2 W. J. BLOK AND DON PIGOZZI There are a number of important logics to which Tarski's method cannot be directly applied, or when it can does not give the expected result. Modal logics, such as the Lewis systems S i , S2, and S3, which do not have the rule of necessitation h j = h D^, are examples of this kind. In this case the relation defined in (1) is not a congruence on the algebra of formulas because h f » tp does not imply \~ Df Dtp. Other examples are provided by a family of logics that arise from the consideration of a strict (non-material) form of implication wrhere p ip can be a theorem only in the event ip is actually involved in some concrete way in the deduction ofift. The best known logics of this kind axe the systems R and E of relevance and entailment found in Anderson and Belnap [2]. In these logics there exist theorems tp and tp for which the implication ip ip fails to be a theorem. Hence the set of all theorems cannot coincide with an equivalence class of the relation = defined in (1). The question naturally arises if any of these logics can be algebraized by some method other than the classical one, or if they axe in a sense inherently non-algebraizable. In order to answer questions like this the notion of an algebraizable logic must be made precise. The problem of formulating such a notion with sufficient degree of generality does not seem to have been addressed in the literature.2 We propose one in this paper. We believe that it is a very natural one, and that any logic that fails to meet its criteria can with justification be called inherently non-algebraizable. Such a claim cannot of course be established in any absolute, mathematical sense. But to support it we will present several different characterizations of aigebraizability apart from the defining condition. They represent natural but quite different aspects of the algebraization process. The fact they all characterize the same notion can be viewed as strong evidence that it is the proper one. For our purposes a logic is given by an arbitrary set of axioms and inference rules. Logics specified in this way have been called deductive or logistic systems. More exactly, for us a logic is specified, not just by its set of theorems, but by its consequence relation h thought of as a bmary relation between sets of formulas and individual formulas. We want to be able to consider deductive systems, like those of relevance and entailment, that do not have the deduction theorem in such systems the consequence relation h cannot be defined in terms of the set of theorems. Deductive systems axe defined and their elementary properties reviewed in Chapter 1. The precise definition of an algebraizable logic is given in Chapter 2, Defini- tion 2.10. We first define the notion of algebraic semantics. Roughly speaking, 2 Other approaches to a general theory of algebraic logic can be found however see Andreka, Gergely, and Nemeti [3] and Andreka, Nemeti, and Sain [4]. (A summary of this work can be found in Henkin, Monk, and Tarski [15, Part II]). See also Felscher and Schulte Mbnting [13] and Rasiowa [36].
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