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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]
1.1 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]
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]
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]```
Item type Home library Shelving location Call number Materials specified Status Date due Barcode
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Últimas adquisiciones 03 F677a (Browse shelf) Available A-9292
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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 --
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] --
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] --
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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