ABSTRACT ALGEBRAIC LOGIC AND THE DEDUCTION THEOREM W. J. BLOK AND D. PIGOZZI Contents 1. Introduction 2 1.1. Connections with other work 4 2. Deductive Systems 5 2.1. Preliminaries 5 2.2. Examples 9 2.3. Matrix semantics 19 2.4. Notes 21 3. k-Dimensional Deductive Systems 26 3.1. k-dimensional deductive systems 26 3.2. Matrix semantics for k-deductive systems 29 3.3. Examples 34 3.4. Algebraic deductive systems 37 3.5. The deduction-detachment theorem in k-deductive systems 38 3.6. Notes 39 4. Algebraizable Deductive Systems 43 4.1. Equivalence of deductive systems 43 4.2. Algebraizable deductive systems 47 4.3. Intrinsic characterizations of algebraizability 50 4.4. The Leibniz operator 57 4.5. The deduction-detachment theorem and equivalence 62 4.6. Notes 64 5. Quasivarieties with Equationally Definablepacetoken = Principal Relative Congruences 66 5.1. Equationally definable principal relative congruences 66 5.2. Examples 70 Date: July 26, 2001. 1991 Mathematics Subject Classification. Primary: 03B60, 03G25. Secondary: 08C15, 06D99. Key words and phrases. algebraizable logic, Leibniz congruence, quasivariety, definable principal congru- ences, congruence distributivity. This paper is based on a course of lectures given at the Summer School on Algebraic Logic, Budapest, July, 1994; Vanderbilt University, Nashville, May, 1995; and the X Latin American Symposium on Mathematical Logic, Bogot´a,Columbia, July 24–29, 1995. The second author was supported in part by National Science Foundation grant CCR-9593168 and a grant from the Ministerio de Education y Cienca of the government of Spain. 1 2 W. J. BLOK AND D. PIGOZZI 5.3. First-order definable principal relative congruences 71 5.4. Relative congruence extension and relative congruence distributivity 72 5.5. Algebraizable deductive systems and the DDT 73 5.6. Examples 74 5.7. Notes 75 References 76 1. Introduction In this paper we outline the basic part of the theory of abstract algebraic logic as devel- oped originally in [13, 14] and apply it to an algebraic study of the deduction theorem. In contrast to traditional algebraic logic, where the focus is on the algebraic forms of specific logical systems, we are concerned here with the process of algebraization itself and how it can be applied to a wide class of logical systems that includes among others all sentential logics and equational logics. The original motivation for this work was an attempt to find the proper framework in which to study general properties of the deduction theorem from an algebraic point-of- view. The deduction theorem is the formal expression of one of the most important and useful principles of classical logic: to prove that an implication holds between propositions it suffices to give a proof of the conclusion on the basis of the assumption of the antecedent. It is such a familiar part of ordinary logical argumentation that it is hardly recognizable as being something whose use might be problematic. But in fact it is not part of the usual formalizations of the classical propositional logic and must be proved as a metatheorem. Moreover, while the principle remains valid for intuitionistic logic, it is known to fail in other important logics, for instance, certain systems of modal logic. It also fails in the most common formalization of first-order predicate logic, although a weaker version holds that is often not distinguished from the deduction theorem in its usual form. It turns out that there is a close connection between the deduction theorem and the universal algebraic notion of definable principal congruence relations. The abstract theory of algebraizability outlined here was developed in part to provide the proper context in which to formalize this connection precisely. The ultimate goal was to be able to apply the extensive work on the definability of principal congruence relations in universal algebra to answer some important questions about the validity of the deduction theorem in a variety of logical systems. To motivate our abstract theory of algebraization we consider in Section 2 several dif- ferent logics whose algebraic counterparts are well-known and whose algebraic metatheory constitute the core of classical algebraic logic. These include the classical and intuitionistic sentential logics as well as two modal systems: the system of Kripke and G¨odel’sformu- lation of Lewis’s system S5. The construction of the algebra counterpart of each of these logics follows the same pattern: An equivalence relation on formulas is associated with each theory T of the logic. Two formulas ϕ and ψ are equivalent if there is a proof of the biconditional ϕ ↔ ψ from the logical axioms and rules of inference of the logic together with the nonlogical axioms that define T . This equivalence relation respects the logical connectives; for example, if ϕ and ψ are equivalent respectively to ϕ0 and ψ0, then ϕ ∧ ψ is equivalent to ϕ0 ∧ ψ0. Consequently, a quotient algebra can be formed whose elements ABSTRACT ALGEBRAIC LOGIC 3 are equivalence classes of formulas and whose operations are induced by the logical connec- tives. This classical “Lindenbaum-Tarski process” associates a class of algebras with each logic: Boolean algebras in the case of classical propositional logic, and Heyting algebras, modal algebras, and monadic algebras respectively for the intuitionistic propositional logic, Kripke’s modal logic, and G¨odel’s version of S5. Each class of algebras can be viewed as the algebra counterpart of its corresponding logic in the sense that there is a close correspon- dence between the deductive theory of the logic and the equational theory of the algebras. (We mean “equational theory” in a somewhat more general sense than it is normally meant in algebra. We will be concerned with the quasi-identities that a class of algebras satisfies rather than just the identities.) There are logics for which the classical Lindenbaum-Tarski process does not seem to work, and as an example we consider another formalization of S5, due to Carnap. In this case the equivalence relation between formulas defined by the biconditional ↔ does not preserve the necessity connective 2. It is not clear however if the failure of the classical Lindenbaum- Tarski process to produce an algebra counterpart is due to some inherent deficiency of the logic, or if there is a generalization of the process that will work. It is partly in an attempt to answer questions of this kind that the theory of abstract algebraic logic was developed, and it can be viewed as an attempt to abstract the Lindenbaum-Tarski process. The logics considered in Section 2 are all examples of what we call 1-dimensional de- ductive systems (1-deductive systems for short). Roughly speaking these correspond to sentential logics with a “Hilbert-style” axiomatization. They encompass a wide class of logical systems, many of which are not normally conceived of as sentential logics. For ex- ample the first-order predicate logic can be viewed this way when appropriately formalized. Equational logic cannot be naturally formalized as a 1-deductive system. For the purpose of providing a context in which 1-deductive systems and equational logic can be treated uniformly, we introduce in Section 3 the notion of a k-dimensional deductive (k-deductive) system for each nonzero natural number k and discuss its semantics. The equational logic of every quasivariety of algebras can be naturally formalized as a 2-deductive system. We call such systems algebraic. k-dimensional deductive systems are sufficiently general to en- compass not only all 1-deductive and equational logics, but also a class of logics that are closely related to “Gentzen-style” formalizations. The k-deductive systems provide a natural universe in which to study abstract algebraic logic, and, in particular, to define precisely what it means for an arbitrary deductive system to be algebraizable, that is, to have a natural algebra counterpart. The definition is given in Section 4 and is based on a general notion of equivalence of arbitrary deductive systems that is motivated by the case studies in Sections 2 and 3. A k-dimensional deductive system and an l-dimensional deductive system are said to be equivalent if the consequence relation of each is formally definable in that of the other, and the two definitions are inverses of one another in a natural sense. We then define a k-deductive system S to be algebraizable if it is equivalent in this sense to an algebraic deductive system, i.e., the formalization of the equational logic of some quasivariety K as a 2-deductive system. The quasivariety K is taken to be the natural algebra counterpart of S. Moreover, we prove that there can be at most one quasivariety in this relation to S, so that the algebra counterpart of a k-deductive system, if it exists, is an intrinsic property of the system. Our definition of being algebraizable is justified by showing that each of the deductive systems considered 4 W. J. BLOK AND D. PIGOZZI in Sections 2 and 3, that we knew beforehand to have algebra counterparts, are indeed algebraizable in this sense. One of the most important aspects of traditional algebraic logic is the way in which it can be used to reformulate metalogical properties of a particular logical system in algebraic terms. Abstract algebraic logic allows us to study such connections in a general context. Known metalogical or algebraic results can then be applied to obtain new results in the other domain. In Section 3 we introduce an abstract version of the deduction theorem applicable to arbitrary k-dimensional deductive systems. In the special case of an algebraic 2-deductive system, i.e., the equational logic of a quasivariety K, the abstract deduction theorem turns out to be equivalent to a property of K that is well-known in universal algebra: that of having equationally definable principal relative congruences (EDPRC).