Algebraic Valuations as Behavioral Logical Matrices

Abstract Valuation Semantics

“…Our aim was to provide a consistent revision of the initial proposal in [6,7], that would overcome its limitations and allow for a smooth connection with the behavioral approach to the algebraization of logics, where logical matrices are not suited. Namely, using abstract valuations, we were able to treat both the Suszko and Leibniz congruences in a symmetric way, thus providing a solid foundation for defining and studying the canonical semantics of a logic, in a way that can be seen as a natural extension of the reduced matrix models of a logic, as reinforced by the results obtained.…”

“…As a first step toward coping with the inadequacy of logical matrices in the behavioral setting, the work in [6,7] explored the possibility of replacing logical matrices by an algebraically well-behaved version of valuation semantics that could serve as the semantic counterpart of the behavioral approach to the algebraization of logics, and obtained some bridge results generalizing the role played by matrix semantics in abstract algebraic logic, in the lines of [3,9,12,21]. Still, some difficulties could be noticed in [7] regarding the fact that the valuation, as a semantic unit, has a 'local' character when contrasted with the 'global' character of a logical matrix, which gives rise to a collection of valuations, one for each possible assignment to the logical variables.…”