Abstract Algebraic Logic an Introductory Chapter

Home , Abstract algebraic logic, Algebraic logic, Craig interpolation

Abstract Algebraic Logic An Introductory Chapter

Josep Maria Font

1 Introduction ...... 1 2 Some preliminary issues ...... 5 3 Bare algebraizability ...... 10 3.1 The equational consequence relative to a class of algebras ...... 11 3.2 Translating formulas into equations ...... 13 3.3 Translating equations into formulas ...... 15 3.4 Putting it all together ...... 15 4 The origins of algebraizability: the Lindenbaum-Tarski process ...... 18 4.1 The process for classical logic ...... 18 4.2 The process for algebraizable logics ...... 20 4.3 The universal Lindenbaum-Tarski process: matrix semantics ...... 23 4.4 Algebraizability and matrix semantics: deﬁnability ...... 27 4.5 Implicative logics ...... 29 5 Modes of algebraizability, and non-algebraizability ...... 32 6 Beyond algebraizability ...... 37 6.1 The Leibniz hierarchy ...... 37 6.2 The general deﬁnition of the algebraic counterpart of a logic ...... 45 6.3 The Frege hierarchy ...... 49 7 Exploiting algebraizability: bridge theorems and transfer theorems ...... 55 8 Algebraizability at a more abstract level ...... 65 References ...... 69

1 Introduction

This chapter has been conceived as a brief introduction to (my personal view of) abstract algebraic logic. It is organized around the central notion of algebraizability, with particular emphasis on its connections with the techniques of traditional algebraic logic and especially with the so-called Lindenbaum-Tarski process. It

Josep Maria Font Departament of Mathematics and Computer Engineering, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain. e-mail: [email protected]

1 2 Josep Maria Font goes beyond algebraizability to offer a general overview of several classifications of (sentential) logics—basically, the Leibniz hierarchy and the Frege hierarchy— which have emerged in recent decades, and to show how the classification of a logic into any of them provides some knowledge regarding its algebraic or its metalogical properties. In the final section, a more abstract view of algebraizability is introduced. When attempting to describe the essential ideas of abstract algebraic logic for readers who are new to the topic, the first issue to be addressed is: What is abstract algebraic logic? If in general algebraic logic can be described as the branch of mathematics that studies the connections between logics and their algebra-based semantics, then abstract algebraic logic can be described as the more general or abstract side of that study. The expression algebra-based semantics is used in a deliberately ambiguous and informal sense.1 It is intended to refer to any kind of semantics where the mathematical objects where the formulas of the language are interpreted are either just plain algebras, or algebras endowed with some additional structure (a particular element or subset, a family of subsets, or an order relation), and where the interpretations are given by the homomorphisms from the formula algebra to the algebras that underlie the models. The study of these connections may have different goals. • To describe them: It is natural that this was, historically, the first aspect of the subject to be developed, starting with Boole’s pioneering work (1847). The oldest connections to be described take the form of completeness theorems; later on other, stronger connections were discovered, such as the notion of algebraizability. The task of finding meaningful descriptions (preferably algebraic descriptions) of the reduced models of a logic and of its algebraic counterpart (in what- ever sense) also belongs to this aspect of the subject. • To explain them: This involves developing general theories regarding the connections and exploring them in many different ways, to obtain a deeper understanding of how and why the different kinds of connections work, of the relations between them, of how large their domain of application is, etc. Ideally, each general theory should provide criteria that justify the choice of a specific algebra-based semantics as the algebraic counterpart of a logic (either for arbitrary logics, or only for those of a certain class which the theory is intended to explain). • To exploit them: Once a connection has been described, the most usual way of exploiting it is to use powerful and well-developed algebraic techniques to prove properties of the logic; but there are also cases of the reverse process. This oc- curred at each stage in the evolution of algebraic logic, in different ways. One of the distinctive features of the abstract algebraic logic way of doing this, in contrast to the more traditional way, is that its results concern not just an individual

1 The term algebraic semantics, which has been much used for years in informal comments in the same vague sense as my “algebra-based semantics”, has acquired a formal meaning in the modern theory (Deﬁnition 3.4). I consider it to be good practice to limit its usage to this strictly technical sense. Abstract Algebraic Logic 3

logic, but classes of logics that are treated in a uniform way;2 often these classes are defined by some abstract characteristic (such as the classes in the hierarchies reviewed in Section 6) or constitute a set of extensions of a particular logic (as in the applications of Theorem 4.2 mentioned at the end of Section 7). Naturally, these three aspects are inextricably intertwined. In particular, the general 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 insight into the general notions themselves, their relations, their applicability, their scope and their limits. This empirical work is an important guide for the more abstract work. The framework where the connections we are interested in here have been found to be tighter and more fruitful, and where the algebra-based semantics is exclusively algebraic, is undoubtedly that of algebraizable logics. This justifies devoting more than half of this chapter to describing that framework, its variants and extensions, and its roots in the earliest milestone in modern algebraic logic: the construction of the Lindenbaum-Tarski algebras in order to obtain the first general algebraic completeness theorems. After the preliminary Section 2, algebraizability is gradually described in Sec- tions 3 to 5 by relating it to the progressive generalization of the Lindenbaum- Tarski construction, starting from its well-known application to classical propositional logic and extending it to increasingly larger classes of logics. These sections, which are decidedly introductory and discuss the details of (some of) the key proofs, should be accessible to any reader with a general background in mathematical logic. A certain tendency to identify abstract algebraic logic with the theory of algebraizability initiated by Blok and Pigozzi has been observed in the literature; however, this is misguided, as algebraizability is only the core of the subject. More- over, the full development of the theory of algebraizability itself does indeed need the more general approach to algebraic logic provided by the classical theory of matrices (the next kind of algebra-based semantics); the results concerning algebraizability that appear in this context form the crowning stage of a very elaborated framework, where the classical theory is extended in newer and more abstract directions. Some of those directions were developed by Blok and Pigozzi themselves (the theory of protoalgebraic logics), some earlier on by Czelakowski (the theory of equivalential logics), and some later on by others (such as Herrmann’s extension to non-finitary logics, or Raftery’s theory of truth-equational logics). Abstract algebraic logic has led to the development of tools and techniques together with a strong and convoluted mathematical construction with deep results that provide the whole construction with unity. Moreover, it turns out that some less standard logics do show some algebraic behaviour, but it is not naturally captured by the theory of matrices; this brings generalized matrices (a more complicated algebra-based semantics) into the picture.

2 This feature has also been highlighted as distinctive of (some of) the modern studies of non- classical logics, notably of modal logics; see for instance Chagrov and Zakharyashev (1997, p. 109). 4 Josep Maria Font

The essentials of all these issues, in a brief overview or survey style, are described in Section 6 (which is anticipated in Section 4.3). In particular, that survey includes a description of several of the classification criteria3 that have arisen, and which have given rise to the so-called Leibniz and Frege hierarchies. Section 6.3 also reviews the main relations between the two hierarchies and mentions a few recent results and some open problems. Revealing a general notion behind the Lindenbaum-Tarski process is only part of the interest in the notion of algebraizability. It is certainly true that the general notion helped to make it clear that some obscure logics are actually amenable to algebraic treatment, or to provide a sound foundation for claims of the form “such-and-such class of algebras is the algebraic counterpart of such-and-such logic”. However, the importance of the notion also lies in its consequences, that is, in how it can be applied to obtain a precise formulation of the equivalences, informally called bridge theorems, that hold between certain metalogical properties (such as the deduction theorem or the interpolation property) and certain purely algebraic properties (such as having equationally definable principal congruences or several forms of amalgamation); in some cases, these equivalences had previously been reported in the literature for restricted groups of particular logics. Although most of this would be the natural continuation of Section 5, it is treated in Section 7, after the exposition of the hierarchies, because it has been discovered that some of the equivalences mentioned are best dealt with within the framework of a class in the Leibniz hierarchy that is larger than the class of algebraizable logics. Finally, in Section 8, one of the new directions along which the theory is expanding, a rather abstract one, is just hinted at. It is impossible to describe all the topics and directions of research that have been pursued in the last two decades. Among these4 are: categorical abstract algebraic logic (Voutsadakis, 2003); applications to many-sorted logics and in particular to behavioural logics (Caleiro and Gonçalves, 2013); the algebraization of Gentzen systems, hypersequent systems and other more general formalisms (Raftery, 2006a); the use of generalized matrices as models of Gentzen systems, and the interplay with their roleˆ as generalized models of sentential logics as described in Section 6.2 (Font, Jansana, and Pigozzi, 2001); the interplay with equality-free model theory (Nurakunov and Stronkowski, 2013); the implicational hierarchy (Cintula and Noguera, 2010); the algebraic study of logics that preserve degrees of truth (Bou et al., 2009); the algebraization of logics by classes of algebras of a different signature (Russo, 2013); etc. Something similar happens when considering applications to the study of particular metalogical properties, and of particular classes of logics and their algebraic counterparts: the choice of the topics to be discussed in detail was difficult and often dictated—besides personal taste—by the space constraints of a chapter of this kind. This Introduction opens with an attempt to describe what abstract algebraic logic is. An alternative way of grasping this is to disregard the introductory discussion

3 Looking back at history, it is easy to see that finding useful classifications has been one major driving force in the development of several areas of mathematics, and of all sciences. 4 The references given here for each case are just pointers to facilitate contact with the topic, rather than the earliest or the main work on it. Abstract Algebraic Logic 5 and just to go ahead and read the technical contents of the chapter. I hope that, as a sample of the topic, it speaks for itself, and sufficiently supports the statement that, simply: abstract algebraic logic is the algebraic logic of the twenty-first century.

Further reading

Only proofs of basic results in the ﬁrst sections are given or sketched; detailed proofs of all the statements can be found in the monumental monograph by Czelakowski (2001), or in my textbook (Font, 2016); or also in the references given for selected results. The Handbook of Mathematical Fuzzy Logic contains a long chapter (Cin- tula and Noguera, 2011) whose aim is, in the author’s own words, “to present a marriage of Mathematical Fuzzy Logic and (Abstract) Algebraic Logic”; it also contains proofs and its (more specialized) point of view may appeal particularly to readers of the present volume. As a matter of fact, the seminal monograph by Blok and Pigozzi (1989) still makes a good introduction to the core of the subject, as does the unpublished (but available) Blok and Pigozzi (2001). Some survey papers that may complement this chapter with different views are Andreka,´ Nemeti,´ and Sain (2001), Font, Jansana, and Pigozzi (2003) and Raftery (2011b), as well as the encyclopædia articles by Pigozzi (2001) and by Jansana (2011). For very precise historical information on technical points, see the notes at the end of each chapter of Czelakowski (2001); in general, Anellis and Houser (1991), Blok and Pigozzi (1988), Burris and Legris (2015), Czelakowski and Malinowski (1985) and Surma (1982) provide a wider historical context.

2 Some preliminary issues

The simplest, neatest kind of algebra-based models are just plain algebras. The notion of algebraizability expresses a relation between a (sentential) logic and a class of algebras. Before presenting that notion, we have to agree on what we understand by each of these two components, and this entails fixing some terminology and notations. Let L be an algebraic language, that is, a similarity type with no relational symbols. The set of terms of L is defined in the usual way, starting from a certain count- ably infinite set V of variables. In our context, the terms are also called formulas and constitute an algebra of type L denoted by Fm (or FmL if necessary); it is the absolutely free algebra of type L generated by V. The algebraic operations of L are called connectives when their logical function is privileged; the usual such operations are ¬,∧,∨,→, etc. Substitutions of formulas for variables are, algebraically speaking, the endomorphisms of the formula algebra. Besides formulas or terms, we can form equations, which are just pairs hα,βi of formulas, denoted by α ≈ β , for α,β ∈ Fm; and quasi-equations, which are first- 6 Josep Maria Font

5 V order formulas of the form i

Deﬁnition 2.1. A (sentential) logic of type L is a pair L = hL,`L i where L is an algebraic language and `L ⊆ P(Fm)×Fm is a relation that satisﬁes the following properties, for all Γ ∪ {ϕ} ⊆ Fm:

If ϕ ∈ Γ , then Γ `L ϕ. (I)

If Γ `L ϕ, then ∆ `L ϕ for any ∆ ⊆ Fm such that Γ ⊆ ∆. (M)

If Γ `L ϕ, then ∆ `L ϕ for any ∆ ⊆ Fm such that ∆ `L ψ for all ψ ∈ Γ . (C)

If Γ `L ϕ, then σ(Γ ) `L σ(ϕ) for every substitution σ. (S)

Ordinary inﬁx notation for the binary relation `L has been used. The labels are self-explanatory: (I) is for “identity”, (M) is for “monotonicity”, (C) is for “cut” and (S) stands for structurality or “substitution invariance”. A relation satisfying (I), (M) and (C) is a closure relation. Thus, it is safe to identify a logic with a substitution-invariant closure relation on the algebra6 of formulas. The relation `L is called the derivability relation or the consequence relation of the logic. A theorem of the logic is a formula ϕ such that /0 `L ϕ (one usually writes just `L ϕ ); in semantically deﬁned logics, the term tautology is used as a synonym of “theorem”. A theory is a set of formulas that is closed under consequence; that is, a set Γ ⊆ Fm such that Γ `L ϕ implies ϕ ∈ Γ . The set of all theories of L is denoted by ThL .

Thus, in some contexts you have to be precise as to which consequence relation you are talking about, lest you (or your readers) fall into some unexpected misunderstandings. Probably the best known, paradigmatic example of this situation is that of modal logics based on classical (non-modal) logic. Let us agree that a set S of formulas (in a language expanding the classical language with a unary connective representing “necessity”, “provability”, or an analogous modal notion) is called a normal modal system whenever it contains all instances of theorems of classical, non-modal logic and all formulas (α →β)→(α →β), and is closed under substitutions (if α ∈ S then σ(α) ∈ S for all substitutions σ ), under modus

5 See the comment on usage of ∧ and → on page 9. 6 Set-theoretically, it is a relation on the set of formulas, but since condition (S) takes the algebraic structure of formulas into account, it makes sense to say it is a relation on the algebra of formulas. Abstract Algebraic Logic 7 ponens (if α ∈ S and α → β ∈ S then β ∈ S) and under necessitation (if α ∈ S then α ∈ S). Then, two consequence relations are naturally associated with S, ` both having all the formulas of S as axioms: the local consequence `S has modus g ponens as its only rule, and the global consequence `S has both modus ponens and necessitation as its rules of inference; the terms “local” and “global” refer to the kind of completeness they satisfy, with respect to classes of Kripke-style frames, when these are available. It is clear that both consequences have the same set of g ` theorems, namely S, and that the theories of `S are just the theories of `S that are closed under necessitation. While the two logics have the same algebraic counterpart,7 they show quite different behaviour in several respects; for instance, as far as their algebraizability and their classification in the hierarchies of abstract alge- g ` braic logic are concerned (except for trivial cases, `S is algebraizable while `S is ` g only equivalential, and `S is fully selfextensional while `S is not even selfextensional). Thus, when speaking about modal logic in the context of abstract algebraic logic one has to clarify whether one refers to the local or to the global consequence. Other logical contexts, where consequence relations do not usually play a prominent role,ˆ demand a similar clarification before discussing algebraizability or, in general, for their study with the tools of abstract algebraic logic. In some cases, the need to deal with a consequence relation rather than with a set of formulas may introduce a certain degree of artificiality. For instance, in the domain of relevance logic, reaching a consensus to discuss axiomatic systems is easier when these are intended only to generate the theorems of some logic (i.e., when only proofs from axioms are considered) than when they are intended to define a consequence relation (i.e., when proofs from arbitrary assumptions are considered). This is a price one has to pay for entering the universe of abstract algebraic logic. A reasonable compromise to reconcile the general theory with everyday practice in some fields, which is useful when dealing with a large family of particular logics, is to take the sets of formulas that are the theorems of the axiomatic extensions of a base logic as the “logics” under study; in this way there is always a consequence relation associated with each such “logic”, albeit hidden in the background.8 It is not uncommon to find work in the literature that states that the notion of algebraizability, and the whole theory, concerns only syntactically defined logics; this is an important and most regrettable misunderstanding. In fact, the notion does not depend on the way the logic (that is, its consequence relation) is presented: it can be syntactic (defined by an axiomatic system in the Frege-Hilbert style, a Gentzen calculus, a tableaux system, a labelled deductive system, etc.) or semantic (and there is also a large variety of semantic devices: truth-tables and logical matrices, algebras, ordered algebras, topological models, relational semantics, etc.); but once defined,

7 This exemplifies why it is imperative that algebra-based semantics that are more general than just plain algebras be considered in some domains, and hence in the general theory. 8 Two modern treatises where this strategy is explicitly adopted, and where algebraizability plays some role,ˆ are Kracht (2007) and Galatos et al. (2007, pp. 72,88). 8 Josep Maria Font the fact that a logic is or is not algebraizable depends only on the consequence relation itself, not on its presentation.9 In contrast, in Galatos, Jipsen, Kowalski, and Ono (2007, p. 9), we can read that “one and the same logic can be algebraizable in one presentation and not algebraizable in another”. This seems to contradict the previous paragraph, but the contra- diction is only apparent, because the terms “logic” and “presentation” are used in different senses: in the previous paragraph the term “logic” was used in the technical sense of Definition 2.1, and by different “presentations” I meant different ways of defining the consequence relation; in the sentence just quoted, the term “logic” is used in a liberal, informal way, so that several formal systems of incomparable frameworks are viewed as different “presentations” (or “formalizations”) of “the same logic”, in the sense we say, for instance, that classical logic can be formalized or presented both as a sentential logic (in the technical sense of Definition 2.1) and as a two-sided, multiple-conclusion sequent calculus. In this chapter, algebraizability is described primarily as concerning sentential logics in the sense of Definition 2.1; but in Section 8, we see that it can be applied to Gentzen systems as well, mutatis mutandis, so that the situation hinted at in the sentence quoted applies. To mention just a very simple example: the conjunction- disjunction fragment of classical logic is not algebraizable when it is formalized as a sentential logic (Font and Verdu,´ 1991), but it is algebraizable when it is formalized as a Gentzen system (Rebagliato and Verdu,´ 1993). Logics defined by syntactic means are normally finitary; that is, they satisfy, for all Γ ∪ {ϕ} ⊆ Fm,

Γ `L ϕ if and only if there is a ﬁnite Γ0 ⊆ Γ such that Γ0 `L ϕ . (F)

Logics defined semantically need not have this property, and accordingly in this chapter it is not considered as part of the definition; but notice that many works (in particular, those by Blok and Pigozzi themselves) do consider it so. Definition 2.1 establishes a crisp boundary delimiting which formalizations of logic may be covered by the algebraizability paradigm, and more generally by abstract algebraic logic as evolved from the algebraizability paradigm and related studies. Before going on, let me say a word about what is not covered by Definition 2.1, and hence by the algebraizability paradigm; notice that almost every alternative for- malization of the notion of a logic has been studied algebraically to some degree, but such studies either do not fit into the framework of this chapter, or have not attained a comparable degree of sophistication. Technically, logics understood as Gentzen-style systems of several kinds (one-conclusion, multiple-conclusion, one- sided, many-sided, etc.) do not fall under Definition 2.1, but as already said, many

9 Quite a different issue is whether one is able or not, or whether it is easy, to prove that a given logic is algebraizable, or to disprove this. This indeed may depend heavily on its presentation. Indeed, there are better tools to achieve this for syntactically presented logics (see Theorems 4.1 and 5.3) than for semantically presented ones (see Theorem 4.14); but, more generally, Moraschini (2016) has proved that the general classification problem, either in the Leibniz hierarchy or in the Frege hierarchy, is undecidable for axiomatically presented logics (hence a fortiori for arbitrary logics), while it is decidable for logics defined by a finite set of finite matrices of finite type. Abstract Algebraic Logic 9

of the basic ideas can be adapted to fit these formalisms, since they define consequence relations on sets of “formulas” of more complicated grammatical structure. More radical is the difference in the case of “symmetric” consequence relations, and of non-monotonic logics. But probably the most important large area that may be missing here is the algebraization of predicate logics. As Blok and Pigozzi (1989, Appendix C) say: “The problem of algebraizing predicate logic is of a different character than the problem for propositional logics because the standard deductive systems for predicate logic are not structural [. . . ]”. In fact, the algebraic study of predicate logics has a long tradition10 and has produced a vast literature as well as many remarkable results, some of great technical difficulty; but the approach is based on another general definition of the notion of a logic, which incorporates a semantic component. The chapter by Andreka,´ Nemeti,´ and Sain (2001) is a good guide to and place to start the study of this area.

Some more points on notation

Note the usage of ≈ for formal equations, and of ∧ and → for the first-order conjunction and implication needed to write quasi-equations; these two symbols should not be confused with the sentential connectives of conjunction (∧) and implication (→), which may or may not be present in the language L of a particular sentential logic under consideration. ConA is the set of congruences of an algebra A. Hom(A,B) is the set of all homomorphisms from algebra A to algebra B, and End(A) := Hom(A,A) is the set of all endomorphisms of A. Sequences of the form ha1,...,ani or han : n ∈ ωi are denoted by ~a, while “~a ∈ ~A” is shorthand for “ai ∈ A for every i”; this is used for variables, formulas and elements of arbitrary algebras. If δ denotes a formula, writing it as δ(z,~z) expresses the assumption that the variables occurring in it belong to the set {z,z1,...,zn}, and that a special roleˆ is assigned to z (the remaining variables are then referred to as parameters); this notation is mainly used so as to be able to write δ(α,~β) to denote the result of a substitution (defined by z 7→ α and zi 7→ βi ) on δ(z,~z), and to describe the interpretations of this formula in a practical way: if h ∈ Hom(Fm,A) A satisfies h(z) = a and h(zi) = ci , then h(δ) is also denoted by δ (a,~c). The same notation (more familiar in the interpretation of terms in universal algebra and model theory) is extended to sets of formulas and of equations in the natural way; recall

10 To the best of my knowledge, the term “abstract algebraic logic” first appears in the literature in the title of Section 5.6 of Henkin, Monk, and Tarski (1985), which is devoted to a general study of the connections between theories of classical first-order logic and classes of cylindric algebras. A related approach, through polyadic algebras, is offered by Halmos (1962), who is generally credited for having coined the term “algebraic logic”. A third, essentially different approach to the algebraic study of predicate logics exploits Mostowski’s idea of interpreting quantifiers as infinite meets and joins in ordered structures (rather than as independent algebraic operations); over time this has become the most popular choice for the algebraic study of particular first-order non-classical logics for which a successful algebraic study of their sentential fragment exists: see Rasiowa (1974), Cintula and Noguera (2015) and Hajek´ (1998). 10 Josep Maria Font that equations are just pairs of formulas, so that if E(x) ⊆ Eq is a set of equations in a single variable x and a ∈ A, then EA(a) ⊆ A × A; more precisely, if E(x) = A A A αi(x) ≈ βi(x) : i ∈ I , then E (a) = hαi (a),βi (a)i : i ∈ I . The consequence `L of a logic relates sets of formulas to single formulas. This relation is extended, with the same notation, to a relation between two sets of formulas, by putting, for Γ,Γ 0 ⊆ Fm,

0 def 0 Γ `L Γ ⇐⇒ Γ `L ϕ for all ϕ ∈ Γ .

The relation `L induces an equivalence relation, the interderivability relation of L , deﬁned as def α a`L β ⇐⇒ α `L β and β `L α , which is also denoted by ΛL from Section 6.3 on. This relation is naturally extended to sets of formulas:

0 def 0 0 Γ a`L Γ ⇐⇒ Γ `L Γ and Γ `L Γ .

More details of the necessary background in logic can be read in Chapter 1 of Galatos et al. (2007); moreover, that chapter and the one by Raftery in this volume contain the basic background in universal algebra needed in abstract algebraic logic.

3 Bare algebraizability

There is a host of logics L for which there exists a class of algebras K such that the following algebraic completeness theorem holds. For all Γ ∪ {ϕ} ⊆ Fm,

Γ ` ϕ ⇐⇒ for all A ∈ K and all h ∈ Hom(Fm,A), L (1) h(γ) = 1 for all γ ∈ Γ implies h(ϕ) = 1.

Here, 1 denotes a special element in the algebras of the class K. When this element is an algebraic constant of the class K, a logic L satisfying (1) is called the assertional logic of K. Table 1 recaps some of the best-known examples, but you most probably know of other cases. For many years, the algebraic study of many logics rested mainly upon this kind of algebraic completeness and its consequences. The genius of Blok and Pigozzi was to realize that this completeness result can be generalized and enhanced in three directions: • Expressing (1) as a relation between two substitution-invariant closure relations: the consequence relation `L of the logic and another relation K , associated with the class of algebras K. Abstract Algebraic Logic 11

L K Classical (sentential) logic Boolean algebras Intuitionistic logic Heyting algebras Dummett-Godel¨ logic Linear Heyting algebras Łukasiewicz’s inﬁnitely-valued logic MV-algebras Global consequence of modal system K Normal modal algebras Global consequence of modal system S4 Closure algebras Global consequence of modal system S5 Monadic algebras BCK logic BCK algebras Table 1 Some logics and their algebraic counterparts obtained by the traditional method.

• Expressing (1) in terms of a translation of formulas into equations (namely, the translation α 7−→ α ≈ 1), and generalizing this to a suitable notion of a structural transformer. • Observing that this translation can be, so to speak, inverted, in the sense that there is another translation that turns every equation α ≈ β into a set of formulas ∆(α,β), that a kind of completeness dual to (1) holds, and that moreover the two translations are mutually inverse in a precise, technical sense. The three directions are discussed separately, before reaching the real, general deﬁ- nition.

3.1 The equational consequence relative to a class of algebras

The key concept here can be expressed as follows.

Deﬁnition 3.1. Let K be a class of algebras (of a common similarity type). The equational consequence relative to K is the relation between sets of equations and equations deﬁned, for any Θ ∪ {ε ≈ δ} ⊆ Eq, as

def Θ K ε ≈ δ ⇐⇒ for all A ∈ K and all h ∈ Hom(Fm,A), (2) h(α) = h(β) for all α ≈ β ∈ Θ implies h(ε) = h(δ). T For an algebra A, A means {A} , so that K = A∈K A . Sometimes writing A α ≈ β [[h]] instead of h(α) = h(β) helps to emphasize the model-theoretic aspect of this relation.

An important point is not to confuse11 this relation with Birkhoff’s equational consequence (denoted, as usual, by ). The latter does not depend on a particular class of algebras K (i.e., it is absolute) and is deﬁned as

11 This warning complements the very pertinent one in Galatos et al. (2007, pp. 68–69) concerning the relation between Birkhoff’s consequence and ﬁrst-order consequence. 12 Josep Maria Font

def Θ ε ≈ δ ⇐⇒ for all algebras A of the similarity type, (3) A α ≈ β for all α ≈ β ∈ Θ implies A ε ≈ δ .

As usual, A α ≈ β means that h(α) = h(β) for all h ∈ Hom(Fm,A). These two consequences are really different, as may be guessed from the different scope of the quantiﬁers “for all h” present in (2) and implicit in (3). Both consequences and K satisfy the replacement rule

α ≈ β for each formula δ(z,~z), (4) δ(α,~z) ≈ δ(β,~z) and while Birkhoff’s consequence satisﬁes the substitution rule

α ≈ β σ(α) ≈ σ(β) for each substitution σ , the consequences K are invariant under substitutions in a sense similar to condition (S) in Deﬁnition 2.1, namely that

if Θ K ε ≈ δ then σ(Θ) K σ(ε) ≈ σ(δ), for each substitution σ .

In contrast, is not invariant under substitutions (example: x ≈ y x ≈ z, but x ≈ x 6 x ≈ z) and in general K does not satisfy the substitution rule (example: x ≈ y 2K x ≈ z, when K contains at least one non-trivial algebra). The sets of equations that are closed under Birkhoff’s consequence are called equational theories, and correspond to varieties, the classes of algebras that are closed under isomorphisms, subalgebras, products12 and homomorphic images; these are actually the classes of algebras defined by a set of equations. When K is a quasivariety (a class of algebras defined by quasi-equations, or, equivalently, a class that is closed under isomorphisms, subalgebras, products and ultraproducts), then K is finitary, in a sense similar to condition (F) on page 8. Note that when Θ is finite, the definition (2) can be expressed in terms of modelling quasi-equations by V {αi ≈ βi : i < k} K ε ≈ δ ⇐⇒ K ∀~x αi ≈ βi → ε ≈ δ i

Γ `L ϕ ⇐⇒ {γ ≈ 1 : γ ∈ Γ } K ϕ ≈ 1. (5) In light of this, the completeness result acquires a more algebraic rather than semantic tone, and at the same time one that makes several generalizations possible and natural, as discussed next.

12 For simplicity, closure under the product operator includes the case of the product of an empty family, which is deﬁned to be a trivial algebra. Abstract Algebraic Logic 13

As in the case of logics, initially the consequence K is a relation between a set of equations and a single equation. This relation is extended, keeping the symbol, to a relation between two sets of equations, by putting, for Θ,Θ 0 ⊆ Eq,

0 def 0 Θ K Θ ⇐⇒ Θ K ε ≈ δ for all ε ≈ δ ∈ Θ ; the associated equivalence relation =||=K and its extension to sets are deﬁned similarly to a`L : 0 def 0 0 Θ =||=K Θ ⇐⇒ Θ K Θ and Θ K Θ

3.2 Translating formulas into equations

The use of the equational consequence of the relevant class of algebras in order to convert the algebraic completeness theorem (1) into the equivalence (5) was made possible by “rewriting” each formula α as the equation α ≈ 1. For this to make sense, this 1 should be a symbol of the language; either a primitive constant or a constant term in the relevant class of algebras. We often say that we have “translated” formulas into equations, but technically, the function is called a transformer;13 moreover, in general we take it that it turns a formula into a set of equations rather than a single equation (you will see why later on). Observe also that the translation has been achieved by taking a “basic” equation x ≈ 1 and then replacing x by α in it. Putting all these features together, we arrive at the following notion.

Deﬁnition 3.2. A transformer from formulas to equations is a function τ : Fm → P(Eq). A transformer is structural when it commutes with substitutions in the sense that for all substitutions σ and all ϕ ∈ Fm, στ(ϕ) = τσ(ϕ).

For this definition to make sense, substitutions are extended to equations in the natural way, i.e., by defining σ(α ≈ β) := σ(α) ≈ σ(β), and to sets of equations S by taking unions, i.e., by defining σ(Θ) := α≈β∈Θ σ(α ≈ β); then, the expression στ(ϕ) makes sense. Meanwhile, transformers are also extended to sets of formulas by taking unions, S i.e., τ(Γ ) := ϕ∈Γ τ(ϕ). A function between the two power sets satisfies this if and only if it commutes with unions; thus, it is equally useful to think of a structural transformer as a function τ : P(Fm) → P(Eq) that commutes with unions14 and with substitutions. It is easy to see that:

13 The term “translation” is much overused in the literature. 14 Considering the natural lattice structure of these power sets (with set inclusion as the order relation), a function that commutes with unions is just a residuated function from P(Fm) to P(Eq). The residuation view is the key to the more abstract approaches to the notion of algebraizability that are brieﬂy touched upon in Section 8. 14 Josep Maria Font

Lemma 3.3. A transformer τ : Fm → P(Eq) is structural if and only if it is deﬁn- able from a set of equations in a single variable E(x) ⊆ Eq by putting τ(ϕ) := E(ϕ) for all ϕ ∈ Fm.

In many cases, the set τ(ϕ) is actually unitary, as in the previous example, where τ(α) = {α ≈ 1}, obtained from the equation x ≈ 1. In many cases where there is no constant 1 in the language, the equation x ≈ x→x works as well; but there are more complicated examples; for instance, Blok and Pigozzi (1989) show that relevance logic R satisﬁes a result like (5) with x ∧ (x → x) ≈ x → x playing the roleˆ of x ≈ 1. Those authors gave the general form of (5) a name:

Deﬁnition 3.4. Let L be a logic, K a class of algebras, and τ a structural transformer. The class K is a τ -algebraic semantics for the logic L , and L is the τ -assertional logic of the class K, when for all Γ ∪ {ϕ} ⊆ Fm,

Γ `L ϕ ⇐⇒ τ(Γ ) K τ(ϕ). (ALG1) A logic has an algebraic semantics when it has a τ -algebraic semantics for some structural transformer, τ . The equations in a single variable of the set E(x) that deﬁnes τ are called the deﬁning equations of the algebraic semantics.

The algebraic completeness theorem (5) can be rephrased as saying that

“K is an algebraic semantics for L with defining equation x ≈ 1”, and we can view Table 1 as listing algebraic semantics of a few well-known logics. In this way, the term “algebraic semantics” has acquired a very precise, technical meaning. Note, however, that the term is very often used in a more ambiguous sense: as indicating that there is some kind of relation between a certain logic and a certain class of algebras, but not necessarily in the sense of Definition 3.4; as an example of this consider the local consequence associated with modal system S4: it is also related to the class of closure algebras, as we see on page 27, but this class is not, technically, an algebraic semantics for that logic. Observe that if K is an algebraic semantics for a logic L , then any other class 0 K such that K = K0 is one as well, with the same defining equations. Thus, for instance, the single two-element Boolean algebra 2, or the class of all finite Boolean algebras, are also algebraic semantics for classical logic (together with the class of all Boolean algebras). The results of Blok and Rebagliato (2003) show that an algebraic semantics for a logic can be quite weird; the bare notion has received little attention and seems to be of limited interest by itself. Only when coupled with the inverse translation to be discussed in the next section does it give rise to a more interesting notion. Abstract Algebraic Logic 15 3.3 Translating equations into formulas

The perspective from which we view an algebraic semantics as a relation between `L and K expressed by a structural transformer makes it all the more natural to consider the symmetric situation: if ∆(x,y) ⊆ Fm is a set of formulas in two variables, then by putting ρ(α ≈ β) := ∆(α,β), and taking unions, one obtains a function ρ : P(Eq) → P(Fm) which commutes with unions and with substitutions in the obvious sense; that is, a structural transformer from equations to formulas. Then the dual relation to (ALG1) would be that, for all Θ ∪ {ε ≈ δ} ⊆ Eq,

Θ K ε ≈ δ ⇐⇒ ρ(Θ) `L ρ(ε ≈ δ). (ALG2) There is no name for logics that satisfy this property alone; it holds in all equivalential logics (one of the classes in the Leibniz hierarchy), and it actually characterizes them if supplemented with a rather natural property of the set ∆(x,y); see Theo- rem 6.2. The “reverse algebraic completeness” expressed by (ALG2) is certainly found in classical logic and Boolean algebras, or in intuitionistic logic and Heyting algebras, although it is not usually expressed through K . Those logics (and many others) satisfy (ALG2) relative to the corresponding class of algebras with ∆(x,y) = {x ↔ y}; the case where Θ is the empty set may look more familiar: it says that an equation ε ≈ δ holds in all Boolean (resp. Heyting) algebras if and only if the formula ε ↔δ is a theorem of classical (resp. intuitionistic) logic, and these are well-known facts. It is easy to see that if a logic L satisfies (ALG2) with respect to a class K through a transformer ρ , then any fragment of L whose language contains the language of the formulas that define ρ also satisfies it, with respect to the appropriate class of algebraic reducts of the algebras in K. In the examples of classical and intuitionistic logic, this applies to all their fragments containing ↔ (obvious); but also to all those containing implication →, if instead of {x ↔ y} we consider ∆(x,y) = {x → y,y → x}, which performs the same function. This explains why, in the general case, it is natural to take transformers as mapping an equation to a set of formulas, and symmetrically a formula to a set of equations.

3.4 Putting it all together

Algebraizability adds, to the existence of two transformers going back and forth between the logic and the equational consequence of the class of algebras, the requirement that these transformers are mutually inverse modulo the consequences. This is reﬂected in the two additional conditions found in the real deﬁnition:15

15 This is actually an extension of the original notion, due to Blok and Pigozzi (1989); see Section 5 for details. 16 Josep Maria Font

Deﬁnition 3.5. A logic L is algebraizable when there are a class K of algebras and structural tranformers τ : P(Fm) → P(Eq) and ρ : P(Eq) → P(Fm) such that for all Γ ∪ {ϕ} ⊆ Fm and all Θ ∪ {ε ≈ δ} ⊆ Eq, the following conditions hold:

Γ `L ϕ ⇐⇒ τ(Γ ) K τ(ϕ) (ALG1)

Θ K ε ≈ δ ⇐⇒ ρ(Θ) `L ρ(ε ≈ δ) (ALG2) ϕ a`L ρ τ(ϕ) (ALG3) ε ≈ δ =||=K τ ρ(ε ≈ δ) (ALG4)

The equations E(x) that define the transformer τ are called the defining equations, and the formulas ∆(x,y) that define the transformer ρ are called the equivalence formulas. The equivalent algebraic semantics of an algebraizable logic L is the largest class K satisfying the above properties.16 Observe that, by substitution invariance, it is enough to require that conditions (ALG3) and (ALG4) hold just for variables (instead of arbitrary formulas): x a`L ρ τ(x) (ALG3) x ≈ y =||=K τ ρ(x ≈ y) (ALG4) The definition is highly symmetric, at the price of being redundant, because

(ALG1) + (ALG4) ⇐⇒ (ALG2) + (ALG3).

The more direct approach to algebraizability is through (ALG1) and (ALG4): while (ALG1) is a natural completeness theorem of the logic L relative to the class of algebras K, the condition (ALG4) concerns the class K alone. Moreover, the consequence x ≈ y K τ ρ(x ≈ y) is trivially equivalent to the statement that K τ ρ(x ≈ x) . Thus, a logic is algebraizable if and only if it has a τ -algebraic semantics K and a reverse transformer ρ such that in each A ∈ K, the equations of the set τ ρ(x ≈ x) and the entailment τ ρ(x ≈ y) A x ≈ y hold. The latter property is a kind of “rule of thumb” for identifying algebraizability: in the candidate class of algebras, identity should be characterized by some set of equations that collectively has the form of the equations appearing in the algebraic completeness, applied to a set of two-variable formulas. For instance, there is a very large class of algebras associated with non-classical logics where the quasi-equation (x → y ≈ 1)∧ (y → x ≈ 1) → x ≈ y holds; namely, the class of implicative algebras introduced by Rasiowa (1974). There is a large group of logics (the so-called implicative logics of Section 4.5)

16 In the original notion, any class K satisfying the four conditions was called an equivalent algebraic semantics of L . Raftery (2006a) started the practice of reserving the name for the largest of all such classes, which is hence unique; see the observations on its existence and character at the end of this section. Abstract Algebraic Logic 17 that have an implication connective → and which satisfy an algebraic completeness theorem like (5) with respect to a class of implicative algebras; the preceding considerations show that all these logics are, indeed, algebraizable. This group is already very large, but there are also many logics that are algebraizable without being implicative: • The equivalence fragments of classical logic and of intuitionistic logic are algebraizable, with E(x) = {x ≈ x ↔ x} and ∆(x,y) = {x ↔ y}. • All the substructural logics associated with varieties of residuated lattices in the book by Galatos et al. (2007) are algebraizable, with E(x) = {x ∧ 1 ≈ 1} and ∆(x,y) = {x\y,y\x}. This constitutes a very large group, including many logics in the linear logic family, most fuzzy logics, Łukasiewicz many-valued logics, and so on. Only some of the logics in this large group are also implicative. • The relevance logic R mentioned above is algebraizable, with E(x) = {x ∧ (x → x) ≈ x→x} and ∆(x,y) = {x→y,y→x}. Technically, this logic does not belong to the preceding group, because it does not contain a constant that plays the roleˆ of 1, but is very close to it. In Section 5, some less standard examples of algebraizable logics are listed. As a consequence of Theorem 4.1 below, it is easy to see that any extension17 of an algebraizable logic is algebraizable as well, with the same transformers; any fragment18 whose language contains the sets E(x) and ∆(x,y) is also algebraizable; and any expansion,18 provided it satisfies a simple condition on ∆ , is again algebraizable. Thus, from each of the preceding examples, a host of new ones is automatically produced; for instance, a host of logics that build on classical logic by expanding its language, such as many logics of the modal family (dynamic, tempo- ral, etc.), are automatically algebraizable if the connectives added satisfy condition (Re∆ ) of Theorem 4.1. A few final observations are in order here concerning the class called in Defi- nition 3.5 the equivalent algebraic semantics of L . Observe that the class K that appears in conditions (ALG1)–(ALG4) does so only through its relative equational consequence K ; therefore, any other class that generates the same consequence would satisfy the same conditions and might thus replace it. It is not difficult to prove that there is always a largest one among all such classes, and that it is a generalized quasivariety19; that is, a class of algebras that can be axomatized by a set of

17 0 A logic L is an extension of a logic L when they have the same language and `L ⊆ `L 0 . 18 If L and L 0 are logics in languages L and L0 respectively, with L ⊆ L0 , then L 0 is an expansion of L when `L ⊆ `L 0 .A conservative expansion is an expansion such that `L = 0 `L 0 ∩ P(FmL) × FmL ; in this case, it is also said that L is the L-fragment of L . 19 Generalized quasivarieties are also called “σ -quasivarieties” in the literature. Due to the fact that we have assumed a fixed set V of variables, generalized quasivarieties as defined above are characterized as the classes of algebras that are closed under isomorphisms, subalgebras, products, and the operation U introduced in Blok and Jonsson´ (2006): U(K) := A : If B is a subalgebra of A generated by a set of cardinality 6 |V|, then B ∈ K . Thus, in particular, generalized quasivarieties are “SP-classes” or prevarieties: these are the classes that are closed under just isomorphisms, subalgebras and products, and can be characterized as 18 Josep Maria Font equations and generalized quasi-equations (infinitary formulas that are like quasi- equations but with a possibly infinite conjunction of equations in the antecedent of the implication). In fact, such a class is the only generalized quasivariety that satisfies the algebraizability conditions for a given logic. Moreover, it is also easy to show20 that this class is independent of the transformers. Thus, it is a uniquely determined algebraic object associated with an algebraizable logic. Later on, in the proof of Theorem 4.1, we see how to construct it explicitly, starting from the logic and any pair of transformers that show algebraizability; see also footnote 22.

4 The origins of algebraizability: the Lindenbaum-Tarski process

It is common to hear or read that the notion of algebraizability arises from a generalization of the Lindenbaum-Tarski procedure of proving the completeness of classical logic with respect to the class of Boolean algebras, in the sense of (1) or (5), and similar statements. In order to understand why this is so, let us review how that original process works.

4.1 The process for classical logic

In this subsection, L denotes classical logic (or another logic with similar properties; see below), and we assume it is defined by some axiomatic system that has modus ponens for the implication connective (MP→) x, x → y `L y among its inference rules; while K denotes the class of Boolean algebras (or the corresponding class for other logics). The proof of the completeness theorem (1) by the Lindenbaum-Tarski method has two separate halves which have very different theoretical importance: (⇒) This is called the Soundness Theorem, and is proved by what is commonly called routine checking; i.e., by checking that if h ∈ Hom(Fm,A) for a Boolean algebra A, then h(ϕ) = 1 for all axioms ϕ , and that all inference rules preserve the property of “being evaluated to 1”. In the case of (MP→) this means checking that if h(ϕ) = 1 and h(ϕ →ψ) = 1 then h(ψ) = 1. All this work is purely algebraic and uses the properties of Boolean algebras in an essential way; for instance, checking (MP→) amounts to checking that in any Boolean algebra, if 1 → a = 1, then a = 1 (and this holds simply because 1→a = ¬1∨a = 0∨a = a). Then, induction on the those defined by (a possibly proper class of) generalized quasi-equations in a language having a proper class of variables (see for instance Hodges, 1993, § 9.2). Note that the term implicational (or implicative) class has been used in the literature for ordinary quasivarieties, for generalized quasivarieties, and for prevarieties. 20 This can be proved directly, just from the definition of algebraizability; but see also Corol- lary 4.15. Abstract Algebraic Logic 19 length of proofs in the axiomatic system defining L completes the proof.

(⇐) This is proved by contraposition: one assumes that Γ 0L ϕ , and one falsi- ﬁes the right-hand side of (1) by constructing a particular Boolean algebra A and a particular h ∈ Hom(Fm,A) such that h(Γ ) ⊆ {1} while h(ϕ) 6= 1. This is strictly speaking the so-called Lindenbaum-Tarski process, and it can be broken down into several different steps:

(LT1) One starts from the assumption that Γ 0L ϕ , and one considers the theory Γ 0 of L generated by Γ . Thus, Γ ⊆ Γ 0 while ϕ ∈/ Γ 0 . (LT2) One defines a relation, denoted here by ΩΓ 0 , by putting α ≡ β (ΩΓ 0) if and only if α → β ∈ Γ 0 and β → α ∈ Γ 0. (LT3) One shows that ΩΓ 0 is a congruence of Fm. This requires usage of specific axioms or theorems of L ; for instance, that (α → β) → (¬β → ¬α) is a 0 theorem, plus (MP→), shows that ΩΓ is a congruence with respect to ¬. (LT4) One shows that the quotient algebra Fm/ΩΓ 0 is a Boolean algebra. Again, this depends on using very particular axioms and theorems of the logic. A key point in this step is to realize that the order relation of the quotient 0 0 algebra is determined in the following way: α/ΩΓ 6 β/ΩΓ if and only if α → β ∈ Γ 0 . (LT5) One shows that for any α ∈ Fm, α ∈ Γ 0 if and only if α/ΩΓ 0 ∈ Γ 0/ΩΓ 0 . It is easy to see that what is required for this to hold is just that if β ∈ Γ 0 and 0 0 β ≡ α (ΩΓ ) then α ∈ Γ , and this in turn follows from (MP→), given the definition of ΩΓ 0 in (LT2). (LT6) Finally, one shows that all formulas in Γ 0 are mutually equivalent, so that Γ 0 constitutes a single element in the quotient algebra, and that this element 0 is the top element 1 ∈ Fm/ΩΓ . Both facts follow from applying (MP→) to the formula α → (β → α), which is a theorem of classical logic (and of many similar logics). Note that after this, the property in (LT5) becomes: For any α ∈ Fm, α ∈ Γ 0 if and only if α/ΩΓ 0 = 1. The “process” is now finished. The quotient algebra Fm/ΩΓ 0 is the desired particular Boolean algebra, by (LT4), and the canonical projection π ∈ Hom(Fm,Fm/ΩΓ 0) defined by π(α) := α/ΩΓ 0 for all α ∈ Fm is the desired particular homomorphism. Then, by (LT1) and (LT5) as rewritten after (LT6), π(Γ ) ⊆ π(Γ 0) = {1} while π(ϕ) 6= 1. This completes the proof. Exactly the same process works for many other logics, including those in Table 1. One class of logics for which the process works with absolutely no modifications (for the corresponding class of algebras) is described in Section 4.5. You may realize that the proof works even if Γ 0 is not exactly the theory generated by Γ ; actually, taking any theory containing Γ and not containing ϕ works. This leaves room for suitable modifications of the choice of Γ 0 in (LT1), which may produce an algebra Fm/ΩΓ 0 with particular properties, and leads to different 20 Josep Maria Font completeness theorems, with respect to restricted classes of algebras.21 An extreme and popular choice for classical logic is to take Γ 0 to be a maximally consistent theory containing Γ but not ϕ (the property that guarantees the existence of such a theory is commonly called “Lindenbaum’s Lemma”); then the quotient algebra is (isomorphic to) the two-element Boolean algebra 2, and we obtain completeness with respect to just this algebra; this proof is virtually equivalent to (one of) the usual textbook completeness proofs for classical logic with respect to two-valued truth tables. In Section 4.3, we see that in fact the process can be generalized in such a way that it applies to absolutely every logic. In some sense, one may say that this is a starting point of abstract algebraic logic; in particular, we see that it is not necessary for the logic to be presented as an axiomatic system, as the previous exposition might suggest. But before that, in Section 4.2, we will see that the process can be performed, mutatis mutandis, for algebraizable logics, in order to find their equivalent algebraic semantics. Then in Section 4.4, we see that this adapted process can actually be considered as a version of the universal process, by introducing appropriate definability conditions.

4.2 The process for algebraizable logics

It may at first seem that in order to establish the algebraizability of a logic, one must previously know of a candidate class of algebras K that satisfies one of the two equivalent pairs of conditions in the definition (notice that K appears in three of the four conditions). The following fundamental result, usually called the syntactic characterization, offers a way to establish algebraizability using conditions exclusively on the logic; moreover it gives a way to find the corresponding equivalent algebraic semantics. It has been used very often in particular cases, and its proof is particularly instructive, as it clearly shows that the five conditions it contains are exactly those that are needed for the Lindenbaum-Tarski process to work in this generalized way.

Theorem 4.1. A logic L is algebraizable if and only if there are sets of equations E(x) ⊆ Eq and of formulas ∆(x,y) ⊆ Fm such that L satisﬁes the following ﬁve conditions:

`L ∆(x,x) (R∆ )

∆(x,y) `L ∆(y,x) (Sym∆ )

∆(x,y) ∪ ∆(y,z) `L ∆(x,z) (Trans∆ )

21 Examples of this strategy are the completeness of the basic fuzzy logic BL with respect to linear BL-algebras (Hajek,´ 1998) and its generalization to the so-called implicational semilinear logics (Cintula and Noguera, 2010), which include all the usual fuzzy logics. Abstract Algebraic Logic 21

n [ ∆(xi,yi) `L ∆(λ x1 ...xn,λ y1 ...yn) for all λ ∈ L, with n = arλ (Re∆ ) i=1 x a`L ∆ E(x) (ALG3)

Then, E(x) is the set of deﬁning equations and ∆(x,y) is the set of equivalence formulas; and the equivalent algebraic semantics of L is the class K(L ,τ,ρ) := A : τ(Γ ) A τ(ϕ) whenever Γ `L ϕ, (6) and E ∆(x,y) A x ≈ y .

Recall that τ(α) := E(α) and ρ(α ≈ β) := ∆(α,β), and that these functions are extended to sets of formulas by taking unions. There is no harm in mixing the two notations, as in (6). Indeed, sometimes one is more illustrative than the other; for instance, the ﬁve conditions were displayed in terms of the sets of formulas to highlight their character of syntactic conditions on the logic alone.

Sketch of the proof of Theorem 4.1. Notice that the four conditions (R∆ )–(Re∆ ) are the “ρ -transforms” of some basic, universal properties of the consequence K . Therefore, if L is algebraizable, those conditions follow from these properties by (ALG2); for instance, x ≈ y K y ≈ x implies (Sym∆ ), the replacement rule (4) implies (Re∆ ), and so forth. As for condition (ALG3), it already appears in the definition of algebraizability. Thus, the five conditions hold. Now assume that L satisfies the five conditions. We are going to review the proof that in this case, L satisfies conditions (ALG1) and (ALG4) with respect to the class K(L ,τ,ρ) defined in (6). Notice that by (ALG1) and (ALG4), if L is algebraizable with respect to some class K by the transformers τ and ρ , then K ⊆ K(L ,τ,ρ); therefore, after showing that L is indeed algebraizable with respect to K(L ,τ,ρ), this class turns out to be the largest one,22 that is, it is the equivalent algebraic semantics of L . To lighten notation, let K denote K(L ,τ,ρ) in the rest of this proof. Condition (ALG4) concerns only the class K, and one of its halves appears explicitly as the second condition on A in definition (6) of K; as for the other half, it amounts to showing that for each A ∈ K, A E ∆(x,x) , but this is a consequence of applying the first condition in (6) to (R∆ ). Thus, what we have to see is that (ALG1) holds. Again, its (⇒) half is just a consequence of the first point in definition (6) of K, and we are left with the proof of its (⇐) half. And this is what an appropriate generalization of the Lindenbaum- Tarski process achieves. Let us review it, without entering into the details, keeping as parallel with the steps in Section 4.1 as possible.

22 This argument only shows that K(L ,τ,ρ) is the largest class for τ and ρ ; Corollary 4.15 shows that this class is actually independent of the transformers, so it is the absolute largest. Notice that (6) describes K(L ,τ,ρ) as the class of all algebras that satisfy a certain set of generalized quasi-equations; thus, this class is indeed a generalized quasivariety, as claimed on page 17. 22 Josep Maria Font

(LT1) One starts from the assumption that Γ 0L ϕ , and one considers the theory Γ 0 of L generated by Γ . Thus, Γ ⊆ Γ 0 while ϕ ∈/ Γ 0 . (LT2) One defines the relation α ≡ β (ΩΓ 0) if and only if ∆(α,β) ⊆ Γ 0 . (LT3) One shows that ΩΓ 0 is a congruence of Fm; this results from mechanical application of conditions (R∆ )–(Re∆ ). Observe that each formalizes one of the properties required: reflexivity, symmetry, transitivity, and compatibility with the operations, respectively. (LT4) One shows that the quotient of the formula algebra under this congruence belongs to the target class, i.e., that Fm/ΩΓ 0 ∈ K. To this end, one has to show that Fm/ΩΓ 0 satisfies the two conditions in (6). Both are shown by using the property in the next step (LT5) and the trick that any h ∈ Hom(Fm,Fm/ΩΓ 0) has the form h = π ◦ σ for some substitution σ , where π : Fm → Fm/ΩΓ 0 is the canonical projection. 0 0 (LT5) One shows that for any α ∈ Fm, α ∈ Γ if and only if Fm/ΩΓ τ(α) [[π]]. This is easily shown by using exclusively (ALG3); as a matter of fact, this property is equivalent to the statement that the first sentence in this point holds for all theories Γ 0 of L . The “process” stops here. The quotient algebra Fm/ΩΓ 0 is the desired particular algebra in K, by (LT4), and the canonical projection π ∈ Hom(Fm,Fm/ΩΓ 0) is the desired particular homomorphism. Applying (LT5) to the facts of (LT1), we can 0 0 conclude that Fm/ΩΓ τ(Γ ) [[π]] while Fm/ΩΓ 6 τ(ϕ) [[π]]. This shows that τ(Γ ) 6 K τ(ϕ) and completes the proof of (ALG1) by contraposition. The parallelism between this proof and the particular one given above for classical logic is clear; observe that the present (LT5) condenses steps (LT5) and (LT6) of the case of classical logic, because the equality α/ΩΓ 0 = 1 appearing there in 0 step (LT6) can be rewritten as the condition Fm/ΩΓ α ≈ 1 [[π]]. In the case of an arbitrary algebraizable logic, there is no need to show that all formulas in Γ 0 are equivalent under ΩΓ 0 , and it makes no sense to speak of the top element of the algebras in K, as they might not have one. Among the examples of substructural logics mentioned near the end of Section 3.4, there are many where these additional properties do not hold. Since every algebraizable logic satisfies the five conditions, the second part of the proof confirms the common claim that algebraizable logics satisfy an algebraic completeness theorem proved by a natural generalization of the Lindenbaum-Tarski process. Moreover, the same proof clearly shows that the completeness also holds (as does the algebraizability) for just the class of algebras

0 0 Fm/ΩΓ : Γ ∈ ThL . (7)

After Theorem 4.11, we see that these algebras deserve the name Lindenbaum- Tarski algebras of the logic L , and they are uniquely determined by it (i.e., they do not depend on the particular set of formulas ∆ used to define ΩΓ 0 ). The properties mentioned at the end of Section 3.4 follow immediately from Theorem 4.1: since all the conditions (R∆ )–(Re∆ ) are formulated in terms of the Abstract Algebraic Logic 23 consequence `L , they are preserved by extensions; they are preserved by fragments, provided that the language contains the connectives that occur in the sets E and ∆ ; and finally, they are preserved by expansions, whenever the new connectives satisfy (Re∆ ). The “canonical” character of the association of a class K(L ,τ,ρ) with each algebraizable logic L , coupled with the first fact just mentioned about the algebraizability of the extensions of an algebraizable logic, produces an important property:

Theorem 4.2. Let L be an algebraizable logic with equivalent algebraic semantics the generalized quasivariety K. Then there is a dual isomorphism between the lattice of all the extensions of L and the lattice of all sub-generalized quasivarieties of K; all these logics are algebraizable and have the corresponding generalized quasivariety as their equivalent algebraic semantics.

Proof. Clearly, if L 0 is an extension of L , then K(L 0,τ ,ρ) ⊆ K(L ,τ ,ρ) = K, and both are generalized quasivarieties. If L is algebraizable, then by Theo- rem 4.1, L 0 is also algebraizable, with the same transformers. The function L 0 7→ K(L 0,τ ,ρ) is order-reversing. Now let K0 ⊆ K be a generalized quasivariety. Since K satisfies (ALG4), K0 also satisfies it (it is a property satisfied individually by each algebra in the class), for the same transformers. Therefore, if we define a logic L 0 from K0 by condition (ALG1) with the same transformers, L 0 is automatically algebraizable with respect to K0 , and its equivalent algebraic semantics, which is the unique generalized quasivariety that satisfies (ALG1) and (ALG4), must perforce be K0 . Finally, that L 0 is an extension of L is a consequence of (ALG1) plus K0 ⊆ K. Thus, the function K0 7→ L 0 also reverses order. The two functions are clearly in- verses to one another. Thus, once an algebraizable logic is found and its equivalent algebraic semantics is identified, one can deal with the lattice of its extensions, in a uniform way, by studying the lattice of sub-generalized quasivarieties of the base class of algebras. This generalizes a number of well-known situations, where attention is restricted to the axiomatic extensions of a particular logic and to the lattice of subvarieties of a particular variety. For more applications of Theorem 4.2, see the end of Section 7.

4.3 The universal Lindenbaum-Tarski process: matrix semantics

After having reviewed the Lindenbaum-Tarski process for classical logic and for algebraizable logics, it is now time to encounter its generalization to absolutely every logic; as is to be expected, this does not rely on the existence of certain sets of formulas or equations with particular properties, but on a more abstract construction which requires a few basic notions of matrix semantics for sentential logics. Logical matrices are constituted by an algebra together with a subset of its universe, and in this sense they are clearly “algebra-based” objects. 24 Josep Maria Font

Definition 4.3. Let L be an algebraic language. An (L-)matrix is a pair hA,Fi where A is an algebra (of type L) and F ⊆ A; this subset is called the filter or truth filter of the matrix. A matrix is a model of a logic L (in the same language) when the following holds, for all Γ ∪ {ϕ} ⊆ Fm:

Γ ` ϕ =⇒ for all h ∈ Hom(Fm,A), L (8) h(γ) ∈ F for all γ ∈ Γ implies h(ϕ) ∈ F.

The class of all models of L is denoted by ModL . A set F ⊆ A is an L -filter when hA,Fi ∈ ModL , and the set of all L -filters of A is denoted by FiL A. Comparison with (1) shows that the Soundness Theorem, i.e., the (⇒) half of (1), in the case of classical logic and Boolean algebras, can be rephrased by saying that in each Boolean algebra the set {1} is a filter of classical logic. In fact, every lattice filter of a Boolean algebra is a filter of classical logic; every lattice filter of a Heyting algebra is a filter of intuitionistic logic; and so on.23 Filters and models can be considered on any algebra, and in particular on the algebra of formulas. Since the endomorphisms of the formula algebra are the substitutions, a quick application of property (S) in Definition 2.1 proves a crucial fact: Lemma 4.4. The filters of a logic on the formula algebra are its theories. And this quickly gives: Theorem 4.5 (Wojcicki,´ 1969). Every logic is complete with respect to the class of all its models. That is, for all Γ ∪ {ϕ} ⊆ Fm:

Γ ` ϕ ⇐⇒ for all hA,Fi ∈ ModL and all h ∈ Hom(Fm,A), L (9) h(γ) ∈ F for all γ ∈ Γ implies h(ϕ) ∈ F.

Proof. Part (⇒) is just the definition of model. Part (⇐) is proved by a radical simplification of the Lindenbaum-Tarski process, as follows. Assume that Γ 6`L ϕ ; just do step (LT1) and obtain the theory Γ 0 generated by Γ ; take hFm,Γ 0i as the required model (it is, by Lemma 4.4) and the identity function from the formula algebra into itself as the required homomorphism; this falsifies the right-hand side of (9). The real generalization of the process requires, of course, a factorization step, and the choice of the right class of algebras (actually, of matrices) so that part (⇒) and step (LT4) work. There is a common misunderstanding, according to which the factorization can only be performed for logics where there is some formula or set of formulas that defines a congruence, as in the two versions of step (LT2) previously shown, or else in those logics where the interderivability relation a`L is itself a congruence. But this is not the case: there is always a natural, canonical way of factorizing every model of every logic. To this end, the following notions are crucial:

23 The logical usage of the term “ﬁlter” is undoubtedly inspired by its algebraic and topological origins. Sometimes the term “deductive ﬁlter” is used to emphasize the difference. Abstract Algebraic Logic 25

Deﬁnition 4.6. Let hA,Fi be an L-matrix. A congruence θ ∈ ConA is compatible with F when for all a,b ∈ A, if a ∈ F and a ≡ b (θ), then b ∈ F . The Leibniz congruence of an L-matrix hA,Fi is deﬁned as

Ω AF := max{θ ∈ ConA : θ is compatible with F}.

A matrix is reduced when its Leibniz congruence is the identity relation. The class ∗ of all reduced matrices that are models of a logic L is denoted by Mod L . The largest compatible congruence always exists, because the ordinary supremum of a family of compatible congruences (which exists because ConA is a complete lattice) is also compatible. Notice that Ω AF is a purely algebraic object, and does not depend on any logic. The following characterization, due independently to Shoesmith and Smiley (1978) and to Czelakowski (1980), extends an idea that is already present in Łos´ (1949):

Theorem 4.7. For any L-matrix hA,Fi and any a,b ∈ A, a ≡ b (Ω AF) if and only if for all δ(x,~z) ∈ Fm and all ~c ∈ ~A, δ A(a,~c) ∈ F ⇔ δ A(b,~c) ∈ F. Though the notion is older than its namesake, it was named after Leibniz by Blok and Pigozzi (1989), based on their reading of this characterization24 as stating that the Leibniz congruence is a ﬁrst-order analogue of the deﬁnition of identity in second-order logic. The Leibniz congruence has also been called the indiscernibility relation or, in a more linguistic-oriented view and for the particular case of the formula algebra, the synonymity relation. The main basic properties we need are:

Lemma 4.8. Let hA,Fi be an L-matrix and L a logic. Then: 1. For all a ∈ A, a ∈ F if and only if a/Ω AF ∈ F/Ω AF. 2. The matrix hA/Ω AF,F/Ω AFi is reduced. 3. hA,Fi ∈ ModL if and only if hA/Ω AF,F/Ω AFi ∈ ModL . Actually, the ﬁrst point is equivalent to the property that Ω AF is compatible with F . The other two points show that any model of a logic produces a reduced model by factorization through the Leibniz congruence. In particular, when applying this to the models of theories of the logic (Lemma 4.4), one ﬁnds the so-called Lindenbaum-Tarski models of a logic L :

∗ Fm Fm ∗ LTMod L := hFm/Ω Γ, Γ /Ω Γ i : Γ ∈ ThL ⊆ Mod L . (10)

The inclusion follows from Lemma 4.4 and points 2 and 3 of Lemma 4.8. Then:

Theorem 4.9 (Wojcicki,´ 1973). Every logic L is complete with respect to any class ∗ ∗ of matrices M such that LTMod L ⊆ M ⊆ Mod L . That is, for all Γ ∪{ϕ} ⊆ Fm,

24 Actually, of a stronger one, formulated in the framework where L-matrices are considered as structures for a first-order language whose constants and function symbols are those in L, and which has only one relation symbol, a unary one, interpreted as the filter of the matrix. This deeply influential idea, due to Bloom (1975), is further explained on page 44. 26 Josep Maria Font

Γ ` ϕ ⇐⇒ for all hA,Fi ∈ M and all h ∈ Hom(Fm,A), L (11) h(γ) ∈ F for all γ ∈ Γ implies h(ϕ) ∈ F.

In particular, every logic is complete with respect to the class of its Lindenbaum- Tarski models and with respect to the class of all its reduced models. ∗ Proof. Part (⇒) follows from the assumption that M ⊆ Mod L , and part (⇐) is proved by the real generalization of the Lindenbaum-Tarski process: 0 (LT1) One starts from the assumption that Γ 0L ϕ , and one considers Γ , the theory of L generated by Γ . Thus, Γ ⊆ Γ 0 while ϕ ∈/ Γ 0 . (LT2) One considers the Leibniz congruence Ω FmΓ 0 given by Definition 4.6. (LT3) According to its own definition, Ω FmΓ 0 is a congruence of Fm. (LT4) The quotient matrix hFm/Ω FmΓ 0, Γ 0/Ω FmΓ 0i belongs to M by the as- ∗ sumption that LTMod L ⊆ M. (LT5) For any α ∈ Fm, α ∈ Γ 0 if and only if α/Ω FmΓ 0 ∈ Γ 0/Ω FmΓ 0 , by point 1 of Lemma 4.8. In particular, by (LT1) plus (LT5), the canonical projection π ∈ Hom(Fm,Fm/Ω FmΓ 0) defined by π(α) := α/ΩΓ 0 for all α ∈ Fm satisfies that π(Γ ) ⊆ π(Γ 0) = Γ 0/Ω FmΓ 0 while π(ϕ) ∈/ Γ 0/Ω FmΓ 0 . Taking (LT4) into account, this falsifies the right-hand side of (11), and finishes the proof. In Theorem 4.11 we see that the congruences denoted by ΩΓ 0 in steps (LT2) on page 19 and on page 22 actually coincide with what has just been denoted by Ω FmΓ 0 , following Definition 4.6; this justifies the choice of the previous notation. You may have observed that, as in the case of an arbitrary algebraizable logic, the general step (LT5) is enough to finish the proof, and there is no need for step (LT6), which only makes sense in particular cases, such as that of classical and similar logics. Matrix semantics provides a first notion of the algebraic counterpart of a logic L (a second, more general notion is found in Section 6.2): it is the class25 ∗ ∗ Alg L := the algebraic reducts of the matrices in Mod L . ∗ The Completeness Theorem 4.9, for M = Mod L , can be restated in terms of the class of algebras as follows:

Γ `L ϕ ⇐⇒ h(Γ ) ⊆ F implies h(ϕ) ∈ F , ∗ for all A ∈ Alg L , all h ∈ Hom(Fm,A), (12) A and all F ∈ FiL A such that Ω F is the identity relation. In the best behaved cases, the conditions in the last row can be formulated men- tioning only the class of algebras; for instance, if L is classical logic, they amount

25 Sometimes the algebras in Alg∗L are called the Leibniz-reduced algebras of L . Abstract Algebraic Logic 27 ∗ to F = {1}, so that Mod L = hA,{1}i : A is a Boolean algebra , and (12) becomes (1), our starting point. Similarly for the other logics in Table 1 on page 11. Corollary 4.15 shows that this simpliﬁcation to use just algebras is actually possible ∗ whenever L is algebraizable, but it is not possible in general; the class Alg L does not characterize the logic, and matrices cannot be dispensed with. The paradigmatic case of the two consequences associated with any normal system S of modal logic ∗ ` ∗ g has already been mentioned: for them, Alg (`S) = Alg (`S); for S4, this is the variety of closure algebras. In Lemma 6.3, we see that there is a one-to-one nat- ∗ ∗ ural correspondence between Mod L and Alg L for a large class of logics that includes the algebraizable ones as well as many others.

4.4 Algebraizability and matrix semantics: deﬁnability

In this section, we see how the Lindenbaum-Tarski process for algebraizable logics matches the universal process, and we check what was announced before: the algebraizable version arises from the general one by plugging two deﬁnability conditions into the two crucial steps of the process. To do this, we need to consider the general version of a property that was seen to play a key roleˆ in the process for classical logic. Consider the following generalization of modus ponens:

x, ∆(x,y) `L y (MP∆ )

Lemma 4.10. Every algebraizable logic L satisﬁes (MP∆ ) for any of its sets of equivalence formulas ∆ . Proof. By (ALG1), condition (MP∆ ) holds if and only if E(x), E ∆(x,y) K E(y). But by condition (ALG4), this holds if and only if E(x), x ≈ y K E(y), and this is trivially true.

It turns out that coupling (MP∆ ) with four of the ﬁve conditions of Theorem 4.1 amounts to one of the deﬁnability conditions we are after:

Theorem 4.11 (Czelakowski, 1981). Let L be a logic and let ∆(x,y) ⊆ Fm. The following conditions are equivalent.

(i) L satisfies conditions (R∆ )–(Re∆ ) and (MP∆ ). (ii) The set ∆ defines the Leibniz congruence in every model of L , in the sense that for every hA,Fi ∈ ModL and every a,b ∈ A, a ≡ b (Ω AF) if and only if ∆ A(a,b) ⊆ F. (iii) The set ∆ defines equality in every reduced model of L , in the sense that for ∗ every hA,Fi ∈ Mod L and every a,b ∈ A, a = b if and only if ∆ A(a,b) ⊆ F. When these conditions hold, ∆ is called a set of equivalence formulas26 for L .

26 They are also called “congruence formulas”, for obvious reasons. 28 Josep Maria Font

The logics that satisfy any of these equivalent conditions are considered in Def- inition 6.1. It is possible to show that conditions (Sym∆ ) and (Trans∆ ) can be dispensed with in point (i) of this result. By Theorem 4.1 and Lemma 4.10, algebraizable logics satisfy point (i) of Theo- rem 4.11; this confirms that the congruence denoted simply as ΩΓ 0 in steps (LT2) on pages 19 and 22 coincides with Ω FmΓ 0 (thus justifying the initial choice of notation), and that the roleˆ played by conditions (R∆ )–(Re∆ ) in step (LT3) of the algebraizable case corresponds to the fact that in the general case we are factoring out by a congruence. Thus, from this point onwards, I write Ω instead of Ω Fm , even in the general case. Moreover, this coincidence also shows that the congruence ΩΓ 0 defined above does not depend on the particular set ∆ used to define it; and in particular that the algebras in (7) are the algebraic reducts of the Lindenbaum-Tarski models of (10). The other definability condition concerns truth; the set F in a matrix hA,Fi is called the truth filter because when using matrices as semantics for logics as in (8) or (9), the truth condition is “to belong to F ”. Using Theorems 4.5 and 4.9, one finds the following. Theorem 4.12 (Herrmann, 1993). Let L be a logic having a set ∆ of equivalence formulas, and let E(x) ⊆ Eq. The following conditions are equivalent. (i) L satisfies condition (ALG3) for ∆ and E. (ii) The set E defines truth from the Leibniz congruence in every model of L , in the sense that for every hA,Fi ∈ ModL and every a ∈ A, a ∈ F if and only if EA(a) ⊆ Ω AF. (iii) The set E defines truth from equality in every reduced model of L , in the ∗ sense that for every hA,Fi ∈ Mod L and every a ∈ A, a ∈ F if and only if A E(x) [[a]]. Note that the set ∆ does not appear in conditions (ii) and (iii); actually, the equivalence between these two is independent of the assumption about ∆ made in the theorem, and the logics that satisfy either of these two conditions are considered in Definition 6.1. Again, comparison between steps (LT5) in the two versions of the process on pages 22 and 26 shows that in the algebraizable case the truth condition has been made definable by satisfaction of equations E. Thus, coupling Theorem 4.1 with the two preceding ones we obtain: Corollary 4.13. A logic is algebraizable if and only if, in all its reduced models, equality is definable from the truth filter through a set ∆(x,y) in the sense of Theo- rem 4.11(iii) and the truth filter is definable from equality through a set E(x) in the sense of Theorem 4.12(iii). This view of algebraizability as mutual interdefinability of equality and the truth filter allows us to obtain one of the few general results that characterize algebraizability for semantically defined logics (it is, however, of limited application). A finite algebra is primal when for all n ∈ ω , all functions f : An → A are definable by a term in n variables. Using just the definition, it is not difficult to show the following. Abstract Algebraic Logic 29

Theorem 4.14. Let L be a logic defined by a finite matrix hA,Fi such that A is a primal algebra and F 6= /0. Then L is algebraizable.27 To obtain a more graphical rendering of the next property, it is useful to consider the set A Solτ := a ∈ A : A E(x) [[a]] , that is, the set of solutions of the equations E(x) that define the transformer τ ; thus, it is what we could call, with a geometrical analogy, an “algebraic set”. Now, A ∗ condition 4.12(iii) says that F = Solτ for each hA,Fi ∈ Mod L ; from this it is easy to prove that the classes of algebras obtained by the algebraizability approach and by the universal matrix semantics coincide for algebraizable logics.

Corollary 4.15. If L is an algebraizable logic, with transformers τ and ρ , then ∗ A ∗ ∗ Mod L = hA,Solτ i : A ∈ Alg L and Alg L = K(L ,τ,ρ), the equivalent algebraic semantics of L . ∗ ∗ Since Mod L and Alg L are intrinsic28 to L , this is one way of showing that the notion of the equivalent algebraic semantics and the class K(L ,τ,ρ) are independent of the actual transformers. In this case, there is a one-to-one correspondence between reduced models and algebras of the equivalent algebraic semantics of L ; and the algebraic completeness theorem (12) can be expressed solely in terms of the ∗ class of algebras Alg L .

4.5 Implicative logics

Although the proof of Theorem 4.1 was presented as a natural generalization of the Lindenbaum-Tarski process for classical logic and other similar logics, historically it took a long time before this possibility was recognized and the general notion of an algebraizable logic was isolated. A ﬁrst, much earlier generalization isolated the roleˆ of the implication connective as used in Section 4.1; as a consequence, the class of logics it applies to is more restricted than that of the algebraizable ones, but is still very large, and forms the most popular kind of algebraizable logics.

Deﬁnition 4.16. An implicative logic is a logic L in a language L having a binary term, represented here by x → y, such that the following conditions are satisﬁed.

(IL1) `L x → x. (IL2) x → y, y → z `L x → z. (IL3) x1 → y1 ,..., xn → yn ,y1 → x1 ,..., yn → xn `L λ x1 ...xn → λ y1 ...yn , for each λ ∈ L, of arity n > 0.

27 Actually, the logic is BP-algebraizable, in the sense of Deﬁnition 5.1. 28 In the sense that they are uniquely determined by L . In the reverse sense, only Mod∗L deter- mines L , by Theorem 4.9; Alg∗L does not, as already mentioned at the end of Section 4.3. 30 Josep Maria Font

(IL4) x, x → y `L y. (IL5) x `L y → x. Theorem 4.17. All implicative logics are algebraizable, with defining equation E(x) := {x ≈ x → x} and equivalence formulas ∆(x,y) := {x → y, y → x}. Proof. It is enough to check that the mentioned sets E and ∆ satisfy the conditions of Theorem 4.1. For this ∆ , (Re∆ ) is exactly (IL1); (Sym∆ ) is contained in the very definition of ∆ ; (Trans∆ ) and (Re∆ ) are straightforward consequences of (IL2) and (IL3), respectively; and finally, in this case (ALG3) is x a`L {x → (x → x),(x → x) → x}, and this is easily proved using (IL1), (IL4) and (IL5). The converse of Theorem 4.17 does not hold. The class of implicative logics is larger than that of the logics that are algebraizable in that way: the five conditions defining implicative logics are close, but clearly stronger than those in Theorem 4.1; this is apparent, for instance, if we compare (IL4), which is (MP→), with (MP∆ ) for ∆(x,y) = {x → y, y → x}, which is the condition x, x → y, y → x `L y. Both sets of conditions are designed to obtain a smooth and natural generalization of the Lindenbaum-Tarski process; but while in algebraizable logics this is done by generalizing the properties of the equivalence connective, in implicative logics it is based on properties of implication, expressed through the binary relation 6Γ 0 def 0 defined by: α 6Γ 0 β ⇐⇒ Γ `L α → β . It is easy to check that: 0 • (IL1) is equivalent to requiring that for any Γ ∈ ThL , the relation 6Γ 0 is re- flexive. 0 • (IL2) is equivalent to requiring that for any Γ ∈ ThL , the relation 6Γ 0 is transitive.

Under these two conditions, 6Γ 0 is a quasi-order, and it is well known that its symmetrization, which coincides with ΩΓ 0 for ∆(x,y) = {x → y, y → x}, is an equivalence relation compatible with 6Γ 0 ; so that in the quotient, the quasi-order induces an order 6 deﬁned by 0 0 def 0 α/ΩΓ 6 β/ΩΓ ⇐⇒ α 6Γ 0 β ⇐⇒ Γ `L α → β . Taking this into account, we can go on as follows. • (IL3) is equivalent to requiring that for any Γ 0 ∈ ThL , the relation ΩΓ 0 is compatible with the algebraic operations of L; that is, it is a congruence of the formula algebra. Thus, the quotient becomes an algebra of type L. • (IL4) is equivalent to requiring that each Γ 0 ∈ ThL is an up-set of its associated 0 0 quasi-order 6Γ 0 (that is, α ∈ Γ and α 6Γ 0 β imply β ∈ Γ ). • (IL5) is equivalent to requiring that each Γ 0 ∈ ThL constitutes a single equivalence class (i.e., any two formulas in Γ 0 are equivalent modulo ΩΓ 0 ) and this class is the maximum of the order 6 of the quotient. Note that the ﬁrst of these two properties amounts to requiring the condition:

x, y `L ∆(x,y) (G∆ ) Abstract Algebraic Logic 31

for the particular ∆ we are dealing with here; this condition is a consequence of (IL5) but is important in its own right, and reappears in Section 6.1. (In each step, the properties of the preceding steps are assumed to hold.) If we review the Lindenbaum-Tarski process again, we see that steps (LT1)– (LT3) are obtained from conditions (IL1)–(IL3); (LT5) is a consequence of (IL4), which implies that ΩΓ 0 is compatible with Γ 0 ; and in the discussion on page 19 it was already stressed that in the case of classical logic this step is essentially due to (MP→), here (IL4). Step (LT4) depends on the wise definition of the class K with respect to which the completeness is to be proved, and in the case of implicative logics this is actually the class K(L ,τ,ρ) for the transformers specified in The- orem 4.17. By the facts pointed out in the previous paragraph, these algebras are ordered. The stronger form of (IL4) and the additional property (IL5) allow us to obtain here, unlike in the general case of algebraizable logics, step (LT6); that is, that the theory Γ 0 constitutes a single element in the quotient Fm/ΩΓ 0 and that this element is the top element of its order. Thus, in this case the algebraic completeness is totally parallel to that of classical or intuitionistic logic. Actually, Henkin (1950) observed that, after changing only the class of algebras, the process works for the implication fragment of intuitionistic logic. In parallel, Rasiowa and Sikorski (1953) did the same for its positive fragment (i.e., admitting also conjunction and disjunction into the language). The general theory was finally refined29 by Rasiowa (1974) by isolating the conditions of Definition 4.16, which are somehow weaker and are, as discussed before, exactly equivalent to what is required for the process to work in this “implicative version”. Recently, this notion has been weakened and generalized in several natural ways by Cintula and Noguera (2010), as shown in Definition 5.2, where the logics of Definition 4.16 are called precisely Rasiowa implicative. The class of implicative logics is certainly large. Besides all the fragments of classical or intuitionistic logic containing implication (writing x → x instead of 1 if necessary), it contains Johansson’s minimal logic, the global consequences of normal modal logics, Nelson’s constructive logic with strong negation and Rousseau’s version of Post’s finitely-valued logics. Classical predicate logic, when formalized as a deductive system satisfying Definition 2.1, as in Appendix C of Blok and Pigozzi (1989), is also implicative. It is clear that any extension of an implicative logic is also implicative, and that an expansion is implicative if and only if the new connectives satisfy condition (IL3).

29 Rasiowa’s notion assumes two inessential requirements for the language: that it contains no connectives of arity greater than 2, and that → is a primitive connective (rather than an arbitrary term in two variables). Moreover, she considers ﬁnitarity as part of the deﬁnition of a logic. All these restrictions are usually removed in later studies. 32 Josep Maria Font 5 Modes of algebraizability, and non-algebraizability

The notion of algebraizability contained in Definition 3.5 is a result of the original definition in Blok and Pigozzi (1989) after removing some restrictions; namely, that it applied only to finitary logics and to quasivarieties, and that the two transformers were assumed finite. This move was due to Herrmann (1993) and has been adopted in the literature; while the original notion is sometimes called algebraizability in the sense of Blok and Pigozzi.30 The refinements of the notion mostly considered in the literature are the following.

Definition 5.1. An algebraizable logic L is: - finitely algebraizable when the set ∆(x,y) of equivalence formulas is finite. - regularly algebraizable when it moreover satisfies condition (G∆ ). - finitely regularly algebraizable when it is both finitely and regularly algebraizable. - BP-algebraizable when it is finitary and finitely algebraizable.

It may seem strange that the term “finitely” does not mean that the two transformers are finite. This is due to historical reasons (see footnote 33) and to the (straightforward) fact that for finitary algebraizable logics, the set E(x) can always be taken as finite; therefore, for finitary and finitely algebraizable logics both transformers can be taken as finite, as in Blok and Pigozzi’s original definition. This explains the name “BP-algebraizable”. Similarly, when K is finitary, the set ∆(x,y) can always be taken as finite. Regularly algebraizable logics have a single defining equation, of the form x ≈ >, where > is any theorem of the logic with at most one variable x. Since this equation A defines the truth filter Solτ of reduced models, it turns out that for these logics the truth filter of reduced models is a one-element set. Clearly, all implicative logics are in fact finitely regularly algebraizable. It is instructive to review some less standard cases, which show the flexibility of these general notions and the diversity of logics that fall under their scope. • Łukasiewicz’s infinitely-valued logic, defined semantically from the unit real interval with {1} as the truth filter, is known to be non-finitary, but it is still implicative, thus falling under Theorem 4.17. Therefore, in this case, the two transformers are finite; but both the logic and its associated equational consequence (relative to the generalized quasivariety generated by the algebra on the unit real interval, which is known not to be a quasivariety) are non-finitary. • The so-called “Last Judgement” logic of Herrmann (1996) is a finitary modal logic, very close to the local consequence of the minimal normal modal system K, but without the Necessitation Rule even in its weak version (concerning only theorems). It is algebraizable, with the single defining equation ¬x ≈ ¬(x → x) n n and with the infinite set of equivalence formulas (x→y), (y→x) : n > 0 .

30 This phrase is also used to refer, in a more general way, to the idea of characterizing algebraizability as a relation between the consequence of a logic and the equational consequence of a class of algebras through transformers, when compared with more distant approaches. Abstract Algebraic Logic 33

One can show that no finite algebraization is possible, and that the equivalent equational consequence is not finitary. • Raftery (2010) constructed an example of a (non-finitary) logic that is finitely algebraizable but needs an infinite set of defining equations, while its equivalent algebraic semantics is a variety, and hence has a finitary equational consequence. • All the substructural logics determined by a variety of residuated lattices studied by Galatos et al. (2007) are finitely algebraizable, with (x\y) ∧ (y\x) as equivalence formula, and with x ∧ 1 ≈ 1 as defining equation; therefore, the truth filter of their reduced models has the form {a ∈ A : 1 6 a}. If the variety is integral, then 1 (the unit of the monoid operation) is the maximum of the lattice order, and this set reduces to {1}, which means the logic is finitely regularly algebraizable. If the variety is not integral, then the set is not always unitary, and the logic is not regularly algebraizable.

A related, intriguing OPENPROBLEM is that to date, no algebraizable logic is known whose set of defining equations can be finite but cannot consist of a single equation. The dual situation is known to be possible: there are algebraizable logics that admit a finite set of equivalence formulas but not a one-element set (Cintula and Noguera, 2010). Observe that the Lindenbaum-Tarski process, when performed for a regularly algebraizable logic, takes us to the first point in step (LT6) on page 19; namely, that all formulas in Γ 0 are mutually equivalent and hence form a single point in the quotient; this is an immediate consequence of condition (G∆ ). Thus, in all these cases the algebraic models have a single designated element. It need not be, however, the top element. Actually, in general, the algebras need not even be naturally ordered, although this certainly happens in implicative logics. Paradigmatic examples of finitely regularly algebraizable logics that are not implicative are the equivalence fragments of classical and of intuitionitic logic. Actually, there is a qualitative break between implicative logics and algebraizable logics in general, which you may have already noticed: in the former, the equivalence set {x → y, y → x} has the special feature of being the “symmetrization” of a simpler set of formulas, namely {x → y}, to which the notion of an implicative logic assigns definite properties (moreover, this set is a singleton). Extending this feature, Cintula and Noguera (2010) have introduced further modes of algebraiz- ability31 which arise as a consequence of generalized modes of being implicative; briefly, they are as follows.

Deﬁnition 5.2. Let ∇(x,y) ⊆ Fm be a set of formulas in two variables, and put ∇s(x,y) := ∇(x,y) ∪ ∇(y,x). Let L be a logic, and consider the following properties:

`L ∇(x,x) (R∇)

∇(x,y) ∪ ∇(y,z) `L ∇(x,z) (Trans∇)

31 This approach gave rise to other classes of (non-algebraizable) logics, forming the so-called implicational hierarchy, which is not described here. 34 Josep Maria Font

n [ s ∇ (xi,yi) `L ∇(λ x1 ...xn,λ y1 ...yn) for all λ ∈ L, n = arλ > 0 (SRe∇) i=1

x,∇(x,y) `L y (MP∇) s x a`L ∇ E(x) for some E(x) ⊆ Eq (ALG3)

x,y `L ∇(x,y) (G∇)

x `L ∇(y,x) (W∇)

A logic L is algebraically implicational when it satisfies the conditions (R∇)– (MP∇) and (ALG3) for some set ∇ of formulas and some set E of equations. If condition (ALG3) is replaced by (G∇), L is regularly implicational; and if it is replaced by (W∇), it is Rasiowa implicational. When the set ∇ is finite, the adjective finitely is prepended to the name, and when it is a singleton, “implicational” is replaced by implicative.

It is clear how these conditions generalize those previously considered; note, however, the subtle difference between (Re∆ ) and (SRe∇): while the former has ∆ in the assumption, the latter has the symmetrized set ∇s rather than ∇ itself (the “S” in the label is for “symmetrized”). Obviously, (W∇) implies (G∇), and, using (MP∇), it is easy to see that (ALG3) follows from (G∇), with E(x) := {x ≈ >}, where > represents any theorem of the logic in at most the variable x. Thus, informally:

Rasiowa =⇒ regularly =⇒ algebraically

It is also easy to check, using Theorem 4.1, that “algebraically implicational” is the same as “algebraizable”, with ∆ := ∇s as the set of equivalence formulas (hence the same holds for its subclasses), and that “Rasiowa implicative” is the same as “implicative” (Definition 4.16), with → as the single member of ∇. Thus, in all, we have nine classes, of which four are new; they form the poset depicted in Figure 1. The necessary counterexamples establish that they are all different and that no other inclusion relations hold between them (besides the ones in the diagram and the ones implied by it). For instance, the example of the equivalence fragment of classical logic already mentioned is regularly implicative but not Rasiowa implicational; and so on. Although its primary motivation is exclusively syntactical, this approach has some semantic consequences, concerning the relation defined by the set ∇ on the A A matrix models hA,Fi of the logic by putting a 6∇ b if and only if ∇ (a,b) ⊆ F . In the present context, this relation is a quasi-order and the set F is an up-set with respect to it. The quasi-order turns out to be an order if and only if the matrix is reduced. Then the regularity condition (G∇) corresponds to the truth filter F of reduced models being a singleton, while the “Rasiowa” condition (W∇) corresponds to this single element being the maximum of the order. Note, however, that some conditions in Definition 5.2 are stronger than strictly A needed in order to show that the mentioned relation 6∇ is an order in reduced models of the logic. A finer analysis of this issue, (Raftery, 2013a), shows that a set Abstract Algebraic Logic 35

implicative

regularly ﬁnitely Rasiowa implicative implicational

algebraically ﬁnitely regularly Rasiowa implicative algebraizable implicational

ﬁnitely regularly algebraizable algebraizable

algebraizable

Fig. 1 Nine of the ten modes of algebraizability considered so far; the “BP-algebraizable” logics, not shown here, are the ﬁnitely algebraizable ones that are ﬁnitary. The arrow means “included in” (for the classes) or “implies” (for the corresponding properties). Thus, the diagram is “upside down”, with larger classes in lower position; joins in the graph are in fact class intersections.

∇(x,y) has this property if and only if it satisﬁes the properties (R∇), (Trans∇) and

s ∇ (x,y),δ(x,~z) `L δ(y,~z) for all δ(x,~z) ∈ Fm.

These three conditions together with the condition (ALG3) for ∇ turn out to characterize syntactically the class of order algebraizable logics. These are logics that enjoy a tight connection between their consequence relation and a relative “inequa- tional” consequence (i.e., one similar to K but where equality is replaced by order), namely a connection expressed by structural transformers (between formulas and inequations) satisfying properties completely analogous to the (ALG1)–(ALG4) of algebraizability. Roughly speaking, these logics are “algebraized” by a class of ordered algebras, i.e., algebras with an independent order relation,32 which explains the name, and seem to be one solution to Pigozzi’s problem of ﬁnding an abstract characterization of the notion of an implication; see Raftery (2013a) for a thorough discussion and more references. The semantic approach is essential when one wants to ﬁnd counterexamples. While Theorem 4.1 provides a very useful tool for proving that a given logic is algebraizable, its existential character (“there are transformers such that . . . ”) make it in

32 Although this is clearly one kind of algebra-based semantics, it departs from the general framework of this chapter, and its exposition would exceed current space constraints. Let me just add that order algebraizable logics in general need not be algebraizable; and that they are so if and only if the order relation in their equivalent “ordered algebraic semantics” is equationally definable. They are, though, stronger than equivalential, a significant class in the Leibniz hierarchy that appears in Section 6.1. 36 Josep Maria Font general useless (like the definition itself) when one wants to prove that a logic is not algebraizable. To do this, lattice-theoretic characterizations of a universal character have been developed with great success. They exploit the properties of the Leibniz A operator, the function F 7→ Ω F restricted to the family FiL A of the L -filters of an algebra A; i.e., A Ω : Fi A → ConAlg∗ A L L (13) F 7−→ Ω AF ∗ where ConAlg∗L A := {θ ∈ ConA : A/θ ∈ Alg L }. Recall that by Corollary 4.15, the equivalent algebraic semantics of an algebraizable logic L coincides with the ∗ class Alg L of the Leibniz-reduced algebras of L . It turns out that it is possible to characterize the very fact that a logic is algebraizable by the behaviour of this operator. To reduce the burden of definitions and aux- iliary properties, only the characterization concerning BP-algebraizability, which is particularly simple, is reproduced here; other isomorphism theorems appear in Table 2 on page 41 and in Theorem 8.2.

Theorem 5.3. Let L be a ﬁnitary logic and let K be a quasivariety. The logic L is BP-algebraizable and K is its equivalent algebraic semantics if and only if for each algebra A the Leibniz operator Ω A is an isomorphism between the lattice

FiL A of L -ﬁlters of A and the lattice ConKA of congruences of A relative to the quasivariety K.

In general, for any class K of algebras and any algebra A, ConKA is the complete lattice of congruences θ of A such that A/θ ∈ K; if K is a variety and A ∈ K, then simply ConKA = ConA. The isomorphism and its inverse are deﬁnable as suggested by Theorems 4.11 and 4.12:

F 7−→ Ω AF = ha,bi ∈ A × A : ∆ A(a,b) ⊆ F −1 (14) θ 7−→ Ω A θ = a ∈ A : EA(a) ⊆ θ

This result generalizes the well-known isomorphism between filters and congruences in Boolean algebras or in Heyting algebras, as well as that between normal subgroups and congruences in groups, or between ideals and congruences in rings. Perhaps the most noteworthy feature of this result is that the isomorphism holds for every algebra, and not just for algebras in K. This means that there is a lot of freedom when constructing counterexamples that show the non-algebraizability of certain logics by contradicting this isomorphism; that is, when constructing (finite, small) algebras A such that Ω A is for instance non-monotonic, or non-injective, on the L -filters of A; by Theorem 5.3, this would signify a failure of algebraizability, no matter which class K one may have in mind. This is how the non-algebraizability ` of `S5 was first proved (Blok and Pigozzi, 1989, § 5.2.1). From the consequences of Theorem 4.1 stated on page 17, it follows that a non- algebraizability result entails the non-algebraizability of a number of other logics related to the initial one; for instance, from the S5 case, the non-algebraizability of virtually all local consequences of modal logics follows as well. Abstract Algebraic Logic 37

This result can also be used to show that a given class of algebras cannot be logi- fied by the paradigm of algebraizability; that is, that the class is not the equivalent algebraic semantics of any BP-algebraizable logic. This is usually trickier, but is based on the same idea, though in dual perspective: no matter which logic one has in mind, if it should be algebraizable then the lattice structure of the family of its filters on a given algebra would have to be isomorphic, through the Leibniz operator, to that of its relative congruences (if the class is a variety and the chosen algebra belongs to it, this is just its congruence lattice); meanwhile, since the Leibniz operator (and its inverse) is a purely algebraic object independent of the logic, knowledge of the particular algebra and of its congruences may imply that such a situation is impossible. This was done for the first time by Blok for the variety of distributive lattices (see Font and Verdu´ 1991, page 397).

6 Beyond algebraizability

As explained in the Introduction, abstract algebraic logic goes far beyond algebraizability, and develops a richer framework. Within that framework, several larger classes of logics have been identiﬁed and characterized in several ways, and have been seen to be relevant for the study of the correspondences between logical properties and algebraic properties. In some sense, this study is the ultimate goal of the subject. This very sketchy (but not short) section concentrates on the ﬁrst issue (description of the hierarchies), while the second (consequences, for a logic, of belonging to a certain class) is touched upon in Section 7.

6.1 The Leibniz hierarchy

Strictly speaking, the term “Leibniz hierarchy” refers to a classification of logics according to properties of the Leibniz operator (13). However, this should be taken with a pinch of salt: there is not a single unifying scheme that uses the Leibniz operator and encompasses all the classes of logics usually considered in the hierarchy. The classification includes classes defined by related properties, mostly concerning their classes of reduced models (the notion of a reduced model being itself defined in terms of the Leibniz congruence), and for whose study the Leibniz operator is still a useful tool. The core of the hierarchy is of course the class of algebraizable logics. Corol- lary 4.13 presents algebraizability as the conjunction of two definability conditions: the definability of the Leibniz congruence in terms of the truth filter, and the definability of the truth filter in terms of the Leibniz congruence. Each of these conditions defines per se a new, larger class of logics. Definition 6.1. A logic is equivalential when there is a set of formulas ∆(x,y) ⊆ Fm that defines the Leibniz congruence in every model of the logic in the sense of 38 Josep Maria Font

Theorem 4.11(ii). If the set is finite, then the logic is called finitely equivalential. A logic is truth-equational when there is a set of equations E(x) ⊆ Eq that defines truth in every model of the logic in the sense of Theorem 4.12(ii).

Theorem 4.11 establishes equivalent conditions for the definition of an equivalential logic; notably the purely syntactic property that the logic satisfies conditions (R∆ )–(Re∆ ) and (MP∆ ). It is also easy to see that conditions (ii) and (iii) of Theorem 4.12 are equivalent in general, without the assumption regarding ∆ present there; so this establishes an equivalent definition of truth-equational logics. Thus, for both properties, the definability requirement can be restricted to the reduced models of the logic; in this case, since the Leibniz congruence becomes the equality relation in these models, it is equality that is definable from the truth filter (for equivalential logics) and dually the truth filter is definable from equality (for truth-equational logics). Observe that Corollary 4.13 can now be rephrased as saying that a logic is (finitely) algebraizable if and only if it is both (finitely) equivalential and truth- equational.33 Equivalential logics were introduced by Prucnal and Wronski´ (1974), and the first in-depth study of them is Czelakowski (1981). One can obtain the following characterization, announced on page 15.

Theorem 6.2. A logic L is equivalential if and only if there is a set of formulas ∆(x,y) ⊆ Fm and a class of algebras K such that L satisfies (MP∆ ) and the structural transformer ρ defined by ∆ satisfies condition (ALG2) with respect to K. In ∗ this situation, one can always take K := Alg L .

The view of the equational definition of truth in models of a logic as a version of Beth’s definability theorem is an idea of Herrmann (1993). Truth-equational logics were formally introduced and first studied by Raftery (2006b); from the definitions, it is easy to see that they enjoy a natural one-to-one correspondence between ∗ ∗ Mod L and Alg L :

Lemma 6.3. Let L be a truth-equational logic, and let τ be the transformer deﬁned by the set of equations E(x) of Deﬁnition 6.1. Then: ∗ ∗ A 1. hA,Fi ∈ Mod L if and only if A ∈ Alg L and F = Solτ . ∗ A ∗ 2. A ∈ Alg L if and only if hA,Solτ i ∈ Mod L . Note that we are moving from more restricted classes of logics to larger, weaker ones. Truth-equational logics form one of the very large classes of the hierarchy; in fact, this class includes all assertional logics, but it does not include all equivalential logics.

33 Finitely equivalential logics appeared earlier than finitely algebraizable ones. Thus, “finitely” applied to algebraizable logics was adopted to indicate the finiteness only of ∆ , in order to obtain the mentioned equivalence. A dual notion of a “finitely truth-equational logic” is not considered in the literature. Abstract Algebraic Logic 39

Assertional logics are defined at the beginning of Section 3 (page 10). Using Proposition 39 of Raftery (2006b), it is easy to show that a logic L is assertional if and only if its reduced models are unital; that is, the truth filter of the matrices ∗ in Mod L is a one-element set. From this it follows that they are truth-equational, with a truth definition of the form x ≈ >, where > is any theorem of L with at most the variable x, and that if L is an assertional logic (i.e., the assertional logic ∗ of some class K), then it is the assertional logic of the class Alg L . The other very large class in the hierarchy is that of protoalgebraic logics, introduced by Blok and Pigozzi (1986). This class can be described in so many ways that it is difficult to choose one to begin with; here is the first, original definition.

Definition 6.4. A logic L is protoalgebraic when for each theory Γ ∈ ThL , indiscernibility modulo Γ implies interderivability modulo Γ ; that is, when for any α,β ∈ Fm, if α ≡ β (ΩΓ ), then Γ, α `L β and Γ, β `L α . All the other classes of logics considered up to now, except assertional and truth- equational logics, are contained in the class of protoalgebraic logics. The importance of this class is due to its many different characterizations from quite different points of view, and to the pleasant algebraic properties of their matrix models. One of the key points is a matrix version of the correspondence theorem of universal algebra, which means that the filters of the logic present, in a certain sense, nice behaviour similar to that of the congruences of algebras. Only two classes of logics in the Leibniz hierarchy remain to be introduced. They were defined in a different framework, but thanks to the discovery of truth- equational logics, they are now easy to describe.

Deﬁnition 6.5. A logic is weakly algebraizable when it is both protoalgebraic and truth-equational; and it is regularly weakly algebraizable when it is weakly algebraizable and assertional, or equivalently, when it is protoalgebraic and assertional.

Weakly algebraizable logics were introduced34 in Font and Jansana (1996), and studied thoroughly in Czelakowski and Jansana (2000), and also in Czelakowski (2001), where regularly weakly algebraizable logics appeared. There are few proper (i.e., non-algebraizable) examples of logics in these classes in the literature. The best known, the logic of ortholattices described in Czelakowski and Jansana (2000), belongs to the latter class, as do all orthologics that are not orthomodular; while the “logic of Andreka´ and Nemeti”´ that appears in Blok and Pigozzi (1989, Appendix B) belongs to the former but not to the latter. Neither of those logics is algebraizable. Figure 2 depicts the organization of the eleven main classes in the hierarchy;35 most of the relations between them follow easily from the deﬁnitions given, and

34 They were first considered by Czelakowski, in unpublished lectures in 1993, under the name “algebraizable in the weak sense”; a term also used in Czelakowski (2001). 35 Other classes might be considered in the hierarchy, though in a looser sense. These include those obtained by restricting all the classes to their finitary members (among them is the class of BP-algebraizable logics), and some of the classes of Definition 5.2, for instance the “Rasiowa” classes. 40 Josep Maria Font

ﬁnitely regularly algebraizable

ﬁnitely regularly algebraizable algebraizable

ﬁnitely regularly weakly algebraizable equivalential algebraizable

weakly equivalential algebraizable assertional

protoalgebraic truth-equational

Fig. 2 The main classes in the Leibniz hierarchy and their relations. The arrow means “included in” or “implies”. Intersections of classes correspond to joins in the graph. there are counterexamples showing that all inclusions are proper and that there are no inclusions other than those shown in the ﬁgure and those implicit in it (by transitivity of inclusion). Moreover, the graphical joins correspond to intersections of classes; for instance, we see that an assertional logic is equivalential if and only if it is algebraizable, and if and only if it is regularly algebraizable,36 and so on. Now it is time to describe the characterizations that give unity to the hierarchy, and best explain both its structure and the relations between the classes. They are displayed in several tables, in a rather informal and compact way.

Order-theoretic characterizations

These are the properties that give the core of the hierarchy its distinctive abstract character, and concern properties of the Leibniz operator Ω A as a function, mostly relating the order structure of its domain FiL A with that of its codomain ConAlg∗L A. These sets are ordered under set inclusion, and one must take into ac- A count that for any algebra A, the function Ω is always onto ConAlg∗L A. Table 2 summarizes the relevant results. The properties in the table with a not-so-obvious meaning are deﬁned as follows: • Ω A commutes with endomorphisms when for any h ∈ End(A) and any F ∈ A −1 −1 A FiL A, Ω h F = h Ω F . The name of the property is shorthand for “commutes with inverse images under endomorphisms”, which is certainly more ac-

36 And also if and only if it is order algebraizable (see page 35). Abstract Algebraic Logic 41

L is . . . if and only if for every A, Ω A is . . . (and: if and only if over Fm, Ω is . . . ) Protoalgebraic monotone Equivalential monotone and commutes with endomorphisms Finitely equivalential continuous Truth-equational completely order-reﬂecting Weakly algebraizable monotone and injective (i.e., an isomorphism) Algebraizable an isomorphism that commutes with endomorphisms Finitely algebraizable a continuous isomorphism Table 2 The order-theoretic characterizations.

curate; besides being more practical, there is a technical reason for using this shorter name (by analogy with the result in Lemma 8.1). • A Ω is continuous when, for any family {Fi : i ∈ I} ⊆ FiL A that is upwards- S A S S A directed and such that i∈I Fi ∈ FiL A, it holds that Ω i∈I Fi = i∈I Ω Fi . Notice that when L is finitary, the condition that the union of the family is a filter is automatically satisfied. Clearly, continuity implies monotonicity. A • Ω is completely order-reflecting when, for any family {Fi : i ∈ I} ∪ {G} ⊆ T A A T FiL A, if i∈I Ω Fi ⊆ Ω G, then i∈I Fi ⊆ G. One important feature of the sets of properties of the Leibniz operator in each row of Table 2 is that each set holds as stated (i.e., for all algebras) if and only if it holds for just the operator as considered on the formula algebra Ω : ThL → ConAlg∗L Fm; in this case, the endomorphisms are the substitutions. Results of this kind (i.e., asserting that some property holds in the formula algebra if and only if it holds, mutatis mutandis, in all algebras) are called transfer theorems in abstract algebraic logic. They are often far from trivial, and they have an important impact in the theory. Other examples of transfer theorems appear in Section 7. Actually, Theorem 5.3 is a specialized version of the result in the last row of ∗ Table 2: observe that by the result in the last row of Table 5, when the class Alg L is a quasivariety, if the Leibniz operator is an isomorphism, then it is automatically continuous. The four classes not present in Table 2 are characterized by adding another property of the Leibniz operator, of a different character. It is easy to see that37 a logic L is assertional if and only if it has theorems and x ≡ y (ΩΓ ) for every Γ ∈ ThL such that x,y ∈ Γ . By adding to this condition those for being protoalgebraic, equivalential or finitely equivalential, we obtain characterizations, all in terms of the Leibniz operator, for the three classes with “regularly” in their name; moreover, once protoalgebracity is assumed, the condition can be simplified by writing just x ≡ y ΩCL {x,y} , where CL {x,y} is the theory of L generated by the set {x,y}.

37 According to Czelakowski (1981), this fact was ﬁrst stated, essentially, by Suszko in unpublished ∼ lectures (here it is expressed in modern terms). Using the Suszko operator ΩL , to be deﬁned in (17) ∼ on page 47, the condition can be more compactly written as x ≡ y ΩL CL {x,y} . 42 Josep Maria Font

Deﬁnability characterizations

Some deﬁnability characterizations have already been encountered in previous definitions or results. The basic ones are summarized in Table 3.

L is . . . if and only if . . . A Protoalgebraic Ω F is definable from F ∈ FiL A by some ∆(x,y,~z) with parameters, for every A A Equivalential Ω F is definable from F ∈ FiL A by some ∆(x,y) without parameters, for every A Truth-equational The truth filter is definable in Mod∗L by some E(x) ⊆ Eq ∗ Assertional The truth filter is definable in Mod L by x ≈ >, where > is an algebraic constant of Alg∗L (38) Table 3 The main definability characterizations.

It should be understood that in each case, the definability condition has a common form in all the relevant algebras or matrices (i.e., this is about uniform definability). By putting several of these conditions together, we obtain characterizations of four more classes; while by requiring the non-parameterized set ∆(x,y) to be finite, we obtain the three remaining classes (those with “finitely” in their name). The only new notion here appears in the first row (the parameterized case): • If ∆(x,y,~z) ⊆ Fm is a set of formulas in two variables x,y and possibly other A variables ~z, called parameters, then Ω F is definable from F ∈ FiL A by ∆(x,y,~z) when, for any a,b ∈ A, a ≡ b (Ω AF) if and only if S∆ A(a,b,~c) : ~c ∈ ~A ⊆ F . When parameters are absent, this property reduces to that of Theo- rem 4.11.

Syntactic characterizations

Theorems 4.11 and 4.12 show that under certain assumptions the definability conditions are equivalent to the logic satisfying certain properties for the sets of formulas or equations involved. The simplest to state are summarized in Table 4; their main interest is that they concern just the consequence relation of the logic. By demanding that the set ∆ be finite, we obtain parallel characterizations for the classes obtained by prepending “finitely” to the three39 last ones. The second and

38 This condition is equivalent to saying that the truth filter in Mod∗L is a singleton; but as a “definability” property this formulation looks weaker. 39 There is no class of “finitely protoalgebraic” logics in the literature. One partial reason may be that for finitary logics, the set ∆(x,y) of Table 4 can always be chosen as finite in the protoalgebraic case, but not necessarily in the other cases. There is also the technical issue mentioned in footnote 40. Abstract Algebraic Logic 43

L is . . . if and only if there is ∆(x,y) ⊆ Fm such that `L satisﬁes . . .

Protoalgebraic (R∆ ) + (MP∆ )

Equivalential (R∆ ) + (MP∆ ) + (Re∆ ) Algebraizable (R∆ ) + (MP∆ ) + (Re∆ ) + (ALG3) for some E(x) ⊆ Eq Regularly algebraizable (R∆ ) + (MP∆ ) + (Re∆ ) + (G∆ ) Table 4 Some of the syntactic characterizations. third row correspond to Theorems 4.11 and 4.1, respectively; I already commented that the conditions (Sym∆ ) and (Trans∆ ) can be dispensed with in these results. More surprising is the fact that (ALG3) can also be dispensed with in the presence of (G∆ ), as shown in the table. The first row of the table deserves comment. It says that a logic L is protoalgebraic if and only if there is a set ∆(x,y) of formulas in at most two variables such that the logic satisfies the “law of identity” or “reflexivity” `L ∆(x,x) and the rule of “modus ponens” x, ∆(x,y) `L y. In particular, any logic L in a language containing a binary connective → such that `L x → x and x, x → y `L y is protoalgebraic; this explains why the class of protoalgebraic logics is so huge! In fact, most of the examples of non-protoalgebraic logics in the literature are implication-less, such as the implication-less fragment of intuitionistic logic (Blok and Pigozzi, 1989), the fragment of classical logic with just conjunction and disjunction (Font and Verdu,´ 1991), Dunn-Belnap’s four-valued logic (Font, 1997), or positive modal logics (Jansana, 2002). Other examples of non-protoalgebraic logics have a very weak implication, such as some weak subintuitionistic logics (Suzuki et al., 1998; Celani and Jansana, 2001), Wojcicki’s´ “weak relevance” logic WR (Font and Rodr´ıguez, 1994) or a large number of logics that preserve degrees of truth with respect to certain classes of residuated lattices (Bou et al., 2009). Notice that there is nothing in conditions (R∆ ) and (MP∆ ) that is specific to implication. In fact, the equivalence fragments of classical logic and of intuitionistic logic are finitely regularly algebraizable, hence protoalgebraic, and satisfy these conditions with ∆(x,y) = {x ↔ y}. There are some relations between the sets ∆ satisfying the conditions in Table 3 and those satisfying the conditions in Table 4. By Theorem 4.11, any set ∆(x,y) satisfying the conditions in the second row of Table 3 itself satisfies the second row of Table 4, and conversely. As for the first row of each (the protoalgebraic case), the relations are more complicated. If ∆(x,y,~z) defines the Leibniz congruence as in the first row of Table 3, then the set ∆ 0(x,y) := ∆(x,y,~x) satisfies the first row of 40 Table 4, i.e., (R∆ ) and (MP∆ ). The converse process requires a more complicated transformation: if ∆(x,y) satisfies the first row of Table 4, then the set ∆p(x, y,~z) := S∆δ(x,~z),δ(y,~z) : δ(x,~z) ∈ Fm satisfies the first row of Table 3. For a related, simpler relation, under stronger assumptions, see Lemma 6.17.

40 Observe that the set ∆p is always infinite, irrespective of whether ∆ is finite or not. Even if a protoalgebraic logic is finitary, the existence of a finite set defining the Leibniz congruence with parameters cannot be guaranteed. Compare this with the fact mentioned in footnote 39. 44 Josep Maria Font

Model-theoretic characterizations

These classify a logic by closure properties of certain classes of matrices or algebras related to it under some model-theoretic operations, and are summarized in Table 5.

L is . . . if and only if . . . ∗ Protoalgebraic Mod L is closed under PSD ∗ Equivalential Mod L is closed under S and P ∗ Finitely equivalential Mod L is closed under S, P and PU, i.e., it is a “quasivariety” and ﬁnitary of matrices

L is . . . if and only if L is truth-equational and . . . ∗ Weakly algebraizable Alg L is closed under PSD ∗ Algebraizable Alg L is closed under S and P ∗ Finitely algebraizable Alg L is closed under S, P and PU, i.e., it is a quasivariety Table 5 Two groups of model-theoretic characterizations. The classes Mod∗L and Alg∗L always contain the trivial matrices (resp., algebras) and are closed under isomorphisms.

This is a good place to observe that matrices are just relational structures hA,Fi, where besides the algebraic reduct A there is only one relation, a unary one, interpreted by the subset F ; thus, they are structures for a very simple first-order language, and it is all the more natural that many tools and results of model theory can be successfully used. This observation was first made by Bloom (1975) and besides its application to the study of matrices for sentential logics, has also prompted a certain amount of work on the model theory of equality-free sentences, which has generalized techniques and benefited from intuitions coming from algebraic logic. A recent contribution to this programme is Nurakunov and Stronkowski (2013); for more details and older references see Font et al. (2003, Section 4.3).

The operators S, P, PU and PSD appearing in Table 5 are those of taking sub- matrices/subalgebras, products, ultraproducts and subdirect products, respectively, of a given class of matrices/algebras. The characterizations in the second group may seem slightly unsatisfactory, as they need an extra assumption with no model- theoretic character (that the logic is truth-equational); but they highlight precisely the fact that for truth-equational logics the characterizations need not involve matrices, but just plain algebras, and this is also an interesting feature. Note the presence of ﬁnitarity in the last row of the upper half of the table, and its absence in the last row of the lower half. The results summarized in the tables in this section are, in fact, the outcome of several important and technically involved theorems that belong to the established core of abstract algebraic logic. It is clear that they cover interesting cases not falling under the algebraizability paradigm. In the case of algebraizability, the order-theoretic and the syntactic characterizations (Tables 2 and 4) are sometimes called intrinsic (see Blok and Pigozzi, 1989, Chapter 4), meaning that they establish Abstract Algebraic Logic 45 algebraizability by properties of the logic alone, without previous knowledge of the class of algebras that constitute the algebraic counterpart of the logic. In this sense, the syntactic characterizations (Table 4) are less intrinsic, as they still need knowledge of the transformers; while the order-theoretic ones (Table 2) are truly intrinsic, and their versions limited to the formula algebra are even more so, as they involve only properties of the Leibniz operator (which is a purely algebraic object) on the theories of the logic, without any additional component.

6.2 The general deﬁnition of the algebraic counterpart of a logic

Research on the Leibniz hierarchy has fully confirmed that the theory of matrices is an adequate tool for the algebraic study of logics in this hierarchy. Recall that this theory defines (page 26) the class ∗ Alg L := A : there is F ∈ FiL A such that the model hA,Fi is reduced as the algebraic counterpart of the logic L . This fits particularly well with the theory of algebraizable logics, where it produces the equivalent algebraic semantics ∗ (Corollary 4.15). However, it is not clear that Alg L is always the class of algebras most naturally associated with an arbitrary logic L ; particularly when it is non- protoalgebraic. This was first noticed by Font et al. (1991), in the study of CPC∧∨ , the fragment of classical logic with only conjunction and disjunction. Those authors ∗ proved that Alg CPC∧∨ is neither the class of distributive lattices, nor that of its bounded members, as one might expect, but a strange subclass with no other logical or algebraic significance (and which is not even a quasivariety). Soon, other examples of a similar situation arose.41 This prompted the introduction of a more general definition of the algebraic counterpart of a logic, which was achieved by considering a more general kind of algebra-based models, already introduced by Wojcicki´ (1969) in essence. A generalized matrix is a pair hA,C i where A is an algebra and C is a closure system42 of subsets of A, the universe of A. Observe that ordinary matrices hA,Fi can be viewed as generalized matrices of the form hA,{F,A}i, and that a logic L can also be viewed as the generalized matrix hFm,ThL i. This suggests that generalized matrices may be a very flexible, convenient tool. In particular, they

41 This includes almost all the non-protoalgebraic logics mentioned on page 43. A strikingly simple example is that of CPC∧ , the fragment of classical logic with only conjunction: while this logic ∗ is naturally associated with the variety of semilattices, it is not difﬁcult to show that Alg CPC∧ contains just the one- and two-element semilattices (Font and Moraschini, 2014, Corollary 5.3). 42 A closure system on a set A is a family C of subsets of A that contains A and is closed under intersections of arbitrary non-empty families. The sets FiL A and ThL are always closure systems. Originally, the C in a generalized matrix was not assumed to be a closure system, but an arbitrary (non-empty) family of subsets; in this form they were rediscovered by Dunn and Hardegree (2001), who called them “atlases”. There is no essential difference between the two alternatives as far as their roleˆ as models of logics is concerned. 46 Josep Maria Font

incorporate a general semantic notion in a natural way: it is best expressed through the closure operator43 C that is associated with the closure system C by putting, for X ⊆ A, C(X) := T{F ∈ C : X ⊆ F}; then hA,C i is a generalized model (g-model for short) of a logic L when for all Γ ∪ {ϕ} ⊆ Fm, Γ `L ϕ ⇒ for all h ∈ Hom(Fm,A), h(ϕ) ∈ C h(Γ ) . (15)

Thus, g-models incorporate a semantics for the consequence relation of a logic. This notion of a model appears as strikingly natural if one wants to privilege the view of logics as consequence relations. Moreover, it is more neutral as to the meaning of the objects in the model, since it does not depend on the designation of some particular subset of the algebra as representing “the truth”. The comparison with matrix semantics is straightforward, because trivially hA,C i is a g-model of L if and only if C ⊆ FiL A. Thus, the ﬁnest g-model of L on a given algebra A is the generalized matrix hA,FiL Ai; g-models of this form (there is exactly one on each algebra) are called basic full g-models. Recall (Lemma 4.4)

that FiL Fm = ThL ; therefore, each logic L , when viewed as the generalized matrix hFm,ThL i, is its own basic full g-model on the formula algebra. The basic full g-models hA,FiL Ai can be viewed as an “algebraic image” of the logic, and a good deal of the research in this branch of abstract algebraic logic is concerned with the study of which properties of the logic (expressible as properties of a generalized matrix) also hold for its basic full g-models; see the discussion on “transfer theorems” and the sample of results in Section 7. The technical tools needed to speak about the use of generalized matrices in abstract algebraic logic start with associating a congruence with each of them in a natural way. The Tarski congruence of a generalized matrix hA,C i is:

A \ A Ω∼ C := Ω G : G ∈ C . (16)

It is easy to see that this is the largest congruence of A that is compatible with all the G ∈ C . A generalized matrix is reduced when its Tarski congruence is the identity relation. The reduction of hA,C i is the generalized matrix hA/Ω∼AC ,C /Ω∼AC i, where C /Ω∼AC := {F/Ω∼AC : F ∈ C }; as is to be expected, it is always reduced (compare with Lemma 4.8). A generalized matrix is a full g-model of a logic L when its reduction is a basic full g-model of L . The corresponding completeness theorems, parallel to Theorems 4.5 and 4.9, are formulated by putting an “ ⇐⇒ ” in (15) and requiring its right-hand side to hold for all generalized matrices in a certain class. It is easy to see that every logic is complete with respect to the following classes: all its g-models; all its reduced

43 A closure operator on a set A is a function C : P(A) → P(A) that satisfies, for all X ,Y ⊆ A, that X ⊆ C(X) = CC(X) ⊆ C(X ∪Y). Several notational shortcuts are popular, such as writing C(a) for C{a}, or C(X ,a) for CX ∪ {a}, and so on; recalling that the argument of C should always be a subset of A helps to avoid misunderstandings. The closure operator associated with the A A closure system FiL A is denoted by FgL ; thus, for any X ⊆ A, FgL (X) is the smallest L -filter of A containing X . The closure operator associated with ThL is denoted by CL . A closure operator C is finitary when for any X ⊆ A, C(X) = S{C(Y) : Y ⊆ X , Y finite}. Abstract Algebraic Logic 47 g-models; all its basic full g-models; all its full g-models; and all its reduced full g-models, which coincide with the reduced basic full g-models (it is easy to see that a reduced full g-model must be basic). Here I am concerned only with the use of g-models to define the algebraic counterpart of a logic in the most general case, and to set up another hierarchy of logics, the so-called Frege hierarchy. The Tarski operator can be used to define another operator on filters of a logic ∼A on each algebra A, the Suszko operator: the function ΩL : FiL A → ConA that assigns to each F ∈ FiL A the congruence

∼A ∼A \ A ΩL F := Ω {G ∈ FiL A : F ⊆ G} = {Ω G : G ∈ FiL A , F ⊆ G}. (17) This operator can also be considered on the formula algebra, i.e., defined on the theories Γ of the logic; in this case, the superscript indicating the algebra is omitted, ∼ and we write simply ΩL Γ . In contrast with the Tarski and the Leibniz operators, which are purely algebraic objects, the Suszko operator is strictly relative to the logic L , as reflected in the ∼A notation. A matrix hA,Fi is Suszko-reduced when ΩL F is the identity relation. The Suszko operator makes sense because in some contexts (particularly when working in parallel with the Leibniz operator) it seems desirable to have operators on L -filters rather than on closure systems of L -filters. It was thoroughly studied in general by Czelakowski (2003)44 and it was one of the main tools in the study of truth-equational logics by Raftery (2006b). The more general notion of the algebraic counterpart of a logic can be defined with either the Tarski operator or the Suszko operator. It is the class of L -algebras, defined (among others) in any of the following equivalent ways: AlgL := A : there is C ⊆ FiL A such that the g-model hA,C i is reduced = A : the basic full g-model hA,FiL Ai is reduced = A : there is F ∈ FiL A such that the model hA,Fi is Suszko-reduced . There are at least three groups of reasons why this more general notion is more ∗ relevant for arbitrary logics than that of Alg L . I The first is that in the cases where the old definition works, the two coincide: Theorem 6.6. 1. A logic L is protoalgebraic if and only if the Leibniz and the Suszko operators ∼A A coincide on its filters, i.e., ΩL F = Ω F for all F ∈ FiL A and all A. ∗ 2. If L is protoalgebraic, then Alg L = AlgL . 3. If L is algebraizable, then AlgL is the equivalent algebraic semantics of L . 44 That author attributes the definition and first characterization of this operator to Suszko, in unpublished lectures. 48 Josep Maria Font

In particular, for implicative logics, the class AlgL coincides with the class ∗ Alg L as originally deﬁned by Rasiowa; in fact, the name “L -algebras” was coined by her, and we now see that it can be used in general without risking confusion. Notice that the converses of the implications in points 2 and 3 of Theorem 6.6 do not hold: there are many non-protoalgebraic (hence non-algebraizable) logics L for ∗ which Alg L and AlgL coincide; among them are all those where the former is a quasivariety, in particular a variety.45 This fact follows from the most basic relations between the two classes, which are summarized in the following result. ∗ Theorem 6.7. For every logic L , AlgL = IPSD(Alg L ). Moreover: 1. The two classes generate the same quasivariety and the same variety. This variety is called46 the intrinsic variety of L , and is denoted by VL . ∗ 2. Alg L ⊆ AlgL ⊆ VL . ∗ 3. Alg L is a variety if and only if it coincides with VL . In such a case, it also coincides with AlgL . ∼ 4. The variety VL is the variety generated by the quotient algebra Fm/Ω L , where Ω∼L := T ΩΓ : Γ ∈ ThL is the Tarski congruence of the logic viewed as a generalized matrix.

The significance of VL as an algebraic semantics for L is in general weak, because there is no general theory47 that asserts that the algebraic counterpart of a logic should be a variety. However, if one insists on having a variety associated with a logic, then VL is the natural choice. This is reinforced by the following facts (the first is a consequence of Theorem 4.7, and the second is proved using the first).

Lemma 6.8. Let L be a logic. Then:

1. VL α ≈ β if and only if for all δ(x,~z) ∈ Fm, δ(α,~z) a`L δ(β ,~z). 2. If L is complete with respect to a class M of reduced matrices (or of reduced generalized matrices), then the variety generated by the class of algebraic reducts of the (generalized) matrices in M is VL .

I The second group of reasons is of an empirical character: in all particular cases examined in the literature where the two classes differ, it has been found that the class of algebras naturally associated with L according to other criteria or moti- ∗ vated in other ways, even if only intuitively, is AlgL and not Alg L . For instance, AlgCPC∧∨ is the variety of distributive lattices, AlgCPC∧ is the variety of semilattices, and so on. Incidentally, concerning Theorem 6.7.2, neither of the inclusions is an equality in general, and there are even logics for which the three classes in it are different.

45 To the best of my knowledge, the earliest example of this kind in the literature is Wojcicki’s´ “weak relevance” logic WR (Font and Rodr´ıguez, 1994). 46 This name was ﬁrst used by Pynko (1999). 47 Recall that the equivalent algebraic semantics of a BP-algebraizable logic is in general a quasivariety. The best-known example of a BP-algebraizable logic whose equivalent algebraic semantics is not a variety is BCK logic (Wronski,´ 1983). Abstract Algebraic Logic 49

I The third group of reasons is that the approach based on generalized matrices, and in particular the study, begun by Font and Jansana (1996), of the notion of a full g-model and of the structure of the set of full g-models of a logic on a given algebra has generated a deep and rich theory. Moreover, that theory establishes connections between areas of abstract algebraic logic that are in principle unrelated. The ﬁrst important result in this area is yet another “isomorphism theorem”: Theorem 6.9. For any logic L and any algebra A, the Tarski operator, that is, the function C 7→ Ω∼AC , establishes a dual isomorphism between the complete lattice of all full g-models of L on A and that of all congruences of A relative to the class AlgL (both ordered under set inclusion). One of the remarkable things about this result is its generality (it holds for all logics whatsoever), which contrasts with the algebraizability assumptions needed for Theorem 5.3 and for the isomorphisms in Table 2. There is not enough space here to enter into this general theory; some of the results in Sections 6.3 and 7 actually belong to it.

6.3 The Frege hierarchy

This hierarchy is organized around several replacement properties that a logic and its basic full g-models may have. These properties can be formulated in general for an arbitrary generalized matrix; however, here it is better to go directly from the first general definitions to the relevant particular cases (but see also Definition 7.3). The Frege relation ΛAC of a generalized matrix hA,C i is defined, for any a,b ∈ A, as

A def a ≡ b Λ C ⇐⇒ C(a) = C(b).

In general, this relation is an equivalence relation, but it need not be a congruence of A; the Tarski congruence Ω∼AC is the largest congruence of A below ΛAC (this could be an alternative deﬁnition of the Tarski congruence). Moreover, this idea A produces the Frege operator: the function ΛC that assigns, to every F ∈ C , the Frege relation of the closure system {G ∈ C : F ⊆ G}. That is,

A A ΛC F := Λ {G ∈ C : F ⊆ G}.

Each logic L deﬁnes natural specializations of these constructions to the formula algebra and to arbitrary algebras. If we consider the generalized matrix given by all the theories ThL , then we obtain the relations

ΛL := ΛFmThL , and Fm 0 0 ΛL Γ := Λ {Γ ∈ ThL : Γ ⊆ Γ } for each Γ ∈ ThL .

Observe that ΛL is the interderivability relation of L , also denoted by a`L . Sim- ilarly, given any algebra A and considering the closure system FiL A, we obtain 50 Josep Maria Font

A A Λ L := Λ FiL A, and A A ΛL F := Λ {G ∈ FiL A : F ⊆ G} for each F ∈ FiL A.

Thus, ΛAL identiﬁes the elements of the algebra that generate the same L -ﬁlter. In general, all these relations are equivalence relations, but need not be congru- ∼ ∼ ences; Ω L is the largest congruence below ΛL , ΩL Γ is the largest congruence ∼A A ∼A below ΛL Γ , Ω FiL A is the largest congruence below Λ L , and ΩL F is the A largest congruence below ΛL F . Now, the four classes in the hierarchy can be introduced in the following compact way.

Deﬁnition 6.10. Let L be a logic. 1. L is selfextensional when ΛL is a congruence; that is, when ΛL = Ω∼L . 2. L is Fregean when, for each Γ ∈ ThL , Λ Γ is a congruence; that is, when ∼ L ΛL Γ = ΩL Γ . 3. L is fully selfextensional when, for each algebra A, ΛAL is a congruence; that A ∼A is, when Λ L = Ω FiL A. A 4. L is fully Fregean when for each algebra A and each F ∈ FiL A, ΛL F is a A ∼A congruence; that is, when ΛL F = ΩL F . It is easy to use the characterization of the Leibniz congruence in Theorem 4.7 and the expressions (16) and (17) to obtain what could be an alternative, more “logical” deﬁnition.

Lemma 6.11. A logic L is selfextensional if and only if for all α,β ∈ Fm,

if α a`L β then δ(α,~z) a`L δ(β ,~z) for all δ(x,~z) ∈ Fm.

A logic L is Fregean if and only if for all Γ ∈ ThL and all α,β ∈ Fm,

if Γ, α a`L Γ, β then Γ, δ(α,~z) a`L Γ, δ(β ,~z) for all δ(x,~z) ∈ Fm.

These properties may be viewed as formal counterparts of the idea of substitu- tivity of equivalents. As discussed more extensively by Czelakowski and Pigozzi (2004), these properties can be considered, to a certain extent, as abstract counterparts of Frege’s principle of compositionality of denotation (i.e., of truth-value, according to Church) when the notion of logical equivalence is understood as identity of truth-value. The abstract algebraic logic approach to the Fregean principles was pioneered by Pigozzi (1991), building on the seminal ideas of Suszko (1975). Under this view, selfextensional logics would also deserve to be called “Fregean”; but the term “selfextensional” had already been coined by Wojcicki´ (1979). More- over, starting with Font (1993), the term “Fregean” has been ascribed to the more restricted class of logics having the property in its stronger form;48 the existence of

48 In a few early papers the term “Fregean” means the same as “protoalgebraic and Fregean” in the present terminology. Abstract Algebraic Logic 51 strong relations with classes of algebras also called “Fregean” in the literature (see Theorem 6.18) has consolidated this choice. The other two kinds of logics are deﬁned by requiring that these replacement properties transfer to all basic full g-models.49 Fully selfextensional logics have also been called congruential in a few recent papers; but be aware that the latter term has also been used in the past, either for Fregean logics, or for ﬁnitely equivalential logics. As a matter of fact, the four classes in this hierarchy deserve to be considered as “congruential” or “Fregean” in different degrees; these considerations justify naming the hierarchy after Frege. The obvious relations between the four classes of logics are those displayed in Figure 3.

fully Fregean

fully Fregean selfextensional

selfextensional

Fig. 3 The poset of the classes in the Frege hierarchy. The arrow means “included in” or “implies”. In this case, the graphical join is not class intersection (but see Corollary 6.20).

The following examples show that the four classes are distinct and no relations other than those shown hold in general:

• All two-valued logics are Fregean. Classical and intuitionistic logics, as well as many of their fragments (for instance, all those with either implication or equivalence, but also the one with only conjunction and disjunction), are fully Fregean. • Logics that are fully selfextensional (hence selfextensional) but not Fregean (hence, not fully Fregean either): Among the protoalgebraic logics, the local consequences of normal modal logics; among the non-protoalgebraic ones, Dunn- Belnap’s four-valued logic (Font, 1997); Wojcicki’s´ “weak relevance” logic WR (Font and Rodr´ıguez, 1994); the positive modal logics (Jansana, 2002); and the local consequence of the weakest subintuitionistic logic (Celani and Jansana, 2001). • A logic that is Fregean (hence selfextensional) but not fully selfextensional (hence not fully Fregean either) was constructed by Babyonyshev (2003). • Two ad hoc examples of logics that are selfextensional but neither Fregean nor fully selfextensional were constructed by Albuquerque et al. (2016a). Interest- ingly, one of these logics is implicative.

49 Equivalently, to all full g-models; hence the “fully” in the names. 52 Josep Maria Font

• Logics that are not selfextensional, and hence are totally outside the Frege hierarchy: There are lots of them, even among algebraizable logics, for instance the global consequences of normal modal logics, and the Łukasiewicz-style many-valued logics. At the other end of the spectrum, the paraconsistent weak Kleene logic (Bonzio et al., 2016) and Priest’s logic of paradox (Albuquerque et al., 2016b) are non-protoalgebraic (but truth-equational) examples of non- selfextensional logics. • Logics that are neither selfextensional nor protoalgebraic and are not truth- equational either (hence, logics outside the two hierarchies) are harder to ﬁnd. One example is the “logic of distributive bilattices” studied by Bou and Riviec- cio (2011). Thus, the two hierarchies are, in some sense, orthogonal, as these examples show: there are logics located in a very high position in one of them and totally outside the other. Some relations have been found, however, as can be seen below. The replacement property of selfextensional logics, though weak, is still useful when one is interested in the intrinsic variety of the logic; for in this case, this class does have a clear logical meaning:

Lemma 6.12. A logic L is selfextensional if and only if there is a class K of algebras such that for all α,β ∈ Fm, α a`L β if and only if K α ≈ β . In this case, VL is the largest class K, and the only variety with this property; in particular, VL is the variety generated by any such class K. This rather weak replacement property becomes surprisingly strong when it is coupled with other properties (see Deﬁnition 7.3 for their exact meaning):

Theorem 6.13 (Font and Jansana, 1996). Let L be a ﬁnitary logic that has either a conjunction, or a uniterm deduction-detachment theorem (DDT). If L is selfextensional, then it is fully selfextensional. Moreover, in this situation, the class AlgL is a variety, namely AlgL = VL .

This result is an interesting step in the analysis of THE VARIETY PROBLEM, towards explaining why the algebraic counterpart of so many “real” logics is indeed a variety. The problem50 is perhaps more striking in the context of BP-algebraizable logics, where it was originally formulated by Blok and Pigozzi, because while the general theory guarantees that the equivalent algebraic semantics of those logics is a quasivariety, there are few examples of “real” BP-algebraizable logics whose equivalent algebraic semantics is not a variety.51 Fully selfextensional logics have nicer properties. To start with, the following reformulation of their deﬁnition, in terms of the class AlgL , reveals a particularly interesting view.

50 Of course, this is not exactly a technical problem that admits a deﬁnite answer, but rather one of methodology, of understanding the algebraic behaviour of logics and the limits of existing techniques and results. Often these problems are more interesting than other, purely technical ones. 51 The algebraizable logics whose equivalent algebraic semantics is a variety are sometimes called strongly algebraizable in the literature. Abstract Algebraic Logic 53

Lemma 6.14. For any logic L , the following conditions are equivalent. (i) L is fully selfextensional. (ii) For each A ∈ AlgL , the relation ΛAL is the identity relation. A A (iii) For each A ∈ AlgL , the relation 6L , deﬁned as a 6L b if and only if b ∈ A FgL {a}, for any a,b ∈ A, is an order relation. This “logically deﬁned” ordering relation in the algebras of AlgL has been ex- ploited to develop an abstract version of duality theory (Esteban, 2013; Gehrke et al., 2010). Examples of the nice behaviour of these logics are the following results.

Theorem 6.15 (Font and Jansana, 1996). Let L be a finitary and fully selfextensional logic. Then: 1. AlgL is a quasivariety. 2. L is weakly algebraizable if and only if it is BP-algebraizable. Some general consequences of the stronger replacement property of Fregean logics are easy to obtain. For instance, since always x ≡ y (Λ C (x,y)), Defi- L∼ L nition 6.10.2 implies that these logics satisfy the condition x ≡ y (ΩL CL (x,y)) that appears in footnote 37. Therefore, truth filters of reduced models of these logics on non-trivial algebras are singletons, which offers a practical way to disprove that a given logic is Fregean. Moreover, it turns out that in an important part of the Frege hierarchy, the difference between the assertional logics and the truth-equational ones disappears.

Theorem 6.16 (Albuquerque et al., 2016a). If L is a logic that is Fregean or fully selfextensional, then L is assertional if and only if it is truth-equational. If L is Fregean, then these conditions hold if and only if L has theorems. In the protoalgebraic scenario, to be Fregean becomes a surprisingly strong property. To begin with:

Lemma 6.17. Let L be a protoalgebraic logic, and let ∆(x,y) satisfy properties (R∆ ) and (MP∆ )(as in Table 4 on page 43). If L is Fregean, then it is equivalential, with the symmetrized set ∆s(x,y) := ∆(x,y) ∪ ∆(y,x) as a set of equivalence formulas.

But the strength of Fregeanity for protoalgebraic logics goes farther: it places them signiﬁcantly higher up in the Leibniz hierarchy, and at the absolute top of the Frege hierarchy.

Theorem 6.18. Let L be a protoalgebraic and Fregean logic. Then: 1. If L has theorems,52 then L is regularly algebraizable (Font and Jansana, 1996).

52 There is a protoalgebraic logic without theorems, but only one (in each language) and it is rather trivial: it is the so-called almost inconsistent logic, i.e., the logic without theorems such that α `L β for all α,β ∈ Fm. This logic is protoalgebraic and Fregean in a trivial way, but is not algebraizable. 54 Josep Maria Font

2. If L is ﬁnitary, then it is fully Fregean (Czelakowski and Pigozzi, 2004). 3. If L is ﬁnitary and has theorems, then AlgL a Fregean quasivariety (Czela- kowski and Pigozzi, 2004).

Different points in this result are proved using quite different techniques. In- terestingly, the proof of point 2 is obtained by expressing the strong replacement property as an infinite set of Gentzen-style rules and working with generalized matrices as models53 of rules of this kind. Point 3 involves the notion of a Fregean quasivariety, a quasivariety K that is pointed (there is a term, represented as >, 0 that is constant in K), relatively point-regular (for all A ∈ K and all θ ,θ ∈ ConKA, if >A/θ = >A/θ 0 then θ = θ 0 ) and congruence-orderable (for all A ∈ K and all A A A A A a,b ∈ A, if ΘK(a,> ) = ΘK(b,> ), then a = b); here ΘK(c,d) denotes the smallest congruence of ConKA that identifies c and d. One can also prove the following converse: if K is a Fregean quasivariety, then K = AlgL for a finitary, protoalgebraic and Fregean logic L with theorems; namely, the assertional logic of K. So the study of this class of logics amounts to the study of Fregean quasivarieties. Fregean varieties have been extensively studied in universal algebra with logical motiva- tions (Idziak, Słomczynska,´ and Wronski,´ 2009); they were first incorporated into the abstract algebraic logic landscape by Pigozzi (1991). To close the section, let me mention a recent result on the location of truth- equational logics in the Frege hierarchy, and three corollaries that follow by com- bining this result with some previous ones; the second contains alternative characterizations of the logics in Theorem 6.18.3.

Theorem 6.19 (Albuquerque et al., 2016a). A truth-equational logic is fully selfextensional if and only if it is fully Fregean.

Corollary 6.20. If L is a logic with theorems, then L is fully Fregean if and only if it is both Fregean and fully selfextensional.

Corollary 6.21. If L is a ﬁnitary logic, then the following conditions are equivalent. (i) L is protoalgebraic and Fregean and has theorems. (ii) L is algebraizable and fully selfextensional. (iii) L is weakly algebraizable and fully selfextensional. (iv) L is regularly algebraizable and fully Fregean.

Corollary 6.22. If L is a ﬁnitary and weakly algebraizable logic, then L is Fregean if and only if it is fully selfextensional, and if and only if it is fully Fregean.

Thus, for logics with theorems, the graphical join in Figure 3 on page 51 is indeed a class intersection. Moreover, in signiﬁcant parts of the Leibniz hierarchy, the

53 This dual character of generalized matrices, as g-models of sentential logics and as models of Gentzen-style rules or systems, is one of the features that give generalized matrices their special interest. Abstract Algebraic Logic 55

Frege hierarchy becomes simplified: inside truth-equational logics (Theorem 6.19), it consists of three classes (the selfextensional, the Fregean, and the fully Fregean); and inside finitary and weakly algebraizable logics (Corollary 6.22), it is dramat- ically reduced to just two classes (the selfextensional and the fully Fregean). Al- though some of this was known earlier, the fact that in each case the classes are really different has been shown only recently (Albuquerque et al., 2016a). Some OPENPROBLEMS concerning the relations between the hierarchies that come naturally to the mind are: • Are fully selfextensional and Fregean logics always fully Fregean ? The answer is affirmative for logics with theorems (Corollary 6.20), but neither a proof nor a counterexample in the theorem-less case is known. • Are protoalgebraic and Fregean logics always fully Fregean ? The answer is affirmative for finitary logics (Corollary 6.21), but neither a proof nor a counterexample in the non-finitary case is known. Research around these and similar questions is expected to shed more light on the structure of the hierarchies and the relative strength of each of the classes.

7 Exploiting algebraizability: bridge theorems and transfer theorems

In the title of this section, “algebraizability” is used in an extended, informal way, to refer to any framework within which a general procedure or criterion has been adopted in order to associate, with each logic L (perhaps only for logics of a certain kind), a class of algebras K as the algebraic counterpart of that logic, or, more generally, a class of algebra-based structures, such as matrices or generalized matrices. You are in such a situation when you restrict your attention to algebraizable logics; for each of them, you get its equivalent algebraic semantics. You can also consider matrix semantics, which produces, for an arbitrary logic L , the class of ∗ ∗ matrices Mod L and the classes of algebras Alg L and AlgL ; or generalized matrix semantics, which deﬁnitely points to AlgL but also to the classes of full g-models, etc. To simplify this introductory discussion, we consider only classes of algebras. In a given framework, a bridge theorem is a mathematical result stating that for a certain property P concerning a logic and a certain property P0 concerning a class of algebras, if L is a logic (in the class, if applicable) and K is the corresponding algebraic counterpart, then:

L satisﬁes P if and only if K satisﬁes P0 . (18)

The term “bridge theorem” was coined by Andreka,´ Nemeti,´ and Sain long ago (see 2001, pages 135–136 and 186–188); according to them, results of this kind establish a bridge between two different lands, Logic and Algebra, and allow us to transform 56 Josep Maria Font

problems about a logic into problems about a class of algebras, so that we can use the powerful tools of algebra, a much more intensively studied discipline, to solve the problem, and then go back (crossing the bridge again) and obtain a solution to the original logical problem. In particular, they emphasize that this methodology allows us to cope with an ever-increasing forest of new logics using a more uniform and better known toolbox, that of Algebra. In the first, paradigmatic bridge theorems, P is a typically logical property (such as interpolation) and P0 is typically algebraic, i.e., it is a global property of a class of algebras (such as amalgamation). In other bridge theorems, the property P0 has the form “every algebra in K satisfies property Q”, where Q is a property concerning a single algebra (such as having a distributive congruence lattice); in yet other cases, it is the property Q that refers to the class K, while the restriction “in K” in the theorem can be deleted, and a stronger result is obtained. A particular class of bridge theorems are those where both P and P0 (or P and Q) are essentially the same property, suitably interpreted on each side (such as “to be finitely axiomatizable”). These results have been called transfer theorems, as they can be phrased as “the property P transfers from L to K” (respectively, “to all algebras in K”, or even “to all algebras”, in the stronger cases). An important subclass of transfer theorems, where Q is just “P interpreted in an algebra”, are those where P is a property of a closure system (or of a closure operator), so that it applies to a logic L when we view it as a generalized matrix, through the closure system ThL of its theories or its associated closure operator 54 CL ; and it applies to an algebra A through the closure system FiL A of all the A L -filters of A or its associated closure operator FgL . Thus, transfer theorems of this kind state that

ThL (or CL ) satisfies P if and only if (19) A for each A ∈ K, FiL A (or FgL ) satisfies P, and, as already said, in many cases the limitation “∈ K” can be removed, so that the property on the algebraic side is asserted to hold for all algebras, unconditionally. Bridge and transfer theorems are not limited to frameworks where a class of algebras is associated with a logic. Results with a larger scope refer to properties of classes of matrices (the class of all models of the logic or, more often, the class ∗ Mod L of its reduced models) or classes of g-matrices; for instance, observe that a result like (19) in its strong version (i.e., without the “∈ K”) would be a transfer theorem to all basic full g-models of the logic. One is even tempted to say that bridge theorems and transfer theorems are the ultimate justification of abstract algebraic logic. There are bridge theorems and

54 The operator CL is defined as ϕ ∈ CL (Γ ) if and only if Γ `L ϕ for any Γ ∪ {ϕ} ⊆ Fm. Thus, essentially, the closure operator associated with a logic is just another way of expressing the consequence relation. The properties formulated in its terms, such as those in Definition 7.3, are more naturally stated in terms of the relation; but the general definition uses a closure operator to facilitate application to both the logic and the algebras. Abstract Algebraic Logic 57 transfer theorems of a wide range of degrees of difficulty, and some of them need extensive development of the general theory, or complicated algebraic technicalities (or both). Only a few are reproduced here, as each of them would need several definitions, both on the logical and on the algebraic side, and space is limited. We find results of all the different kinds described above. A large group of bridge theorems are unconditional, that is, they hold for all logics and all algebras. The following result is considered one of the oldest folk- lore55 results of its kind, and contains both a bridge theorem and a transfer theorem, concerning the property of being finitary.

Theorem 7.1. Let L be a logic. The following conditions are equivalent: (i) L is ﬁnitary. (ii) The class ModL is closed under ultraproducts. A (iii) For every algebra A, the closure operator FgL is ﬁnitary. The characterizations in Table 2 (page 41) and in Table 5 (page 44) can be viewed as bridge theorems, because they establish that certain properties of an arbitrary logic (namely, the property of belonging to a certain class of the Leibniz hierarchy) is equivalent to an algebraic or lattice-theoretic property of a class of matrices or of algebras (the property may or may not involve the Leibniz operator). Moreover, as already said there, some of these characterizations still hold when the properties are required to hold just in the formula algebra, so that in fact they also involve transfer theorems.

Theorem 7.2. For arbitrary logics, the following properties, if stated for the formula algebra, transfer to all algebras. • The order-theoretic properties of the Leibniz operator in Table 2 (page 41). • The deﬁnability properties56 of the Leibniz operator in Table 3 (page 42).

Note that the transfer only holds for the properties when grouped as in the tables; for instance, while injectivity does transfer when coupled with monotonicity, injectivity alone is known to transfer when the language is countable, but not in general (Moraschini, 2016). Other (almost) universal transfer properties concern properties linking logical connectives with the closure operator. Properties of this kind have been called Tarski-style conditions by Wojcicki´ (1988); for simplicity I mention just a few of them, and, for some, not the most general conceivable version.

55 This qualification certainly applies to the (easy) equivalence between (i) and (iii); the equivalence between (i) and (ii) seems to have first been published by Zygmunt (1974). 56 It does not make sense to say, literally, that the definability of the truth filter in reduced models of the logic (which appears in the third and fourth rows of Table 3) holds in the formula algebra, because in general there are no reduced models on the formula algebra itself. But it can be refor- mulated so that there is a transfer result for it: by Theorem 25 of Raftery (2006a), it holds in all models that are reductions of arbitrary models of the logic (that is, in all its reduced models) if and only if it holds in the models that are reductions of models on the formula algebra. 58 Josep Maria Font

Definition 7.3. Let C be a closure operator on the universe A of an algebra A, with associated closure system C . • A binary term x ∧ y is a conjunction for C when for all a,b ∈ A,C(a ∧ b) = C(a,b). • C satisfies a deduction-detachment theorem (DDT)57 when there is a set I(x,y) of binary terms such that for all X ∪ {a,b} ⊆ A,IA(a,b) ⊆ C(X) if and only if b ∈ C(X ,a). When the set I(x,y) consists of a single formula, the DDT is called uniterm. • A binary term x ∨ y is a disjunction58 for C when for all X ∪ {a,b} ⊆ A it holds that C(X ,a ∨ b) = C(X ,a) ∩C(X ,b). • C satisfies an inconsistency lemma when there is a sequence hΨn : n > 1i of finite sets of, respectively, n-ary terms, such that for all X ∪ {a1,...,an} ⊆ A, the set 59 A X ∪ {a1,...,an} is C-inconsistent if and only if Ψn (a1,...,an) ⊆ C(X). • C satisfies the congruence property when its Frege relation ΛAC is a congruence of A; i.e., when ΛAC = Ω∼AC . • C satisfies the strong congruence property when for all F ∈ C , the Frege relation A A ∼A ΛC F is a congruence of A; i.e., when ΛC F = ΩL F for all F ∈ C . It is trivial to see that the property of having a conjunction transfers to all algebras for arbitrary logics. Among more interesting results we have the following.

Theorem 7.4. 1. The property of satisfying a DDT transfers to all algebras, for all logics. (Czela- kowski, 1985; Raftery, 2011a; see footnote 61) 2. The property of having a disjunction transfers to all algebras, for ﬁnitary logics. (Czelakowski, 1983)

The classical proofs, for the ﬁnitary case, are based on a common underlying technique, arising from Hilbert-Bernays’ “Præmissentheorem” (1939), and which can be generalized60 as follows.

57 Other, weaker versions of a DDT have been considered in the literature: The parameterized one, which allows the terms of the set I to have parameters; the local one, which deals with a (possibly inﬁnite) family of sets of terms in such a way that only one set is used in each particular application of the theorem; and obviously, the one that is both local and parameterized, which turns out to characterize protoalgebraic logics. A fortiori, this implies that any logic satisfying a DDT is protoalgebraic. The newest kinds of DDT are the graded ones (Font et al., 2001) and the contextual ones (Raftery, 2011a); they have either local or parameterized variants as well. 58 An obvious generalization of this property is obtained when the single binary term is replaced by a set of binary terms (as in the formulation given for the DDT); even more general is the parameterized version, where a ﬁnite number of parameters are allowed in the set of terms. In both cases the property is basically the same. 59 A set X is C-inconsistent when C(X) = A. The inconsistency lemma is a generalization of the intuitionistic form of reductio ad absurdum, where Ψn(x1,...,xn) := {¬(x1 ∧ ··· ∧ xn)}. 60 This generalization corresponds, essentially, to the “operation on rules” considered, at the proof- theoretic level, by Pogorzelski (1981); see also Pogorzelski and Wojtylak (2008). Abstract Algebraic Logic 59

Lemma 7.5. Consider the following property of a closure operator C on some algebra A:

There is a ﬁnite Γ (x,~y) ⊆ Fm such that for all ~a ∈ ~A and all n ∈ ω , if an ∈ A A A C(a0,...,an−1) then for all~b ∈ ~A, Γ (an,~b) ∈ C Γ (a0,~b) ∪ ··· ∪Γ (an−1,~b) . For ﬁnitary logics, this property transfers to all algebras.

The restriction to ﬁnitary logics is very common in this area;61 other properties have more restrictions, either on the logics they apply to, or on the algebras the property is transferred to. For instance:

Theorem 7.6 (Raftery, 2013b). The property of satisfying an inconsistency lemma transfers to all algebras, for ﬁnitary and protoalgebraic logics.

Meanwhile, the transfer of the congruence properties is not universal, not even for finitary logics. Observe that a logic L is selfextensional if and only if it has the congruence property; and it is Fregean if and only if it has the strong congruence property. Some results in the previous section can now be rephrased as follows. • The congruence property transfers to all algebras for finitary logics that have either a conjunction or a uniterm DDT (Theorem 6.13). • The strong congruence property transfers to all algebras for finitary and protoalgebraic logics (Theorem 6.18.2), and for fully selfextensional logics with theorems (Corollary 6.20). The two congruence properties have been known not to transfer in general since Baby- onyshev (2003). Recently (Albuquerque et al., 2016a) the congruence property has been shown not to transfer in any class of the Leibniz hierarchy, as it fails for a finitely regularly algebraizable logic. The two OPENPROBLEMS mentioned at the end of Section 6.3 can also be rephrased as transfer problems: • Does the strong congruence property transfer for theorem-less fully selfextensional logics ? • Does the strong congruence property transfer for non-finitary protoalgebraic logics ? The following bridge theorems concern, for an arbitrary algebra A, the family of the finitely generated L -filters of A:

ω A FiL A := F ∈ FiL A : F = FgL (X), X ⊆ A, X ﬁnite . This family, ordered under inclusion, is a join-semilattice (it is in fact a join-sub- ω semilattice of the ﬁlter lattice FiL A); this applies in particular to the family Th L

61 In some cases where by tradition only finitary logics were considered, the properties may turn out to hold in general. One example is the DDT (Theorem 7.4): Czelakowski’s original proof needs finitarity, but Theorem 3.5 of Raftery (2011a) in particular shows the transfer for arbitrary logics. In other cases finitarity seems to be unavoidable; the removal of finitarity from some transfer or bridge theorems where it appears is an interesting OPENPROBLEM. 60 Josep Maria Font of the finitely generated theories of L . When L is finitary, by Theorem 7.1 the lat- ω tice FiL A is algebraic, and the join-semilattice FiL A has more algebraic content, because in this case “finitely generated” is the same as “compact”.

Theorem 7.7 (Czelakowski, 1985). Let L be a ﬁnitary and protoalgebraic logic. The following conditions are equivalent. (i) L satisﬁes a DDT (with respect to some set I(x,y) of binary terms). ω 62 (ii) For all A, the join-semilattice FiL A is dually residuated. (iii) The join-semilattice ThωL is dually residuated.

The equivalence between (i) and (ii) is a bridge theorem; while the implication from (iii) to (ii) can be viewed as a transfer theorem. The bridge theorem has, as is to be expected, more algebraic consequences, which in turn have logical applications (crossing back over the bridge). For instance, since any algebraic lattice is isomorphic to the lattice of ideals of the join-semilattice of its compact elements, and taking into account that satisfying a DDT implies protoalgebraicity (see footnote 57) we obtain:

Corollary 7.8. If L is a ﬁnitary logic satisfying a DDT, then for every algebra A the lattice FiL A is distributive. Filter distributive logics (logics satisfying the conclusion of the corollary) have been extensively studied63 in abstract algebraic logic. They enjoy some really interesting features, resulting from the generalization of some well-known universal algebraic properties to their matrix models. It turns out that protoalgebraicity greatly enhances the possibilities for such generalizations. Just as a sample of what can be achieved in this line of research, here is one:

Theorem 7.9 (Pigozzi, 1988; Czelakowski and Dziobiak, 1992). Let L be a finitary logic, in a finite language, that is weakly complete64 with respect to a finite matrix. If L is protoalgebraic and filter distributive, then L is finitely axiomatizable, and it is so with a finite number of axioms and a single proper inference rule.

In parallel to Theorem 7.7, and with the same twofold character of a bridge and a transfer theorem, we also have:

Theorem 7.10 (Raftery, 2013b). Let L be a ﬁnitary and protoalgebraic logic. The following conditions are equivalent: (i) L satisﬁes an inconsistency lemma. ω (ii) For all A, the join-semilattice FiL A is dually pseudo-complemented.

62 Dually residuated join-semilattices are sometimes called Brouwerian semilattices. 63 Not only in the context of protoalgebraic logics: any ﬁnitary logic having a disjunction is ﬁlter distributive (Czelakowski, 1984). 64 The notion of weak completeness, as opposed to ordinary or strong completeness, refers to the theorems of the logic rather than to the consequence relation. Abstract Algebraic Logic 61

(iii) The join-semilattice ThωL is dually pseudo-complemented.

Up to now, each transfer theorem concerns a single, particular property. The following result brings together two of the few existing general transfer theorems; that is, theorems regarding a potentially large group of properties, characterized in some deﬁnite way:

Theorem 7.11. Let L be a ﬁnitary and protoalgebraic logic. The following groups of properties transfer from L to all its full g-models on arbitrary algebras:65 1. All properties of the corresponding lattices of theories or of closed sets expressible as a universal sentence of the (ﬁrst-order) language of lattices. 2. All properties that can be expressed as a set of accumulative66 Gentzen-style rules.

Thus, when limited to basic full g-models, point 1 says that for a finitary and protoalgebraic L , the lattice ThL satisfies a universal (first-order) property if and only if for every algebra A, the lattice FiL A satisfies it. This includes some properties whose transfer had previously been shown by particular arguments, such as distributivity (Corollary 7.8). In some sense, the “best” bridge and transfer theorems are obtained for algebraizable logics. In this case, the links between the logic and its equivalent algebraic semantics are expressed syntactically, by means of two transformers, so that it is to be expected that “metalogical” properties that can be expressed solely in terms of `L can be easily “translated” into parallel properties of K . For instance, the following should be immediately obvious:

Lemma 7.12. Let L be an algebraizable logic with finitary transformers, and let K be its equivalent algebraic semantics. Then: 1. L is finitary if and only if K is a quasivariety. If L is BP-algebraizable (thus, the conditions in point 1 hold), then: 2. L is finitely axiomatizable if and only if K is finitely based. 3. Theoremhood in L is decidable if and only if the equational theory of K is decidable. 4. Finite consequences in L are decidable if and only if the quasi-equational theory of K is decidable.

One of the keystones of bridge theorems for BP-algebraizable logics is Theo- rem 5.3, which tells us that for each algebra, the lattice of ﬁlters of the logic is

65 Transfer to basic full g-models is due to both Czelakowski and Pigozzi; point 1 was published in Czelakowski (2001), and point 2 in Czelakowski and Pigozzi (2004). The transfer to arbitrary full g-models follows from the basic case and several results in Font and Jansana (1996). 66 Roughly speaking, a Gentzen-style rule is accumulative when an arbitrary set of “side assumptions” occurs (the same set) in the antecedent (left-hand side) of all the sequents in the rule. 62 Josep Maria Font isomorphic to the lattice of congruences of the algebra relative to the equivalent algebraic semantics of the logic, thus entering a more typically algebraic ﬁeld. This isomorphism can be restricted to the compact67 elements of the two lattices, and ω ω so it maps FiL A to ConK A, the join-semilattice of compact relative congruences. Then, Theorem 7.10 yields:

Theorem 7.13 (Raftery, 2013b). Let L be a BP-algebraizable logic, whose equivalent algebraic semantics is a quasivariety K. The following conditions are equivalent: (i) L satisﬁes an inconsistency lemma. ω (ii) For all A, the join-semilattice ConK A is dually pseudo-complemented. ω (iii) For all A ∈ K, the join-semilattice ConK A is dually pseudo-complemented. The equivalence between (i) and (ii) is just the equivalence between the analogous properties in Theorem 7.10, after considering the isomorphism of Theorem 5.3 (recall that every algebraizable logic is protoalgebraic). That the property in (ii) is still equivalent if limited to algebras in K as in (iii) is a relevant fact; each of the two properties has its strength in different applications, as commented on in footnote 72. The most interesting results arise when the algebraic property is one that has been studied due to independent interest in it within algebra. One of the most famous theorems of this kind is the following.

Theorem 7.14 (Blok and Pigozzi, 2001, but originally proved before 1983). Let L be a BP-algebraizable logic, whose equivalent algebraic semantics is a quasivariety K. The following conditions are equivalent. (i) L satisﬁes a DDT. ω (ii) For all A, the join-semilattice ConK A is dually residuated. ω (iii) For all A ∈ K, the join-semilattice ConK A is dually residuated. (iv) K has equationally deﬁnable principal relative congruences (EDPRC).68

The equivalence between (i) and (ii) follows from Theorems 7.7 and 5.3. Blok and Pigozzi directly proved the equivalence between (i) and (iv) by showing that by algebraizability the DDT for L is transferred to an analogous property for K , and that this property is equivalent to (iv). The equivalence between (iii) and (iv) seems a purely algebraic result, and in fact it is: it holds for every quasivariety, regardless of whether it is the equivalent algebraic semantics of an algebraizable logic or not. However, it is stimulating to know that the ﬁrst version of this equivalence, for the variety case, was obtained by

67 In this situation, the two lattices are lattices of closed sets of certain finitary closure operators, therefore their compact elements are the finitely generated closed sets. 68 EDPRC is the analogue, for quasivarieties and relative congruences, of the property of having equationally definable principal congruences (EDPC) for varieties, defined in Section 8 of Raftery’s chapter in this volume. The property EDPC has been extensively studied in universal algebra. Abstract Algebraic Logic 63

Kohler¨ and Pigozzi (1980) thanks to the intuitions that the link between the DDT and EDPC suggests; with these intuitions, they also obtained an algebraic analogue of Corollary 7.8, showing that EDPC implies congruence distributivity, which solved a problem in universal algebra that had been open for some time. Actually, the very notion of algebraizability was originally developed in order to build a framework where the equivalence between the DDT and EDP(R)C could be rigorously formulated and proved (Pigozzi, 1998). The work on the logical side made it apparent that most of these purely algebraic investigations can be generalized from varieties to quasivarieties if relative congruences enter into the picture (because relative congruences are the algebraic analogue of the logical notions of a theory and a filter; recall Theorem 5.3). The other two most famous bridge theorems, proved under the same assumptions as Theorems 7.13 and 7.14, can be roughly described as follows. • L satisfies Craig’s interpolation property if and only if K satisfies the amalgamation property (Czelakowski and Pigozzi, 1999). • L satisfies Beth’s definability property if and only if epimorphisms in K are surjective (Blok and Hoogland, 2006). I say “roughly” because each of the properties involved in the two results can be formulated in several forms, of different strength; as a consequence, a plethora of important theorems have been obtained. The cited references concern the exact formulation of the results in the present framework, but the connections between these pairs of properties had been observed and studied much earlier by several schol- ars, and continue to be much extended, either restricted to classes of modal or su- perintuitionistic logics (Maksimova), many-valued and fuzzy logics (Montagna) or substructural logics (Wronski,´ Ono, Galatos, Kihara) or in more general contexts69 (Ono, Wronski,´ Nemeti,´ Sain, Madarasz).´ Accurate historical surveys are found in the papers cited, as well as in Andreka,´ Nemeti,´ and Sain (2001, Section 6), but new papers continue to appear. The study of these connections under the perspective of abstract algebraic logic has revealed that more general versions of these bridge theorems hold for equivalential logics (and hence apply to many more particular logics), provided that one deals with matrices instead of plain algebras; in this case, the class of reduced models of the logic plays the roleˆ of the equivalent algebraic semantics. Even more abstract versions of the logical properties have been studied in later papers,70 where it turns out that the categorial aspects of the algebraic properties play a prominent role;ˆ

69 The relations between (the different forms of) interpolation and amalgamation have also been studied in other frameworks, such as equational logic, abstract model theory, and the theory of institutions. The computational applications of interpolation seem to be partly responsible for such interest in this property. 70 For instance: in the traditional versions of interpolation, the relation to be interpolated is either consequence or a connective of implication. In some recent papers, it is equivalence that is interpolated, either understood as a connective or as the more abstract Leibniz congruence. Cabrer and Gil-Ferez´ (2014) study up to eleven different versions of interpolation of all these kinds. 64 Josep Maria Font this makes it possible to study them not just in a category naturally associated with algebras in K, but in equivalent categories.71 Besides their theoretical interest, bridge theorems are particularly useful when applied to particular logics. Most often, they can be used to disprove that a certain logic has a certain property by looking at the corresponding algebraic property, either in general or inside the particular class of algebras or of matrices associated with the logic. In some cases the task is to construct a particular counterexample; in others, to search for known global properties of the class of algebras or matrices.72 These strategies have been successfully applied to several families of sentential logics; the best known concern several versions of the DDT, of the interpolation property, and of the Beth definability property, in fragments of classical or intuitionistic logic, modal logics, many-valued logics, relevance logics, linear logics and substructural logics, several versions of first-order logic (presented as a sentential-like logic), etc. The reverse application is also possible; for instance, syntactic proofs of the interpolation property have been used to infer amalgamation properties in the associated classes of algebras. Let me end this section by emphasizing that applications of algebraizability are not limited to those of bridge theorems. There are logical problems for which there are no precise, general bridge theorems available (yet) but which have been successfully treated by algebraic means, i.e., after crossing the bridge on a less general vehicle. One example of this kind of research, in the area of substructural logics, is contained in the book by Galatos et al. (2007); while the general setting of the book is explicitly the algebraizability paradigm, it takes advantage of several particular properties of the varieties of residuated lattices it deals with (for instance, cut-elimination is proved thanks to several completion results, and in the reverse di- rection semisimplicity of some free algebras is proved by proof-theoretic methods). While the definition of algebraizability establishes a bridge between a single logic and its algebraic counterpart, by Theorem 4.2 the bridge can be extended to cover a large class of logics (its extensions) and their associated classes of algebras. Its applications are often grounded on the following specialization.

Theorem 7.15. Let L be a BP-algebraizable logic whose equivalent algebraic semantics is a variety V. Then there is a dual isomorphism between the lattice of all the ﬁnitary extensions of L and the lattice of all sub-quasivarieties of V; this isomorphism restricts to one between the lattice of all the axiomatic extensions of L and the lattice of all the sub-varieties of V.

This fact allows us to cross the bridge and place the study of some properties of a logic in the land of universal algebra, where the study of lattices of (quasi)varieties

71 This idea was first pursued in Galatos and Raftery (2012), where, thanks to a category equivalence between a class of integral residuated lattices and its non-integral counterpart, some metalogical properties of certain logics having both a fuzzy and a substructural character were indirectly obtained. 72 Observe that properties concerning the class of all algebras, such as those in points (ii) in The- orems 7.13 and 7.14, offer more flexibility when it comes to attaining the first goal, while those restricted to the class K, such as points (iii), may better support the second. Abstract Algebraic Logic 65 has generated a number of very powerful results. Just as an example of a general result that can be obtained in this way: Baker’s Theorem and Jonsson’s´ Lemma can be used to show the following.

Theorem 7.16. If the equivalent algebraic semantics of a BP-algebraizable logic is a congruence-distributive variety generated by a finite algebra, then the logic is finitely axiomatizable and has only finitely many extensions.

It is, however, in the application to particular lattices of logics that this methodology has proved its strength.73 Of course, often the isomorphism can be (and has been) established without appealing to the theory of algebraizability (but, in fact, by using just a particular case of the general proof). However, knowing that the general version holds whenever algebraizability is present should pave the way, and save some work, for similar studies in other areas.

8 Algebraizability at a more abstract level

Soon after its invention, the idea of the algebraizability of a logic, in the sense of its “equivalence” with the equational consequence relative to a class of algebras by means of structural transformers, was extended to ﬁnite-dimensional deductive systems, to Gentzen systems, to logical systems based on hypersequents, etc. It was clear that the whole framework would function wherever one could consider transformers between the syntactic objects involved (formulas, m-formulas, sequents, etc.) and equations, and back again, so that conditions parallel to (ALG1)–(ALG4) would make sense; this would produce the parallel notions of algebraizable Gentzen system (Rebagliato and Verdu,´ 1993), etc. Moreover, it also became clear that this was an instance of a more general idea of equivalence between two consequence relations, and that the fact that one of them is a relative equational consequence was not essential: equivalence between any of the kinds of consequences involved could be considered. This was done by Raftery (2006a) for Gentzen systems, treating algebraizability as a special case of the equivalence between two Gentzen systems of the most general kind. As a bonus, the equivalence between two sentential logics,74 and that between a sentential logic and a Gentzen system, were also studied; he applied this to shed new light on the relations between different formalizations of some substructural logics.

73 The work done by Blok in the 1970s on several lattices of modal logics (see Rautenberg, Za- kharyaschev, and Wolter, 2006, for a survey) is noteworthy, besides its intrinsic interest, for two reasons: he was the ﬁrst to develop this kind of application, and it was one of his main sources of inspiration for the general theory of algebraizability. 74 In the literature, a relation between two sentential logics expressed by means of a transformer in a way similar to (ALG1) has been named in a variety of ways; the term translation is popular, sometimes with either conservative or faithful prepended to it, but these are not used in a uniform way. Notice that the equivalence to which I refer here is stronger than this (the transformer is “invertible”), and that some translations considered in the literature are not structural. 66 Josep Maria Font

The steps from the Lindenbaum-Tarski completeness of implicative logics to algebraizability, and from algebraizability to the equivalence of deductive systems of more general kinds, are generalizations. The next step in the research into algebraizability was an abstraction. To understand its basic idea we need to review the characterizations of algebraizability in Theorem 5.3 and in the next-to-last row of

Table 2 on page 41. These characterizations mention the lattices FiL A and ConKA, but almost nothing has been said about their lattice structure up to now. These sets are easily seen to be closure systems, and hence complete lattices when ordered under set inclusion, with intersection as the inﬁmum operation. As to the supremum of a family, it is computed by closing the union of the family under the closure operator A associated with the closure system; these closure operators are denoted by FgL and A A ΘK respectively; that is, FgL (X) is the smallest L -ﬁlter of A containing X ⊆ A, A and ΘK(θ) is the smallest K-congruence of A containing θ ⊆ A × A. Thus, these suprema are:

_ A [ _ A[ Fi := FgL Fi and θi := ΘK θi i∈I i∈I i∈I i∈I for {Fi : i ∈ I} ⊆ FiL A and {θi : i ∈ I} ⊆ ConKA. Consideration of these lattice structures facilitates a better understanding of the property that the Leibniz operator “commutes with endomorphisms”. Every h ∈ −1 End(A) induces a function h : FiL A → FiL A, because the inverse image of an L -filter is always an L -filter; then the commutativity property defined on page 41 asserts that the Leibniz operator commutes with this function: Ω Ah−1F = h−1Ω AF . But h also induces another function on the set of L -filters, obtained by lifting it A to the power set and closing the result; if we define hL F := FgL (hF) for each F ∈ FiL A, then hL : FiL A → FiL A. Similarly, by considering the extension of h to the Cartesian product A × A, we obtain a function on ConKA, namely hKθ := A 75 ΘK(hθ) for each θ ∈ ConKA. Then one can prove the following.

Lemma 8.1. Let Φ : FiL A → ConKA be a lattice isomorphism, and h ∈ End(A). The following conditions are equivalent.

(i) Φ hL F = hKΦ F for all F ∈ FiL A. −1 −1 (ii) Φ h F = h Φ F for all F ∈ FiL A. When these properties hold for all h ∈ End(A), we say that Φ commutes with endomorphisms of A. By condition (i), such a Φ is a homomorphism of the enriched structures obtained when the lattices FiL A and ConKA are expanded with the functions hL and hK , respectively, considered as unary operations, for all h ∈ End(A). All these constructions, of course, can be carried out in the particular case of the formula algebra.76 The characterization in the next-to-last row of Table 2 (page 41) can be

75 This is a particular case, tailored for this point of the exposition, of a more general and abstract result, which should be easy to guess (Font, 2016, Theorem 1.56). 76 Fm Recall that in this case, FiL Fm = ThL and hence that FgL = CL , and that the endomorphisms of Fm are the substitutions. Abstract Algebraic Logic 67 rephrased by saying that L is algebraizable if and only if, for each algebra A, Ω A is an isomorphism between these expanded lattices. But there is much more. The mere existence of an isomorphism of this kind is enough to guarantee algebraizability:

Theorem 8.2 (essentially, Blok and Pigozzi, 1989, 2001). Let L be a logic and let K be a generalized quasivariety. The following conditions are equivalent: (i) L is algebraizable with equivalent algebraic semantics K. (ii) For every algebra A there is an isomorphism between the expanded lattices

FiL A,hhL : h ∈ End(A)i and ConKA,hhK : h ∈ End(A)i .

(iii) There is an isomorphism between the expanded lattices ThL ,hσ : σ ∈ L End(Fm)i and ConKFm,hσK : σ ∈ End(Fm)i . Observe that in condition (iii) “points” (formulas and equations) have almost dis- appeared; the condition concerns the expanded lattices of closed sets77 of the two closure operators (corresponding to the closure relations `L and K ) whose equivalence through structural transformers constitutes the definition of algebraizability. Another equally essential component should be highlighted before going on: the two closure operators involved are structural, which is also a property that relates them to the endomorphisms of the formula algebra, and which can equivalently be formulated in terms of the closed sets. For instance, it is easy to see that property (S) in Definition 2.1 is equivalent to the condition that σ −1Γ ∈ ThL for all Γ ∈ ThL and all σ ∈ End(Fm); this property is then propagated to the L -filters of arbitrary −1 algebras: if F ∈ FiL A then h F ∈ FiL A for all h ∈ End(A). A result parallel to Theorem 8.2 for the notion of equivalence between two Gentzen systems was obtained by Raftery (2006a, Theorem 6.8). It seems clear that a similar result can be obtained if one develops a theory of equivalence between hypersequent systems, or between finitely-sided sequents, and so on. However, for this, a parallel theory of matrix models, filters, Leibniz operator, etc., has to be developed in each case, with a formalism heavily dependent on the grammatical structure of the languages involved. The subject naturally called for a more abstract setting that would free it from this dependence on the languages on which the deductive systems are defined. This move was made by Blok and Jonsson´ (2006), who considered this kind of isomorphism theorem as capturing the abstract essence of algebraizability. They jumped one abstraction level up, and placed the study in the landscape of structural closure operators on M-sets, that is, sets endowed with the action of a monoid M (the abstraction of the monoid of substitutions of the formula algebra, which acts on the sets of formulas, of equations, of sequents, etc.). The structurality of a closure operator on the universe of an M-set is defined in terms of these actions, and the lattice of closed sets is also expanded by these actions. The next level of abstraction was introduced by Galatos and Tsinakis (2009). It consists in “getting rid of points” completely, by observing that most of the work

77 The relative congruences of the formula algebra, ConKFm, are the closed sets (the “theories”) of the equational consequence K . 68 Josep Maria Font is already done at the level of subsets (the consequences, the filters, etc.), and that the endomorphisms and the transformers, which originally act on points, can be extended to sets by taking unions, so that all the relevant definitions (e.g., structurality, or the four algebraizability conditions) can be equivalently formulated by referring only to subsets. Since complementation plays no roleˆ here, the abstract structures corresponding to power sets are complete lattices, and the abstract correlate of point functions extended to sets by taking unions are residuated functions; some of these functions (those corresponding to the substitutions) act on the elements of the lattice, giving it the structure of a module. Thus, the theory is placed in the framework of modules over complete residuated lattices and closure operators78 over them. In the two abstract approaches just mentioned, the notion of equivalence between structural closure operators appears naturally, and is defined in terms of the existence of an isomorphism between the associated expanded lattices of closed sets or, in the second case, the associated modules of fixed points. A new problem thus arises: that of identifying conditions under which any such isomorphism, assum- ing it exists, is induced by transformers, as the Leibniz operator is in the original setting. This leads to the problem of identifying the abstract correlate of the properties of the algebra of formulas that make the equivalence in Theorem 8.2 possible, and has come to be known as THEISOMORPHISMPROBLEM. Some sufficient, but not necessary, conditions had already been found by Blok and Jonsson´ (2006); the solutions obtained at different levels by Galatos and Tsinakis (2009), and by Font and Moraschini (2015) are formulated in categorial terms, involving the notion of projectivity. These lines of research seem to be expanding further in several directions. Both the idea of equivalence between deductive systems and the isomorphism problem can be studied in other contexts. Let me just mention that Galatos and Gil-Ferez´ (2016) jump up again, to the categories of modules over quantaloids, which allows them to merge this line of research with the already very fruitful one of categorical abstract algebraic logic pursued by Voutsadakis (see 2003, and many others), cover- ing many-sorted logics. Meanwhile, Russo (2013) addresses the issue of considering the equivalence between deductive systems over different languages at these more abstract levels. Furthermore, Moraschini (2016) develops a new concept of translation between equational consequences relative to generalized quasivarieties (again, possibly over different languages) which characterizes the adjunctions between the associated categories, and uses this to prove that every generalized quasivariety is categorially equivalent to the equivalent algebraic semantics of a finitely algebraizable logic (possibly over a different language); in particular, every quasivariety is categorially equivalent in this sense to the equivalent algebraic semantics of a finitary and finitely algebraizable (i.e., BP-algebraizable) logic.

I hope that there is more to come.

78 Now in the purely order-theoretic sense, as opposed to closure operators on power sets as described in footnote 43. Abstract Algebraic Logic 69

Acknowledgements I am most grateful to Tommaso Moraschini and to an anonymous referee, whose many observations concerning details as well as more substantial remarks decidedly im- proved the exposition presented here. While writing this chapter, I was partially funded by the research project MTM2011-25747 from the government of Spain, which includes FEDER funds from the European Union, and the research grant 2009SGR-1433 from the government of Catalo- nia.

References

H. Albuquerque, J. M. Font, R. Jansana, and T. Moraschini. Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic. In J. Czelakowski, editor, Don Pigozzi on Abstract Algebraic Logic and Universal Algebra, series Outstanding Contributions to Logic. Springer-Verlag, 2016a. 25 pp. To appear. H. Albuquerque, A. Prenosil,ˇ and U. Rivieccio. An algebraic view of super-Belnap logics. Sub- mitted manuscript, 2016b. H. Andreka,´ I. Nemeti,´ and I. Sain. Algebraic logic. In D. M. Gabbay and F. Guenthner, editors, Handbook of philosophical logic, second edition, volume 2, pages 133–248. Kluwer Academic Publishers, 2001. I. H. Anellis and N. Houser. Nineteenth century roots of algebraic logic and universal algebra. In H. Andreka,´ J. D. Monk, and I. Nemeti,´ editors, Algebraic logic, volume 54 of Colloquia Math. Soc. Janos Bolyai, pages 1–36. North-Holland, Amsterdam, 1991. S. Babyonyshev. Fully Fregean logics. Reports on Mathematical Logic, 37:59–78, 2003. W. Blok and E. Hoogland. The Beth property in algebraic logic. Studia Logica, 83:49–90, 2006. W. Blok and B. Jonsson.´ Equivalence of consequence operations. Studia Logica (Special issue in memory of Willem Blok), 83:91–110, 2006. W. Blok and D. Pigozzi. Protoalgebraic logics. Studia Logica, 45:337–369, 1986. W. Blok and D. Pigozzi. Alfred Tarski’s work on general metamathematics. The Journal of Symbolic Logic, 53:36–50, 1988. W. Blok and D. Pigozzi. Algebraizable logics, volume 396 of Mem. Amer. Math. Soc. A.M.S., Providence, January 1989. Out of print. Scanned copy available from http://orion.math. iastate.edu:80/dpigozzi/. W. Blok and D. Pigozzi. Abstract algebraic logic and the deduction theorem. Manuscript, 2001. Available from http://orion.math.iastate.edu:80/dpigozzi/. W. Blok and J. Rebagliato. Algebraic semantics for deductive systems. Studia Logica (Special issue on Abstract Algebraic Logic, Part II), 74:153–180, 2003. S. L. Bloom. Some theorems on structural consequence operations. Studia Logica, 34:1–9, 1975. S. Bonzio, J. Gil-Ferez,´ F. Paoli, and L. Peruzzi. On paraconsistent weak Kleene logic. Submitted manuscript (2016). G. Boole. The mathematical analysis of logic. Being an essay towards a calculus of deductive reasoning. Macmillan, Cambridge, 1847. F. Bou, F. Esteva, J. M. Font, A. J. Gil, Ll. Godo, A. Torrens, and V. Verdu.´ Logics preserving degrees of truth from varieties of residuated lattices. Journal of Logic and Computation, 19: 1031–1069, 2009. F. Bou and U. Rivieccio. The logic of distributive bilattices. Logic Journal of the IGPL, 19: 183–216, 2011. S. Burris and J. Legris. The algebra of logic tradition. In E. N. Zalta, editor, The Stanford En- cyclopedia of Philosophy. Spring 2015 edition. Archived as: http://plato.stanford.edu/ archives/spr2015/entries/algebra-logic-tradition/. L. M. Cabrer and J. Gil-Ferez.´ Leibniz interpolation properties. Annals of Pure and Applied Logic, 165:933–962, 2014. 70 Josep Maria Font

C. Caleiro and R. Gonçalves. Abstract valuation semantics. Studia Logica (Special issue on Abstract Algebraic Logic), 101:677–712, 2013. S. Celani and R. Jansana. A closer look at some subintuitionistic logics. Notre Dame Journal of Formal Logic, 42:225–255, 2001. A. Chagrov and M. Zakharyashev. Modal Logic, volume 35 of Oxford Logic Guides. Oxford University Press, 1997. P. Cintula and C. Noguera. Implicational (semilinear) logics I: a new hierarchy. Archive for Mathematical Logic, 49:417–446, 2010. P. Cintula and C. Noguera. A general framework for Mathematical Fuzzy Logic. In P. Cintula, P. Hajek,´ and C. Noguera, editors, Handbook of Mathematical Fuzzy Logic, vol. 1, volume 37 of Studies in Logic. College Publications, London, 2011. P. Cintula and C. Noguera. A Henkin-style proof of completeness for first-order algebraizable logics. The Journal of Symbolic Logic, 80:341–358, 2015. J. Czelakowski. Reduced products of logical matrices. Studia Logica, 39:19–43, 1980. J. Czelakowski. Equivalential logics, I, II. Studia Logica, 40:227–236 and 355–372, 1981. J. Czelakowski. Logical matrices, primitive satisfaction and finitely based logics. Studia Logica, 42:89–104, 1983. J. Czelakowski. Filter distributive logics. Studia Logica, 43:353–377, 1984. J. Czelakowski. Algebraic aspects of deduction theorems. Studia Logica, 44:369–387, 1985. J. Czelakowski. Protoalgebraic logics, volume 10 of Trends in Logic - Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 2001. J. Czelakowski. The Suszko operator. Part I. Studia Logica (Special issue on Abstract Algebraic Logic, Part II), 74:181–231, 2003. J. Czelakowski and W. Dziobiak. A single quasi-identity for a quasivariety with the Fraser-Horn property. Algebra Universalis, 29:10–15, 1992. J. Czelakowski and R. Jansana. Weakly algebraizable logics. The Journal of Symbolic Logic, 65: 641–668, 2000. J. Czelakowski and G. Malinowski. Key notions of Tarski’s methodology of deductive systems. Studia Logica, 44:321–351, 1985. J. Czelakowski and D. Pigozzi. Amalgamation and interpolation in abstract algebraic logic. In X. Caicedo and C. H. Montenegro, editors, Models, algebras and proofs, volume 203 of Lecture Notes in Pure and Applied Mathematics Series, pages 187–265. Marcel Dekker, New York and Basel, 1999. J. Czelakowski and D. Pigozzi. Fregean logics. Annals of Pure and Applied Logic, 127:17–76, 2004. J. M. Dunn and G. M. Hardegree. Algebraic methods in philosophical logic, volume 41 of Oxford Logic Guides. Oxford Science Publications, Oxford, 2001. M. Esteban. Duality theory and abstract algebraic logic. Ph. D. Dissertation, University of Barcelona, 2013. J. M. Font. On the Leibniz congruences. In C. Rauszer, editor, Algebraic Methods in Logic and in Computer Science, volume 28 of Banach Center Publications, pages 17–36. Polish Academy of Sciences, Warszawa, 1993. J. M. Font. Belnap’s four-valued logic and De Morgan lattices. Logic Journal of the IGPL, 5: 413–440, 1997. J. M. Font. Abstract Algebraic Logic - An Introductory Textbook, volume 60 of Studies in Logic. College Publications, London, 2016. J. M. Font, F. Guzman,´ and V. Verdu.´ Characterization of the reduced matrices for the conjunction- disjunction fragment of classical logic. Bulletin of the Section of Logic, 20:124–128, 1991. J. M. Font and R. Jansana. A general algebraic semantics for sentential logics, volume 7 of Lecture Notes in Logic. Springer-Verlag, 1996. Out of print. Second revised edition published in 2009 by the Association for Symbolic Logic, freely available through Project Euclid at http: //projecteuclid.org/euclid.lnl/1235416965. J. M. Font, R. Jansana, and D. Pigozzi. Fully adequate Gentzen systems and the deduction theorem. Reports on Mathematical Logic, 35:115–165, 2001. Abstract Algebraic Logic 71

J. M. Font, R. Jansana, and D. Pigozzi. A survey of abstract algebraic logic. Studia Logica (Special issue on Abstract Algebraic Logic, Part II), 74:13–97, 2003. With an update in vol. 91 (2009), 125–130. J. M. Font and T. Moraschini. Logics of varieties, logics of semilattices, and conjunction. Logic Journal of the IGPL, 22:818–843, 2014. J. M. Font and T. Moraschini. M-sets and the representation problem. Studia Logica, 103:21–51, 2015. J. M. Font and G. Rodr´ıguez. Algebraic study of two deductive systems of relevance logic. Notre Dame Journal of Formal Logic, 35:369–397, 1994. J. M. Font and V. Verdu.´ Algebraic logic for classical conjunction and disjunction. Studia Logica (Special issue on Algebraic Logic), 50:391–419, 1991. N. Galatos and J. Gil-Ferez.´ Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π -institutions. Journal of Pure and Applied Algebra, 2016. To appear. N. Galatos, P. Jipsen, T. Kowalski, and H. Ono. Residuated lattices: an algebraic glimpse at substructural logics, volume 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 2007. N. Galatos and J. Raftery. A category equivalence for odd Sugihara monoids and its applications. Journal of Pure and Applied Algebra, 216:2177–2192, 2012. N. Galatos and C. Tsinakis. Equivalence of consequence relations: an order-theoretic and categorical perspective. The Journal of Symbolic Logic, 74:780–810, 2009. M. Gehrke, R. Jansana, and A. Palmigiano. Canonical extensions for congruential logics with the deduction theorem. Annals of Pure and Applied Logic, 161:1502–1519, 2010. P. Hajek.´ Metamathematics of fuzzy logic, volume 4 of Trends in Logic - Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 1998. P.R. Halmos. Algebraic logic. Chelsea Publ. Co., New York, 1962. L. Henkin. An algebraic characterization of quantifiers. Fundamenta Mathematicae, 37:63–74, 1950. L. Henkin, J. D. Monk, and A. Tarski. Cylindric algebras, part II. North-Holland, Amsterdam, 1985. B. Herrmann. Equivalential logics and definability of truth. Ph. D. Dissertation, Freie Universitat¨ Berlin, 1993. B. Herrmann. Equivalential and algebraizable logics. Studia Logica, 57:419–436, 1996. W. Hodges. Model theory, volume 42 of Encyclopedia of Mathematics and its Applications. Cam- bridge University Press, Cambridge, 1993. P. Idziak, K. Słomczynska,´ and A. Wronski.´ Fregean varieties. International Journal of Algebra and Computation, 19:595–645, 2009. R. Jansana. Full models for positive modal logic. Mathematical Logic Quarterly, 48:427–445, 2002. R. Jansana. Propositional consequence relations and algebraic logic. In E. N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Spring 2011 edition, 2011. Archived as: http://plato. stanford.edu/archives/spr2011/entries/consequence-algebraic/. P. Kohler¨ and D. Pigozzi. Varieties with equationally definable principal congruences. Algebra Universalis, 11:213–219, 1980. M. Kracht. Modal consequence relations. Chapter 8 of P. Blackburn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, volume 3 of Studies in Logic and Practical Reasoning, pages 491–545. Elsevier, Amsterdam, 2007. J. Łos.´ O matrycach logicznych, volume 19 of Prace Wrocławskiego Towarzystwa Naukowege, Ser. B. University of Wrocław, 1949. T. Moraschini. Investigations into the role of translations in abstract algebraic logic. Ph. D. Dissertation, University of Barcelona, June 2016. A. Nurakunov and M. Stronkowski. Relation formulas for protoalgebraic equality free quasivarieties: Pałasinska’s´ theorem revisited. Studia Logica (Special issue on Abstract Algebraic Logic), 101:827–847, 2013. 72 Josep Maria Font

D. Pigozzi. Finite basis theorem for relatively congruence-distributive quasivarieties. Transactions of the American Mathematical Society, 310:499–533, 1988. D. Pigozzi. Fregean algebraic logic. In H. Andreka,´ J. D. Monk, and I. Nemeti,´ editors, Alge- braic Logic, volume 54 of Colloq. Math. Soc. Janos´ Bolyai, pages 473–502. North-Holland, Amsterdam, 1991. D. Pigozzi. Abstract algebraic logic: past, present and future. A personal view. In J. M. Font, R. Jansana, and D. Pigozzi, editors, Workshop on Abstract Algebraic Logic, volume 10 of Quaderns, pages 122–138. Centre de Recerca Matematica,` Bellaterra, January 1998. D. Pigozzi. Abstract Algebraic Logic. In M. Hazewinkel, editor, Encyclopædia of Mathematics, Supplement III, pages 2–13. Kluwer Academic Publishers, Dordrecht, 2001. W. Pogorzelski. On Hilbert’s operation on logical rules I. Reports on Mathematical Logic, 12: 35–50, 1981. W. Pogorzelski and P. Wojtylak. Completeness Theory for Propositional Logics. Studies in Uni- versal Logic. Birkhauser¨ Verlag, Basel, 2008. T. Prucnal and A. Wronski.´ An algebraic characterization of the notion of structural completeness. Bulletin of the Section of Logic, 3:30–33, 1974. A. Pynko. Definitional equivalence and algebraizability of generalized logical systems. Annals of Pure and Applied Logic, 98:1–68, 1999. J. Raftery. Correspondences between Gentzen and Hilbert systems. The Journal of Symbolic Logic, 71:903–957, 2006a. J. Raftery. The equational definability of truth predicates. Reports on Mathematical Logic (Special issue in memory of Willem Blok), 41:95–149, 2006b. J. Raftery. A non-finitary sentential logic that is elementarily algebraizable. Journal of Logic and Computation, 20:969–975, 2010. J. Raftery. Contextual deduction theorems. Studia Logica (Special issue in honor of Ryszard Wojcicki´ on the occasion of his 80th birthday), 99:279–319, 2011a. J. Raftery. A perspective on the algebra of logic. Quaestiones Mathematicae, 34:275–325, 2011b. J. Raftery. Order algebraizable logics. Annals of Pure and Applied Logic, 164:251–283, 2013a. J. Raftery. Inconsistency lemmas in algebraic logic. Mathematical Logic Quarterly, 59:393–406, 2013b. H. Rasiowa. An algebraic approach to non-classical logics, volume 78 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1974. H. Rasiowa and R. Sikorski. Algebraic treatment of the notion of satisfability. Fundamenta Math- ematicae, 40:62–95, 1953. W. Rautenberg, M. Zakharyaschev, and F. Wolter. Willem Blok and modal logic. Studia Logica, 83:15–30, 2006. J. Rebagliato and V. Verdu.´ On the algebraization of some Gentzen systems. Fundamenta Infor- maticae (Special issue on Algebraic Logic and its Applications), 18:319–338, 1993. C. Russo. An order-theoretic analysis of interpretations among propositional deductive systems. Annals of Pure and Applied Logic, 164:112–130, 2013. D. J. Shoesmith and T. J. Smiley. Multiple-conclusion logic. Cambridge University Press, Cam- bridge, 1978. S. Surma. On the origin and subsequent applications of the concept of the Lindenbaum algebra. In Logic, methodology and philosophy of science VI, Hannover 1979, pages 719–734. North- Holland, Amsterdam, 1982. R. Suszko. Abolition of the Fregean axiom. In R. Parikh, editor, Logic Colloquium 1972–73, volume 453 of Lecture Notes in Mathematics, pages 169–239. Springer-Verlag, Berlin, 1975. Y. Suzuki, F. Wolter, and M. Zacharyaschev. Speaking about transitive frames in propositional languages. Journal of Logic, Language and Information, 7:317–339, 1998. G. Voutsadakis. Categorical abstract algebraic logic: equivalent institutions. Studia Logica (Special issue on Abstract Algebraic Logic, Part II), 74:275–311, 2003. R. Wojcicki.´ Logical matrices strongly adequate for structural sentential calculi. Bulletin de l’Academie´ Polonaise des Sciences, Classe III, XVII:333–335, 1969. Abstract Algebraic Logic 73

R. Wojcicki.´ Matrix approach in the methodology of sentential calculi. Studia Logica, 32:7–37, 1973. R. Wojcicki.´ Referential matrix semantics for propositional calculi. Bulletin of the Section of Logic, 8:170–176, 1979. R. Wojcicki.´ Theory of logical calculi. Basic theory of consequence operations, volume 199 of Synthese` Library. Reidel, Dordrecht, 1988. A. Wronski.´ BCK algebras do not form a variety. Mathematica Japonica, 28:211–213, 1983. J. Zygmunt. A note on direct products and ultraproducts of logical matrices. Studia Logica, 33: 349–358, 1974.