An algebraic perspective on valuation semantics∗

The class of so-called non-truth-functional logics constitutes a challenge to the usual algebraic based semantic tools, such as matrix semantics [ LS58]. The problem with these logics is the existence of noncongruent connectives that are not always interpreted homomorphically in a given algebra. The key idea of valuation semantics [dCB94], which arose precisely from the attempt to give some reasonable sort of semantics to these non-truth-functional logics, is to drop the condition that formulas should always be interpreted homomorphically. Nevertheless, although satisfactory, the existing proposal of valuation semantics is not as general as one would expect. It lacks a workable algebraic theory, such as that relating logical matrices to the Blok-Pigozzi theory of AAL [BP89]. Herein, we propose and study an algebraic generalization of the notion of valuation semantics. Our aim is to go deeper in the path of giving an algebraic counterpart to non-truth-functional logics, not only by extending to the many-sorted case, but also by strictly generalizing the existing notion in the one-sorted (propositional) case. Since our notion of valuation semantics arises naturally in semantical considerations from the novel behavioral approach to the algebraization of logics [CGM07], we will study the relation between them. As a byproduct we reinvent a complete valuation semantics for the paraconsistent logic C1 of da Costa, now in an algebraic behavioral disguise.