josep maria font - ubshort contents short contentsvii detailed contentsix a letter to the readerxv...

Abstract Algebraic Logic

An Introductory Textbook

Josep Maria Font

College PublicationsLondon

Josep Maria FontDepartament de Matemàtiques i InformàticaUniversitat de Barcelona (UB)

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ISBN 978-1-84890-207-7

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Short contents

Short contents vii

Detailed contents ix

A letter to the reader xv

Introduction and Reading Guide xix

1 Mathematical and logical preliminaries 11.1 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 712.1 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 1073.1 Transformers, algebraic semantics, and assertional logics . . . . . . 108

3.2 Algebraizable logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.3 A syntactic characterization, and the Lindenbaum-Tarskiprocess 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

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4 The semantics of matrices 1834.1 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 2355.1 Generalized matrices: basic concepts . . . . . . . . . . . . . . . . . 236

5.2 Basic full generalized models, Tarski-style conditions andtransfer 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 3176.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

6.2 Protoalgebraic logics . . . . . . . . . . . . . . . . . . . . . . . . . . 322

6.3 Definability of equivalence (protoalgebraic andequivalential logics) . . . . . . . . . . . . . . . . . . . . . 352

6.4 Definability of truth (truth-equational, assertional andweakly algebraizable logics) . . . . . . . . . . . . . . . . 371

6.5 Algebraizable logics revisited . . . . . . . . . . . . . . . . . . . . . 400

7 Introduction to the Frege hierarchy 4137.1 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 xixOverview of the contents . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiNumbers, words, and symbols . . . . . . . . . . . . . . . . . . . . . . . xxviFurther reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii

1 Mathematical and logical preliminaries 11.1 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


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


x detailed contents

1.4 Finitarity and structurality . . . . . . . . . . . . . . . . . . . . . . . 44

Finitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Structurality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


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


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


2 The first steps in the algebraic study of a logic 712.1 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

The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

The implicative case . . . . . . . . . . . . . . . . . . . . . . . . . 91


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 1073.1 Transformers, algebraic semantics, and assertional logics . . . . . . 108


3.2 Algebraizable logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Uniqueness of the algebraization: the equivalentalgebraic semantics . . . . . . . . . . . . . . . . . . . . . 118


3.3 A syntactic characterization, and the Lindenbaum-Tarskiprocess again . . . . . . . . . . . . . . . . . . . . . . . . . 123


3.4 More examples, and special kinds of algebraizable logics . . . . . . 129

detailed contents xi

Finitarity issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Regularly algebraizable logics . . . . . . . . . . . . . . . . . . . 140


3.5 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . 145

The evaluated transformers and their residuals . . . . . . . . . 146

The theorems, in many versions . . . . . . . . . . . . . . . . . . 147

Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155


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 andits applications . . . . . . . . . . . . . . . . . . . . . . . . 168

Weak versions of the Deduction Theorem . . . . . . . . . . . . . 173


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 1834.1 Logical matrices: basic concepts . . . . . . . . . . . . . . . . . . . . 183

Logics defined by matrices . . . . . . . . . . . . . . . . . . . . . 184

Matrices as models of a logic . . . . . . . . . . . . . . . . . . . . 190


4.2 The Leibniz operator . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Strict homomorphisms and the reduction process . . . . . . . . 199


4.3 Reduced models and Leibniz-reduced algebras . . . . . . . . . . . 203


4.4 Applications to algebraizable logics . . . . . . . . . . . . . . . . . . 214


4.5 Matrices as relational structures . . . . . . . . . . . . . . . . . . . . 220

Model-theoretic characterizations . . . . . . . . . . . . . . . . . 225


5 The semantics of generalized matrices 2355.1 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


5.2 Basic full generalized models, Tarski-style conditions andtransfer theorems . . . . . . . . . . . . . . . . . . . . . . 244

xii detailed contents


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


5.4 The algebraic counterpart of a logic . . . . . . . . . . . . . . . . . . 270

The L-algebras and the intrinsic variety of a logic . . . . . . . . 273


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


5.6 Generalized matrices as models of Gentzen systems . . . . . . . . 304

The notion of full adequacy . . . . . . . . . . . . . . . . . . . . 308

6 Introduction to the Leibniz hierarchy 3176.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

6.2 Protoalgebraic logics . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

The fundamental set and the syntactic characterization . . . . . 323

Monotonicity, and its applications . . . . . . . . . . . . . . . . . 327

A model-theoretic characterization . . . . . . . . . . . . . . . . 331

The Correspondence Theorem . . . . . . . . . . . . . . . . . . . 332

Protoalgebraic logics and the Deduction Theorem . . . . . . . . 334

Full generalized models of protoalgebraic logics andLeibniz filters . . . . . . . . . . . . . . . . . . . . . . . . . 341


6.3 Definability of equivalence (protoalgebraic andequivalential logics) . . . . . . . . . . . . . . . . . . . . . 352

Definability of the Leibniz congruence, with orwithout parameters . . . . . . . . . . . . . . . . . . . . . 352

Equivalential logics: Definition and general properties . . . . . 357

Equivalentiality and properties of the Leibniz operator . . . . . 360

Model-theoretic characterizations . . . . . . . . . . . . . . . . . 362

Relation with relative equational consequences . . . . . . . . . 367


6.4 Definability of truth (truth-equational, assertional andweakly algebraizable logics) . . . . . . . . . . . . . . . . 371

Implicit and explicit (equational) definitions of truth . . . . . . 373

Truth-equational logics . . . . . . . . . . . . . . . . . . . . . . . 374

Assertional logics . . . . . . . . . . . . . . . . . . . . . . . . . . 383

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Weakly algebraizable logics . . . . . . . . . . . . . . . . . . . . 387

Regularly weakly algebraizable logics . . . . . . . . . . . . . . . 395


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


7 Introduction to the Frege hierarchy 4137.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

7.2 Selfextensional and fully selfextensional logics . . . . . . . . . . . . 419

Selfextensional logics with conjunction . . . . . . . . . . . . . . 421

Semilattice-based logics with an algebraizableassertional 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


Summary of properties of particular logics 471Classical 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 497Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Index of logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

Index of classes of algebras . . . . . . . . . . . . . . . . . . . . . . . . . 503

General index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Index of acronyms and labels . . . . . . . . . . . . . . . . . . . . . . . . 512

Symbol index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

A letter to the reader

Verba volant, scripta manent

(Caius Titus)

Dear reader,

Let me be clear from the start: this book is an introductory textbook on abstractalgebraic logic. This straightforward phrase condenses the distinctive features ofthe book: its subject matter, its character and style, and its intended readership.

The subject is abstract algebraic logic or, if you prefer, simply algebraic logic;the Introduction begins (page xix) with a short discussion of the topic itself. Ihesitated over whether to delete the word “abstract” from the book’s title, butfinally decided to keep it, thus emphasizing that the book is about the modernapproach to algebraic logic. I believe, however, that abstract algebraic logic is simplyalgebraic logic for the twenty-first century, and I modestly hope that the book willhelp to spread this view and that one day in the future the term will becomesynonymous with “algebraic logic”.

In character, the book is introductory to the subject. This means I do not intendto cover every aspect or to arrive at the deepest results on the notions I treat.I privilege breadth over depth, intending to present a broad view of a largelyunknown territory without intending to reach all its borders—a task for explorersrather than for the public at large. If you find important results missing (evensome of a basic character), this indicates that it is probably not the book for you,and that you will know where to find them. If you are new to the subject andafter reading or studying the book you become interested in learning more, thefinal part (pages xxviiff.) of the Reading Guide is intended to provide clues tohelp you find your way in the literature.

The book was also conceived as a textbook, that is, mainly with students (orbeginners in the field) in mind, and it takes a bottom-up approach, that is, goingfrom the particular to the general. I do not intend to start by presenting the mostgeneral possible concepts, proving theorems at their highest level of generalityand then providing specificity through down-to-earth cases. I prefer to guidereaders by means of successive steps of generalization and abstraction. Moreover,since there are very few formal courses on algebraic logic in the university

xvi a letter to the reader

curricula round the world, I anticipate that the book will be used mostly forself-study. Advanced readers will have enough good sense to go more quicklythrough any deliberately detailed or verbose explanations and to choose onlywhat may be of interest to them.

However, there are a few more advanced sections or subsections of a surveycharacter, in which I want to offer a taste of some of the farther edges of recentresearch. To do this I have preferred a rather cursory presentation in order tobe able to discuss some beautiful results which would be impossible to fullyprove within this book, in the available space. I hope I have found a reasonablecompromise that will motivate some readers to study more advanced materialfrom other sources.

The book has been written as a workbook: very often you will be asked toprovide proofs of minor, auxiliary results and to complete exercises, which Ihope are not particularly difficult. They have been carefully designed to helpyou gradually master the material; only occasionally do they supplement theexposition of the main text with additional results. You should work them out assoon as you reach them on your way through the book; in some cases the answerappears, more or less hidden, in a later comment, or the properties they containare superseded by later results, but this does not diminish the usefulness of doingthem at the right time. Besides, you should convince yourself that all unprovenstatements not explicitly formulated as exercises are indeed true, providing a realproof if in doubt.

There is a certain amount of repetition: some topics are introduced early butare studied thoroughly only in later chapters. My guess is that a significantgroup of readers will study only part of the book, and yet there are some niceresults which they should be aware of, even if they do not reach the chapters inwhich these results are treated in their proper framework (and where the result inquestion appears again, sometimes proved in a different way, sometimes provedfully for the first time). In my early days as a teacher of statistics I learned apopular dictum in information theory, which says that without redundancy, there isno information.

You will find a few things anticipated: at some points you will be asked touse or consult a theorem that is proved much later in the book. This is mainlydone for illustration purposes, often in examples, rather than to prove otherresults, and I have ensured that this does not compromise the logical flow of themathematical argumentation.

Historical notes and references are included here and there rather unsystematically.I have tried to highlight the origin of the most central ideas and theorems andacknowledge their authors, but only to put the subject in context, rather than aimat an exhaustive historical survey.

A warning is in order: this book is not for beginners in logic. Mastery of a standardfirst course in mathematical logic is essential, and an elementary backgroundin universal algebra, model theory and the theory of ordered sets, lattices andBoolean algebras is highly recommended; among the best textbooks on thesetopics are [24, 51, 78, 89], where all the undefined notions of these fields may

a letter to the reader xvii

be found. A good, succint exposition of the logical and universal algebraicbackground needed to start the study of algebraic logic appears in Chapter 1 of[123]. There are a few sections requiring some proficiency with these subjects,and I have inserted appropriate warnings and given further references.

In addition, to understand the framework, and even to grasp its very firstgeneral notion (that of a sentential logic), it is undoubtedly necessary for you tobuild upon a certain body of knowledge of particular logics. This is instrumentalin order to appreciate the theory: it is a general theory of logics, so an insightinto real logics, how different they can be and what they have in common, isessential. Thus I assume (not as prerequisites in the mathematical sense) thatyou have some previous knowledge of some sentential logics, typically classicallogic and a few non-classical ones. Many of these are described explicitly asexamples, summarizing their properties and giving references; the level of detailvaries, and I have made no attempt at uniformity in this regard. In any case,this book is no substitute for a textbook on non-classical logics; for that, you canread [26, 55, 128, 147, 187, 216, 238] and other relevant chapters of handbookssuch as [27, 60, 119].

§ § §

I have felt the need for a book such as this one mainly when teaching. Thepresent book originates (and hopefully benefits) from previous experience inteaching introductory courses or tutorials to graduate students and scholars ofdifferent background and nationality. My main teaching task in this area hasbeen in the Master and Ph. D. programmes Logic and Foundations of Mathematicsand Pure and Applied Logic, which have been running now for more than twentyyears, and in the more recent Master in Advanced Mathematics, either at theUniversity of Barcelona or as joint programmes of the Catalan universities. I havealso taught shorter courses and tutorials round the world, and have written afew survey-style or historical papers [94, 96, 97, 102, 110]. All this activity hascontributed to shape a certain view on the exposition of the subject. Often, whenasked to recommend reading, I have realized that the material that constitutesthe new approach is scattered over many research and survey papers and a fewmonographs, which are usually either at a rather hard level or presented sketchily,and that no systematic treatment exists at an introductory level.

Moreover, I wanted to convey my personal view of the subject. This explains thechoice of some topics, and the (sometimes non-standard) paths followed to reachsome results. At any rate, I do not claim to choose the “best” way: it is just mytaste, and I offer it for you to try, and to agree or disagree with.

§ § §

This book is undoubtedly a consequence of my intellectual adventure overa whole career, and many people has had a direct, personal influence on myscholarly development. In the first place, I owe a great deal to Ramon Jansanaand Don Pigozzi, with whom I shared an enthusiasm for abstract algebraic logicand the initiatives to spread it, and from whom I have learned a lot about logic,

xviii a letter to the reader

mathematics and scholarship. I also want to thank my mentors, colleagues,coauthors, collaborators and former students from Catalonia (Francesc d’A. SalesVallès, Josep Pla, Ventura Verdú, Antoni Torrens, Gonzalo Rodríguez, Miquel Rius,Àngel Gil, Jordi Rebagliato, Román Adillón, Raimon Elgueta, Pilar Dellunde, JoanRoselló, Francesc Esteva, Lluís Godo, Joan Gispert, Félix Bou, Carles Noguera,Àngel Garcia-Cerdanya), from Spain (Antonio-Jesús Rodríguez, José Luis GarcíaLapresta, José Gil-Férez, David Gracia, María Esteban) and from the rest of theworld (José Coch, Robert Bull, Jerzy Perzanowski, Hiroakira Ono, Kosta Došen,Isabel Loureiro, Wim Blok, Andrzej Wronski, Piotr Wojtylak, Wojciech Dzik,Roberto Cignoli, Janusz Czelakowski, Kate Pałasinska, Ryszard Wójcicki, MikeDunn, Massoud Moussavi, Fernando Guzmán, Wolfgang Rautenberg, SergioCelani, Petr Hájek, Daniele Mundici, James Raftery, Clint van Alten, JacekMalinowski, Jorge Castro, Petr Cintula, Alessandra Palmigiano, Francesco Paoli,Umberto Rivieccio, Tommaso Moraschini, Hugo Albuquerque, Tomáš Lávicka).They have been there, at diferent moments and in different ways: asking questions,answering questions, listening, reading, writing, making resources available,giving advice, or entrusting me with different jobs. I learned something fromeach one of them.

Although this book is a highly personal project, other people have contributedto shape it as you see it now, and I want to express my debt and warm thanks toall of them: they include students who have worked with preliminary versions ofthe text and colleagues whose comments on the first serious draft of the completebook have helped to improve it in significant ways. I am particularly indebted toJanusz Czelakowski, José Gil-Férez, Ramon Jansana, Tomáš Lávicka, TommasoMoraschini and Carles Noguera for all their observations. Last, but not least, Iwant to thank Toffa Evans, Joe Graham and Mike Maudsley for improving myEnglish, and once again, José Gil-Férez for improving my TEX. How I dealt withtheir recommendations, and the final result, is of course my sole responsibility.

I must also thank Dov Gabbay for including the book in his highly-regardedseries Studies in Logic, and Jane Spurr for her editorial assistance.

Teaching at a non-elementary level is intrinsically intertwined with research.Therefore, and also for legal reasons, I must acknowledge (again, with thanks)that my research work over the years has been (partially) supported by grantsthat the Barcelona Research Group on Algebraic Logic and Non-Classical Logics hasreceived from the governments of Spain and of Catalonia, the latest ones beingproject mtm2011-25747 (which includes feder funds from the European Union)and grant 2014sgr-788, respectively.

I hope you enjoy the book.

Sincerely,

Josep Maria FontBarcelona, 29 february 2016

Introductionand

Reading Guide

If algebraic logic can be quickly (and roughly) described as the branch of math-ematics that studies the connections between logics and their algebra-based semantics,then abstract algebraic logic can be described as its more general and abstractside. Three aspects of the study of these connections deserve a special mention:• Describing them: This was historically the first aspect of the subject to be

developed, starting with Boole’s pioneering work [44] in 1847. The oldestconnections to be described took the form of completeness theorems. Morerecently other, stronger connections (which may be considered refined com-pleteness theorems) have been discovered; these include algebraizability andtruth-equationality. The description of the reduced models of a logic and of itsalgebraic counterpart (in the different senses of this term) also belongs to thisaspect of the subject.

• Exploiting them: Mostly, this is done by using more powerful and well-developed algebraic theories to prove properties of a logic, but recently somecases of the reverse process have also been worked out. This of course hasbeen done, in different ways, at each stage in the evolution of algebraic logic.One of the distinctive features of abstract algebraic logic, in contrast to moretraditional work, is that its results concern not just an individual logic, butclasses of logics that are treated uniformly. The general results that supportthese applications have come to be called bridge theorems in the context ofthe third aspect below; as I say in the general discussion at the beginning ofSection 3.6, bridge theorems are the ultimate justification of abstract algebraic logic.

• Explaining them: This consists of developing general theories of these connectionsand exploring them in many different ways to gain a deeper understandingof how and why the different kinds of connections work, what the relationsbetween them are, how large their domain of application is, and similar issues.Ideally, each general theory should support the choice of a specific algebra-based semantics as the algebraic counterpart of a given logic, on the basis ofgeneral criteria that avoid ad hoc justifications.

xx introduction

Naturally, these three aspects are inextricably intertwined. Moreover, thegeneral notions need to be tested against many examples, either natural or ad hoc,not just to obtain properties of particular logics, but in order to gain insights intothe general notions themselves, their relations, their applicability, their scope andtheir limits; this empirical work† is an important guide for the more abstract work.

The expression algebra-based semantics is used in a deliberately ambiguousand informal sense. It is intended to refer to any kind of semantics in whichthe non-linguistic objects (the domains where the formulas of the languageare evaluated) are either just plain algebras, or algebras endowed with someadditional structure (a particular element or subset, a family of subsets, an orderrelation) and in which the evaluations are all the homomorphisms from theformula algebra to the algebras underlying the models. Notice that the termalgebraic semantics, which has been extensively used in a similarly informal wayfor years, does have a formal meaning in the modern theory (Section 3.1); so Ithink it is good practice to limit its usage to this strictly technical sense and use“algebra-based semantics” otherwise.

Algebraic logic is an old subject,‡ whose birth is commonly attributed to thework of Boole, De Morgan, Peirce, Schröder and other scholars in the secondhalf of the nineteenth century. Since then the subject has evolved from a bunchof ad-hoc procedures that establish some links between a particular logic (ini-tially classical sentential logic) and certain algebraic structures (initially Booleanalgebras) to a full-fledged mathematical theory, with its well-defined object, itstypical methods, techniques and constructions, its important general theorems,and its applications to diverse kinds of logics. The development of the subjecthas been slow, particularly the process of self-structuring as a systematicallyorganized corner of mathematical logic. Without attempting to write a detailedhistory, one can say that the subject really took off in the 1920s, and began to besomething more than a mere description of the connections between classical logicand Boolean algebras with the work of Łukasiewicz, Post, Lindenbaum, Tarskiand other—mainly Polish—logicians; the notion of logical matrix introduced atthe time allowed theoretical studies and a potentially universal applicability. Lateron, Tarski and his followers focused on the algebraization of (classical) first-orderlogic. The task of extending the so-called Lindenbaum-Tarski method of algebraizingclassical logic to other sentential logics generated a host of papers and, ultimately,well-known books by Rasiowa and Sikorski [209, 1963] and by Rasiowa [208,1974]. Later on, in the 1970s and the early 1980s, a truly general study of thetheory of logical matrices was undertaken by several scholars—again, mainlyPolish logicians. Czelakowski’s theory of equivalential logics [67, 1981], for the firsttime, provided several characterizations, both semantic and syntactic, of a classof logics to which the Lindenbaum-Tarski method, suitably generalized, could

† The importance of dealing with strange, or seemingly pathological examples should not be under-estimated. Something similar happens in medicine, where the study of subjects with importantneurological disorders provides essential clues to advance in the understanding of the normalbehaviour of the human brain and mind.

‡ Some, viewing history of mathematics from a global perspective, will prefer to say that algebraiclogic is relatively young.

introduction xxi

be applied. This scientific progress is certainly related to the parallel growth ofmodel theory and universal algebra, the two disciplines that have become the mainmathematical tools for algebraic logic. All these trends are well represented inWójcicki’s comprehensive monograph [249, 1988]. Additional historical informa-tion can be obtained in [13, 31, 50, 73, 94, 97, 158, 176, 188, 229, 251]; the succint“biased survey of algebraic logic” of [123, pp. 7–9] is also worth reading.

Since the 1980s the subject has seen important changes, the developmentof new and powerful tools and general frameworks to deal with, classify andcompare a much larger number of sentential logics and to establish connectionsbetween their metalogical and algebraic properties. This more recent evolution isdue mainly to the works of Blok, Pigozzi, Czelakowski, Herrmann, Jansana andRaftery in the more traditional, matrix-based line, and of Bloom, Brown, Suszko,†

Font, Jansana, Torrens and Verdú, in the newer line based on abstract logics orgeneralized matrices. The convergence of these approaches is apparent in [107,1996], and is also described in [110]. The modern form of the subject started tobe called abstract algebraic logic sometime in the 1990s,‡ and this denominationhas been included as entry 03g27 in the 2010 revision of the Mathematics SubjectClassification. Today, the algebraic study of particular sentential logics is conductedthrough reference to some general standards that classify logics inside twohierarchies, the Leibniz hierarchy and the Frege hierarchy, according to severalcriteria. In addition to the results concerning this or that logic or class of algebras,several important general theorems have also been obtained; the building andinvestigation of these hierarchies is probably the most beautiful and impressivemathematical enterprise to date in the field.

One of the distinctive features of abstract algebraic logic is that it transcendsthe mere generalization of the techniques that have been seen to work in a handfulof so-to-speak “classical” (in the sense of “paradigmatic” or “established”) cases;for instance, while Rasiowa in her [208] did generalize the Lindenbaum-Tarskiprocess in the sense that she identified a broad class of logics for which thisprocess can be carried out with almost no changes, she did not look for otherequivalent characterizations of the logics in this class (which would have allowedher to show which logics cannot be treated with her methods). The deeper abstractframework underlying Rasiowa’s studies was first uncovered by Czelakowskiin [67], and more definitively by Blok and Pigozzi in their [30, 32, 34].

As I said before (page xv), abstract algebraic logic is the algebraic logic of thetwenty-first century—and of the last quarter of the twentieth century. Hopefully,with the passing of time the word “abstract” currently added to the traditionalname will simply be forgotten.§

† Suszko’s influence on algebraic logic has been profound, as it has also been on the traditional theoryof matrices; this is evident from his early and fundamental [166, 1958].

‡ The term “universal algebraic logic” was used for a short time; see [190, 1998]. As far as I know,the term “abstract algebraic logic” first appears in the literature in the title of Section 5.6 of [138,1985], which is devoted to a general study of the connections between theories of classical first-orderlogic and varieties of cylindric algebras. Here I am referring to the later usage of the term in thesentential logic framework.

§ Much as today we all designate simply as “algebra” what was initially called “abstract algebra” or

xxii introduction

As a final point on the subject matter of this book, notice that it deals only withsentential logics. So, unless specified otherwise, “logic” always means “sententiallogic”, either at an informal or at a formal, technical level. This does not meanthat first-order logic cannot be treated successfully with algebraic tools. Quiteto the contrary, for many scholars the expression “algebraic logic” is indeedsynonymous with the algebraic study of first-order logic developed either in thestyle of Tarski and his important school† or in the style of Halmos (dealing mainlywith cylindric and relation algebras, or with monadic and polyadic algebras,respectively). Indeed, these scholars may be surprised that this study has onlya marginal place in this book (as a mater of fact, Section 2.5 briefly reviewsthe two radically different ways in which predicate logics can be algebraicallyapproached; there you will find a few references). In general, however, theseworks have a very different character from the algebraic study of sententiallogics. Most of their main results, obtained by very sophisticated mathematicaltechniques, concern the study of particular theories (or particular languages) inclassical first-order logic. By contrast, a general, abstract theory still has to be built,at least with the degree of systematization and sophistication found nowadays inthe algebraic studies of sentential logics; in particular, no hierarchies of logics likethose developed for sentential logics in Chapters 6 and 7 have been constructedto classify non-classical first-order logics according to their algebraic behaviour.

Overview of the contents

As explained before, this book adopts a bottom-up approach; that is, it describesthe different algebra-based semantics at increasing levels of the mathematicalcomplication of the semantics objects. It starts with the algebraization of implica-tive logics, where the truth definition is given by a single element of the algebras,and next moves on to algebraizable logics, where it is given by a set of equations.This is followed by an explanation of matrix semantics, where the truth definitionis given by an arbitrary set, and it concludes with generalized matrix semantics,where it is given by a family of subsets. The mathematical sophistication does notfollow a parallel pattern, though, as the chapter on algebraizability is probablymore difficult than the one on matrices. In any event, the last two chapters on theLeibniz and Frege hierarchies are unavoidably the hardest and more complicatedin the book. A more detailed description of each chapter’s contents follows, withsome important reading tips.

As for Chapter 1, the name says it all: Mathematical and logical preliminaries.Here are most of the general definitions, terminology and notation, and thecommon mathematical background for the entire book; not everything is neededeverywhere, so you may be able to skim the chapter and return to it as and

“modern algebra” in order to emphasize that it differs from the algebra of the nineteenth century.This can be seen from van der Waerden’s famous 1930 book Moderne Algebra, which has simply beenentitled Algebra since its 1970 edition.

† The term “Tarski-style algebraic logic” is sometimes used to distinguish its work from the algebraicstudy of sentential logics, which is then called “Polish-style algebraic logic”, or even “Polish logic”,a “rather irritating cliché” in Wójcicki’s words [249, p. xi].

reading guide xxiii

when needed, depending on your background. Section 1.2 contains a discussionof the general definition of the notion of a sentential logic,† which is in somesense the starting point of algebraic logic. Less common is the detailed attentiondevoted to the theory of closure operators (Sections 1.3–1.5) and to the equationalconsequence relative to a class of algebras (the final part of Section 1.6). Most ofSections 1.1, 1.2 and 1.6 touch upon very general subjects and are here mainly toestablish terminology and notations, but are definitely not designed for learning thesubjects. You are expected to have a working knowledge of some particular logicsand of some classes of algebras (lattices, Boolean algebras), so that the generalscene-setting of this chapter looks familiar to you, and builds on a firm footing ofconcrete examples.

Chapter 2 is mainly (but not exclusively) motivational, and should be easy toread. It starts by reviewing the usual textbook completeness proof of classicaltwo-valued logic, which everyone should be familiar with. Then it shows howthe so-called Lindenbaum-Tarski process arises naturally from the analysis of thisproof, and introduces its more straightforward generalization, the algebraizationof implicative logics. Nevertheless, nobody should miss the first subsection (“Thegeneral case”, pages 88–91) of Section 2.3, as it introduces the key notion of filterof a logic, absolutely indispensable throughout the book. The chapter draws toa close with a discussion (Section 2.4) on how the said process can be extendedmore or less naturally in several directions, and two digressions (Section 2.5)on the algebraic study of first-order logic, the first of which may be viewed as ajustification for why sentential logics are studied in a purely algebraic way, asalgebraic logic does.

Chapter 3 develops most of the central core of abstract algebraic logic, the the-ory of algebraizable logics, presented as based on the idea of equivalence betweenthe consequence of a sentential logic and the equational consequence relative toa class of algebras, effected by a pair of mutually inverse structural transformers.Some proofs are developed as further generalizations of the Lindenbaum-Tarskiprocess (however, connections with the theory of matrices are touched upononly in Section 4.4). Section 3.5, which contains the abstract characterizations ofalgebraizability known as Isomorphism Theorems, is probably the hardest in thischapter. Section 3.6 contains an initial discussion of some central topics to beaddressed in later chapters, such as bridge theorems and transfer theorems, and usesthe Deduction-Detachment Theorem as a case study, offering a first taste of whatis to come. The chapter ends with a brief account (Section 3.7) of how the basicidea of algebraization can be, and has been, extended and generalized to wideror more abstract notions of logic.

Chapter 4 develops the essentials of the theory of logical matrices, the coreof the more classical algebraic logic. In particular the chapter contains the firstgeneral definition of the notion of the algebraic counterpart of a logic and showshow it fits with the constructions done for implicative logics and for algebraizable

† Spoiler alert: in abstract algebraic logic a sentential logic is a substitution-invariant consequencerelation on the set of formulas of some sentential language. I adhere completely to Wójcicki’s wordson the inferential approach to logic [249, pp. xii–xiii].

xxiv reading guide

logics in previous chapters, and how it fails to give a satisfactory account of thealgebraic aspect of some less standard logics. Only notions indispensable for thestudy of the Leibniz hierarchy are given here; for a more complete view of thiswell-developed area you should study [70, 249] and other papers.

Chapter 5 contains an exposition of the theory of generalized matrices. Thematerial is developed in more detail than that in Chapter 4, because the materialin the earlier chapter is better known than the literature on generalized matricesand there are more sources. It is here where the truly general definition of thealgebraic counterpart of a logic is given, together with that of the intrinsic variety of alogic. These classes of algebras are identified in a large number of particular logics,including ones in which the matrix-based definition of the previous chapter seemsto fail. I also include a discussion of why I think that this is the right definitionfor this (informal) notion. The most novel idea in this area, the notion of a fullgeneralized model of a logic, is dealt with in Section 5.5, where another fundamentalIsomorphism Theorem is proved. The chapter ends with Section 5.6, which offersa brief glimpse into the usage of generalized matrices as models of Gentzensystems and how this is related to their role as models of sentential logics.

Chapter 6 is really just an introduction to the study of the Leibniz hierarchy,albeit a detailed one, which should be supplemented by reading and studying [35,70, 72, 201] and other related works. Things here turn more complicated andadvanced, requiring more work on your part; for instance, proofs in the textoften rely on results proposed as exercises, with hints when necessary. Themost important class in the hierarchy (that of algebraizable logics) is treated inChapter 3, and it is here only revisited, particularly as far as its relations to otherclasses in the hierarchy are concerned. Section 6.4 contains some fairly recentresults on assertional logics and truth-equational logics.

The final Chapter 7 on the Frege hierarchy is shorter, because this hierarchy isless complicated than the Leibniz one and has been less studied; it also containssome very recent material. With the two hierarchies already studied, this is thenatural place to establish the relations between them; inevitably, this means thatsome of the key results of the final section (Section 7.3) bring together resultsfrom distant parts of the book and, therefore, may be harder to follow.

Figure 1 depicts the relations between chapters and sections. From this, youcan see several possible partial paths and alternative arrangements of the material,either for reading or for teaching.

An Appendix collects, in a summarized way, the features of the best-knownnon-classical logics (relative to the subject matter of this book), which appear scat-tered over many numbered examples and other comments. The book concludeswith a set of very comprehensive Indices.

2.1 2.2 2.3

2.5 2.4 3.1 – 3.5 4.1 – 4.3 4.5

3.7 3.6 4.4 5.1 – 5.5

6.1 – 6.4 5.6

6.5 7.1 – 7.2

7.3

Figure 1: A chart of the main dependencies in the book (excludingChapter 1). Solid arrows indicate mathematical dependency, though thisshould be taken with a pinch of salt, as not everything in the sourceof an arrow is needed everywhere in its target. Dotted arrows indicatemotivational dependency. The double framed groups of sections mark fourpossible ways to start the study of abstract algebraic logic, with increasinginitial level of generality.

xxvi reading guide

Numbers, words, and symbols

• The book is divided into chapters, sections and subsections, but only thefirst two levels are numbered. All numbered statements (definitions, theorems,examples, etc.) share the same two-part number sequence inside each chapter.The exceptions are exercises, which are numbered independently by chapter,and figures, tables and displayed material. Thus, any three-part label refers toa labelled item in a list contained in an already labelled statement. For instance,“Theorem 4.6.3” and “Theorem 4.6(i)” refer, respectively, to item 3 and to item(i) of Theorem 4.6, which is the sixth numbered statement in Chapter 4.

• The symbol � marks the end of some environment, which should be obviousfrom context: this is most often the end of a proof, but it could be the endof an example, a comment, or a specifically labelled paragraph which mightotherwise be difficult to distinguish from surrounding text. It also indicatesabsence of proof.

• There are three symbols for equality: the symbol = is used to express the factthat the mathematical objects represented by the expressions joined by thesymbol coincide. The symbol := means “equal by definition”, and is used onlywhen introducing a new symbolic expression, defined in terms of an alreadymeaningful one. Finally, the symbol ≈ is used to build formal identities, thatis, “equations” between two expressions of some formal language; in fact, anexpression like α ≈ β is just an alternative, more intuitive notation for the pair〈α , β〉, and it turns into a real equality only when α and β are given valuesfrom some mathematical domain (and only then will the equality becomeeither true or false).

• The symbols ⇐⇒ , ⇒ and ⇐ are used as shorthand for “if and only if”,“implies” and “follows from”, respectively; the last two are also used to markthe two halves of equivalences, in proofs and comments. The symbol def⇐⇒indicates a definition that has the form of an equivalence.

• The parentheses “(” and “)” serve several specific purposes in logico-algebraicsymbolisms, and I have decided to limit their appearances as much as possible.Specific, technical uses include: expressing the values of functions with morethan one argument, as in f (a1, . . . , an); expressing the dependence of a formulaon certain variables, as in α(x), and the results of applying a substitution oran evaluation to the formula so denoted, as in α(ϕ) or αA(a); and expressingrelation under a binary relation, as in a ≡ b (R), which is just shorthand forthe assertion that 〈a , b〉 ∈ R (technically, a binary relation R on a set A is justan R ⊆ A× A); the infix notation aRb is also used. Sequences (in particular,ordered pairs) are denoted by 〈a0, . . . , an−1〉 and so forth, as is popular inset theory; the expression ~a denotes a countable sequence of elements ai (seeSection 1.1).

In particular, no parentheses are used for functions of one argument,e.g., I have put f a instead of f (a), and hC1h−1 ⋃{C2 hXi : i∈ I} insteadof h

(C1(h−1(

⋃{C2(h(Xi)) : i∈ I})))

, and so forth. If you read carefully, noconfusion should arise.

reading guide xxvii

In any event, I use parentheses, as any mathematician does, to freely groupsymbols, in order to either eliminate ambiguity or facilitate legibility.

• The main typographical conventions are: A , B , . . . for sets (but C is reservedfor closure operators, and L for languages) and a , b , c , . . . for their elements;A,B , . . . for algebras; B , C , . . . for families of sets; L for logics;M ,N for ma-trices and g-matrices; K, M, . . . for classes (of algebras, matrices or g-matrices);w , x , y , z for sentential variables; α , β , . . . , ϕ , ψ for formulas (but π , ρ , σ , τ , θare used with other purposes); Γ, ∆, Σ for sets of formulas; f , g , h , . . . forfunctions, etc. No scheme of this kind can be 100% consistent, but I have triedmy best.

• Italic typeface is used for emphasis, and in particular boldface italics are usedwhen introducing a new term, either in a formal definition or in the text,while upright boldface appears in titles of sectional units and in phrases thatperform a similar function in the middle of the text.

• Universal quantification in formal statements is sometimes omitted for simplicityor for lack of space, and should be implicitly understood in the natural way,according to the context (“for every logic”, “for every algebra”, etc.). Obviousindices such as “i∈ I” are also omitted in some symbolic expressions for thesame reasons.

• I firmly believe that all teaching is communication. This book proposes a dialoguebetween the writer (me) and the reader (you). So when I write “we”, I mean“you and me”,† who are supposed to be travelling together, hand in hand,through the material. This I learned from the late Paul Halmos [134, 135],whose wise counsel I always try to follow (only to find time and again that hisstandards are too high for me).

Further reading

If you want to go deeper into any of the topics, or seek a different view, here aresome recommendations.

To start with, there are a few big books, containing lots of information:

• Czelakowski’s [70] is the current‡ encyclopædia of abstract algebraic logic. Thisbook presents in a systematic way most of the published literature, and evenpreviously unpublished work by the author and his co-authors. In Chapters 0

and 1 (including their exercises), it contains general material on algebraic logic,and it brings together many results that are often difficult to find. Beyond thisinitial, more general part, however, it deals only with the class of protoalgebraiclogics and its subclasses, with particular emphasis on the class of algebraizablelogics and on the logical theory of quasivarieties.

† I presume this is not far from what Wilfrid Hodges meant when, in [144, p. xiii], he wrote theapparently tautological, yet significant sentence: “‘I’ means I, ‘we’ means we”.

‡ Given that the book was published in 2001, a number of more recent developments and results haveto be looked for elsewhere.

xxviii reading guide

• Rasiowa’s classic book [208] established the paradigm of algebraic logic forseveral generations of logicians. Chapters I to VII examine several classes ofalgebras, Chapter VIII contains the general theory, and the rest of the bookspells out its application to a number of logics whose algebraic counterpartsare the classes of algebras dealt with in the first part.

• Wójcicki’s [249] is a systematization of “Polish-style” work† in algebraic logicup to the 1980s, and more generally of the study of sentential logics understoodas consequence operators. The book also contains some interesting discussionsof rather philosophical import.

• The recent Handbook of Mathematical Fuzzy Logic [60] contains a chapter byCintula and Noguera [62] whose aim is, in the authors’ own words, “to presenta marriage of Mathematical Fuzzy Logic and (Abstract) Algebraic Logic”. This90+-page exposition starts with very general material and then confines itselfto the large class of “weakly implicative logics”. Although it concentrateson the algebraic study of fuzzy logics, it is written from a very modernperspective and touches on many points of general interest, some of which arenot presented here.

There are also a number of monographs and important papers that are writtenin a fairly systematic and self-contained way and can be read as introductions tothe research lines started in them:

• Blok and Pigozzi’s [32] contains pioneering work on the notion of an algebraiz-able logic, the main ideas and results (for their initial, restricted definition)and an analysis of examples. Blok and Jónsson’s [29] is also a pioneeringcontribution to the building of the most abstract presentation of the idea ofequivalence between structural closure operators that underlies the notion ofalgebraizable logic, as outlined in Section 3.7.

• The short monograph [107] uses the semantics of generalized matrices (herethey are called abstract logics) to study finitary sentential logics, especially theones for which neither algebraizability nor the older theory of matrices yieldssatisfactory enough results, and also to study Gentzen systems. It introducesthe key notion of a full generalized model of a sentential logic and a trulygeneral definition of the notion of the algebraic counterpart of a logic. In somesense it is the culmination, at a higher degree of abstraction, of work done ona less general framework by the Barcelona group over the years. It includes achapter analyzing many examples, although in a rather condensed form.

• The extension of the idea of algebraizability from sentential logics to moregeneral formalisms was started in Rebagliato and Verdú’s [212], which dealswith Gentzen systems. Raftery’s [200] is a more modern and systematicexposition of this idea, and extends it to the equivalence between a Gentzensystem and a sentential logic, building also on the ideas of [29].

† See footnote † on page xxii.

reading guide xxix

Finally, there are quite a few survey papers on recent trends in algebraic logic,something natural in a field that has undergone deep changes:

• Blok and Pigozzi’s [35] is a combination of a survey of some aspects of abstractalgebraic logic and its relation to traditional methods, and a monograph whereimportant results are proved in detail, some for the first time.

• My two papers [94, 97], devoted to the work by Rasiowa, describe her general-ization of the Lindenbaum-Tarski method. The first one also touches on thefirst-order case, and the second one contains an overview of different directionsof research sparked by her work. Incidentally, the second one is where theexpression “Frege hierarchy” first appears in print.

• Three surveys of abstract algebraic logic to which I have contributed are [96,102, 110]. Although they are diverse in character and scope, they all try todescribe the modern ideas and developments as arising from the essentialtechniques and results of traditional algebraic logic.

• Raftery’s [204] concentrates on algebraizability, mentions some of the latesttrends and emphasizes the algebraic side of the topic.

• Jansana’s and Pigozzi’s (independent) encyclopædia articles [150, 191] arerather condensed but non-trivial, modern expositions of the topic.

The book’s Bibliography contains far more references than a standard textbook.In addition to the references needed to follow the central contents and themathematical background, the bibliography also includes specialized works citedin specific places within the book as well as other fundamental works in the area,in order to support the book’s somewhat hybrid character, as a cross between acourse textbook and a survey monograph, with some historical notes.

Bibliography

[1] Agliano, P. Congruence quasi-orderability in subtractive varieties. Journal of theAustralian Mathematical Society 71 (2001), 421–446.

[2] Fregean subtractive varieties with definable congruences. Journal of theAustralian Mathematical Society 71 (2001), 353–366.

[3] Agliano, P., and Ursini, A. On subtractive varieties iii: from ideals to congru-ences. Algebra Universalis 37 (1997), 296–333.

[4] Albuquerque, H. Operators and strong versions in abstract algebraic logic. Ph. D.Dissertation, University of Barcelona, March 2016.

[5] Albuquerque, H., Font, J. M., and Jansana, R. Compatibility operators inabstract algebraic logic. The Journal of Symbolic Logic. To appear.

[6] The strong version of a sentential logic. Manuscript in preparation (2016).[7] Albuquerque, H., Font, J. M., Jansana, R., and Moraschini, T. Assertional

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