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 algebraizabil- ity, 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 ab- stract 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 mathemat- ical 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 old- est connections to be described take the form of completeness theorems; later on other, stronger connections were discovered, such as the notion of algebraizabil- ity. The task of finding meaningful descriptions (preferably algebraic descrip- tions) 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 connec- tions and exploring them in many different ways, to obtain a deeper understand- ing of how and why the different kinds of connections work, of the relations be- tween 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 con- trast 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 gen- eral 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 proposi- tional 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 alge- braizability initiated by Blok and Pigozzi has been observed in the literature; how- ever, 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 alge- braizability 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 di- rections. 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 exten- sion 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 the- ory 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 ap- plied 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 amal- gamation); 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 men- tioned are best dealt with within the framework of a class in the Leibniz hierarchy that is larger than the class of algebraizable logics.