Leibniz filters and the strong version of a protoalgebraic logic

A filter of a sentential logic S is Leibniz when it is the smallest one among all the S-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic S+ defined by the class of all S-matrices whose filter is Leibniz, which is called the strong version of S , in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of S+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from S to S+ . For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz’s finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics.