ALGEBRAIZABLE LOGICS 3 a class K of algebras is called an algebraic semantics for S if the consequence relation \~s of S can be interpreted in the (semantical) equational consequence relation |=^ °f K in a natural way K is called an equivalent algebraic semantics for S if there is an inverse interpretation of |=^ in h^. If S has an equivalent algebraic semantics, we call it algebraizable. S may have many algebraic se- mantics, but the main theorem of Chapter 2 asserts that a deductive system can be algebraized in essentially only one way (Theorem 2.15). In Chapter 3 we study the relationship between the theories of a deductive system and the equational theories of its equivalent algebraic semantics. (A theory of S is any set of formulas that contains all axioms and is closed under the inference rules.) This leads to the first characterization of algebraizability. We consider lattices of theories that have been enriched by operators that correspond in a natural way to the substitution functions on the underlying set of formulas. We prove in Theorem 3.7 that K is the equivalent algebraic semantics for a deductive system S iff there is an isomorphism between the theory lattice of S and the equational theory lattice of K that commutes with the substitution operators. The proof of this theorem is based in part on a well known technique of universal algebra that originated with Mal'cev [27]. Two intrinsic characterizations of algebraizability are given hi Chapter 4, and they have rather different characters. The first one is the main result of the paper and may be of philosophical interest. Let S be a fixed logic and T a theory of S. It is natural to think of formulas p and xp as being equivalent with respect to T if either one can be replaced by the other as a subformula of an arbitrary formula ti without affecting the truth or falsity of d relative to T here the truth or falsity oft? is determined by whether or not d is contained in T. The equivalence relation on formulas defined in this way plays an important part in our work and is denoted by SIT. The definition of SIT is closely related to the well known method of defining the equality relation in second-order logic that goes back to Leibniz, and in Chapter 1.4 we show that SIT is in fact the natural first-order analogue of the second-order Leibniz relation. For this reason SIT is called the elementary Leibniz (equivalence) relation associated with T. In Theorem 4.2 we prove that for S to be algebraizable it is necessary that SIT properly includes SIS whenever T properly includes S, i.e, that the Leibniz operator SI is one-one and order-preserving on the lattice of theories of S. We also prove, in what we consider to be the main result of the paper, that this condition is also sufficient for S to be algebraizable, provided a certain other natural condition holds. An algebraizable logic can also be characterized by the existence of a finite system of binary composite connectives (i.e., formulas in two variables) that collectively have many of the properties of the biconditional of classical logic. This leads to the second intrinsic characterization of algebraizability
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