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

Series Studies in logic. Mathematical logic and foundations v. 60Editor: London : College Publications, c2016Descripción: xxix, 521 p. ; 24 cmISBN: 9781848902077Otra clasificación: 03-01 (03G27)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 i 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] 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 SecHon 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] Hie general case [88] The implicative case [91] Exercises for Section 2.3 [96] 24 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 [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] 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] Exercises for Section [251] 5.3 The Tarski operator and the Susako operator [254] Congruences in generalized matrices [254] Strict homomorphisms [259] Quotients [264] 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 [294] The Galois adjunction of the compatibility relation [300] Exercises for Section 5.5 [302] 5.6 Generalized matrices as models of Gentzen systems [304] The notion of full adequacy [308] 6 Introduction to the Leibniz hierarchy 6.1 Overview [317] 6.2 Protoalgebraic logics [317] Basic concepts [322] The fundamental set and the syntactic characterization [323] Monotonicity, and its applications [327] A model-theoretic characterization [331] The Correspondence Theorem [333] Protoalgebraic logics and the Deduction Theorem [334] Full generalized models of protoalgebraic logics and Leibniz filters [341] Exercises for Section 6.2 [350] 6.3 Definability of equivalence (protoalgebraic and equivalential logics) [352] Definability of the Leibniz congruence, with or without parameters [352] Equivalential logics: Definition and general properties [557] Equivalentiality and properties of the Leibniz operator [360] Model-theoretic characterizations [362] Relation with relative equational consequences [367] Exercises for Section 6.3 [369] 6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) [371] Implicit and explicit (equational) definitions of truth [373] Truth-equational logics [374] Assertional logics [383] Weakly algebraizable logics [387] Regularly weakly algebraizable logics [395] Exercises for Section 6.4 [397] 6.5 Algebraizable logics revisited [400] Full generalized models of algebraizable logics [404] On the bridge theorem concerning the Deduction Theorem [405] Regularly algebraizable logics revisited [407] Exercises for Section 6.5 [411] 7 Introduction to the Frege hierarchy [413] 7.1 Overview [413] 7.2 Selfextensional and fully selfextensional logics [419] Selfextensional logics with conjunction [421] Semilattice-based logics with an algebraizable assertional companion [434] Selfextensional logics with the Uniterm Deduction Theorem [440] Exercises for Sections 7.1 and 7.2 [446] 7.3 Fregean and fully Fregean logics [449] Fregean logics and truth-equational logics [455] Fregean logics and protoalgebraic logics [457] Exercises for Section 7.3 [467] Summary of properties of particular logics [471] Classical logic and its fragments [472] Intuitionistic logic, its fragments and extensions, and related logics [473] Other logics of implication [475] Modal logics [476] Many-valued logics [478] Substructural logics [480] Other logics [482] Ad hoc examples [483] Bibliography [485] Indices [497] Author index [497] Index of logics [500] Index of classes of algebras [503] General index [505] Index of acronyms and labels [512] Symbol index [515]

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01 R456 Historia de la matemática / | 01A70 C771 Mathematicians : | 01A70 L664 Beppo Levi : | 03 F677a Abstract algebraic logic : | 03 H688 A shorter model theory / | 11 B496 The Spectrum of Hyperbolic Surfaces / | 11 El49 Groups acting on hyperbolic space : harmonic analysis and number theory / |

Incluye referencias bibliográficas (p. [485]-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 --

Further reading xxvii --

i 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] --

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 SecHon 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] --

Hie general case [88] --

The implicative case [91] --

Exercises for Section 2.3 [96] --

24 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 [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] --

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] --

Exercises for Section [251] --

5.3 The Tarski operator and the Susako operator [254] --

Congruences in generalized matrices [254] --

Strict homomorphisms [259] --

Quotients [264] --

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 [294] --

The Galois adjunction of the compatibility relation [300] --

Exercises for Section 5.5 [302] --

5.6 Generalized matrices as models of Gentzen systems [304] --

The notion of full adequacy [308] --

--

6 Introduction to the Leibniz hierarchy --

6.1 Overview [317] --

6.2 Protoalgebraic logics [317] --

Basic concepts [322] --

The fundamental set and the syntactic characterization [323] --

Monotonicity, and its applications [327] --

A model-theoretic characterization [331] --

The Correspondence Theorem [333] --

Protoalgebraic logics and the Deduction Theorem [334] --

Full generalized models of protoalgebraic logics and Leibniz filters [341] --

Exercises for Section 6.2 [350] --

6.3 Definability of equivalence (protoalgebraic and equivalential logics) [352] --

Definability of the Leibniz congruence, with or without parameters [352] --

Equivalential logics: Definition and general properties [557] --

Equivalentiality and properties of the Leibniz operator [360] --

Model-theoretic characterizations [362] --

Relation with relative equational consequences [367] --

Exercises for Section 6.3 [369] --

6.4 Definability of truth (truth-equational, assertional and weakly algebraizable logics) [371] --

Implicit and explicit (equational) definitions of truth [373] --

Truth-equational logics [374] --

Assertional logics [383] --

Weakly algebraizable logics [387] --

Regularly weakly algebraizable logics [395] --

Exercises for Section 6.4 [397] --

6.5 Algebraizable logics revisited [400] --

Full generalized models of algebraizable logics [404] --

On the bridge theorem concerning the Deduction Theorem [405] --

Regularly algebraizable logics revisited [407] --

Exercises for Section 6.5 [411] --

7 Introduction to the Frege hierarchy [413] --

7.1 Overview [413] --

7.2 Selfextensional and fully selfextensional logics [419] --

Selfextensional logics with conjunction [421] --

Semilattice-based logics with an algebraizable assertional companion [434] --

Selfextensional logics with the Uniterm Deduction Theorem [440] --

Exercises for Sections 7.1 and 7.2 [446] --

7.3 Fregean and fully Fregean logics [449] --

Fregean logics and truth-equational logics [455] --

Fregean logics and protoalgebraic logics [457] --

Exercises for Section 7.3 [467] --

Summary of properties of particular logics [471] --

Classical logic and its fragments [472] --

Intuitionistic logic, its fragments and extensions, and related logics [473] --

Other logics of implication [475] --

Modal logics [476] --

Many-valued logics [478] --

Substructural logics [480] --

Other logics [482] --

Ad hoc examples [483] --

Bibliography [485] --

Indices [497] --

Author index [497] --

Index of logics [500] --

Index of classes of algebras [503] --

General index [505] --

Index of acronyms and labels [512] --

Symbol index [515] --

MR, MR3558731

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