4 W. J. BLOK AND DON PIGOZZI given in Theorem 4.7 it is an easy consequence of the first. It is especially useful in practice for proving specific deductive systems are algebraizable. The two intrinsic characterization theorems are used in Chapter 5 to in- vestigate the algebraizability of a number of different deductive systems. We prove there exists a large class of modal logics, including Si , S2, and S3, that are not algebraizable in our sense (Corollary 5.6) hence in our view these log- ics are intrinsically non-algebraiz able. (These logics are however protoalgebraic in the sense of [8], and thus amenable to most of the standard methods of alge- braic logic see Chapter 1.4.1.) We also look at the logics of strict implication. We show that relevance logic R is algebraizable (Theorem 5.8), wrhile entail- ment logic E is not (Corollary 5.7). On the other hand, the calculus of pure relevant implication R_, as well as the calculus of pure entailment E_*, fails to be algebraizable (Theorem 5.9). (In contrast the implicational fragments of both classical and intuitionistic propositional calculus are algebraizable.) We also investigate the algebraizability of relevance logic when the so-called min- gle axiom is adjoined, B-C-I and B-C-K logics, and classical equivalence logic (i.e., the {«-»}-fragment of classical propositional logic). The algebraization of predicate logic presents special problems these are discussed in Appendix C. An arbitrary deductive system S can be viewed in a natural way as an elementary (first-order) theory without equality, in fact, as a certain universal Horn theory ES (Chapter 1.3). Algebraizability can then be formulated as an elementary notion, in particular in terms of the definitional equivalence of ES with the elementary theory of a quasi-variety. This is done in Appendix A.

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