational consequence  --
Quasivarieties and generalized quasivarieties 64 Relative congruences  --
The operator U  --
Exercises for SecHon 1.6  --
2 The first steps in the algebraic study of a logic  --
2.1 From two-valued truth tables to Boolean algebras: the Lindenbaum-Tarski process for classical logic  --
Exercises for Section 2.1  --
2.2 Implicative logics  --
Exercises for Section 2.2  --
2.3 Filters  --
Hie general case  --
The implicative case  --
Exercises for Section 2.3  --
24 Extensions of the Lindenbaum-Tarski process  --
Implication-based extensions  --
Equivalence-based extensions  --
Conclusion  --
2.5 Two digressions on first-order logic  --
The logic of the sentential connectives of first-order logic  --
The algebraic study of first-order logics  --
3 The semantics of algebras  --
3.1 Transformers, algebraic semantics, and assertional logics  --
Exercises for Section 3.1  --
3.2 Algebraizable logics  --
Uniqueness of the algebraization: the equivalent algebraic semantics  --
Exercises for Section  --
3.3 A syntactic characterization, and the Lindenbaum-Tarski process again  --
Exercises for Section 3.3  --
3.4 More examples, and special kinds of algebraizable logics  --
Finitarity issues  --
Axiomatization  --
Regularly algebraizable logics  --
Exercises for Section 3.4  --
3.5 The Isomorphism Theorems  --
The evaluated transformers and their residuals  --
The theorems, in many versions  --
Regularity  --
Exercises for Section 3.5  --
3.6 Bridge theorems and transfer theorems  --
The classical Deduction Theorem  --
The general Deduction Theorem and its transfer  --
The Deduction Theorem in algebraizable logics and its applications  --
Weak versions of the Deduction Theorem  --
Exercises for Section 3.6  --
3.7 Generalizations and abstractions of algebraizability  --
Step 1: Algebraization of other sentential-like logical systems  --
Step 2: The notion of deductive equivalence  --
Step 3: Equivalence of structural closure operators  --
Step 4: Getting rid of points  --
4 The semantics of matrices  --
4.1 Logical matrices: basic concepts  --
Logics defined by matrices  --
Matrices as models of a logic  --
Exercises for Section 4.1  --
4.2 The Leibniz operator  --
Strict homomorphisms and the reduction process  --
Exercises for Section 4.2  --
4.3 Reduced models and Leibniz-reduced algebras  --
Exercises for Section 4.3  --
4.4 Applications to algebraizable logics  --
Exercises for Section 4.4  --
4.5 Matrices as relational structures  --
Model-theoretic characterizations  --
Exercises for Section 4.5  --
5 The semantics of generalized matrices  --
5.1 Generalized matrices: basic concepts  --
Logics defined by generalized matrices  --
Generalized matrices as models of logics  --
Generalized matrices as models of Gentzen-style rules  --
Exercises for Section 5.1  --
5.2 Basic full generalized models, Tarski-style conditions and transfer theorems  --
Exercises for Section  --
5.3 The Tarski operator and the Susako operator  --
Congruences in generalized matrices  --
Strict homomorphisms  --
Quotients  --
The process of reduction 266 Exercises for Section 5.3  --
5.4 The algebraic counterpart of a logic  --
The L-algebras and the intrinsic variety of a logic  --
Exercises for Section 5.4  --
5.5 Full generalized models  --
The main concept  --
Three case studies  --
The Isomorphism Theorem  --
The Galois adjunction of the compatibility relation  --
Exercises for Section 5.5  --
5.6 Generalized matrices as models of Gentzen systems  --
The notion of full adequacy  --
--
6 Introduction to the Leibniz hierarchy --
6.1 Overview  --
6.2 Protoalgebraic logics  --
Basic concepts  --
The fundamental set and the syntactic characterization  --
Monotonicity, and its applications  --
A model-theoretic characterization  --
The Correspondence Theorem  --
Protoalgebraic logics and the Deduction Theorem  --
Full generalized models of protoalgebraic logics and Leibniz filters  --
Exercises for Section 6.2  --
6.3 Definability of equivalence (protoalgebraic and equivalential logics)  --
Definability of the Leibniz congruence, with or without parameters  --
Equivalential logics: Definition and general properties  --
Equivalentiality and properties of the Leibniz operator  --
Model-theoretic characterizations  --
Relation with relative equational consequences  --
Exercises for Section 6.3  --
6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics)  --
Implicit and explicit (equational) definitions of truth  --
Truth-equational logics  --
Assertional logics  --
Weakly algebraizable logics  --
Regularly weakly algebraizable logics  --
Exercises for Section 6.4  --
6.5 Algebraizable logics revisited  --
Full generalized models of algebraizable logics  --
On the bridge theorem concerning the Deduction Theorem  --
Regularly algebraizable logics revisited  --
Exercises for Section 6.5  --
7 Introduction to the Frege hierarchy  --
7.1 Overview  --
7.2 Selfextensional and fully selfextensional logics  --
Selfextensional logics with conjunction  --
Semilattice-based logics with an algebraizable assertional companion  --
Selfextensional logics with the Uniterm Deduction Theorem  --
Exercises for Sections 7.1 and 7.2  --
7.3 Fregean and fully Fregean logics  --
Fregean logics and truth-equational logics  --
Fregean logics and protoalgebraic logics  --
Exercises for Section 7.3  --
Summary of properties of particular logics  --
Classical logic and its fragments  --
Intuitionistic logic, its fragments and extensions, and related logics  --
Other logics of implication  --
Modal logics  --
Many-valued logics  --
Substructural logics  --
Other logics  --
Bibliography  --
Indices  --
Author index  --
Index of logics  --
Index of classes of algebras  --
General index  --
Index of acronyms and labels  --
Symbol index  --

MR, MR3558731

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