Leibniz filters and the strong version of a protoalgebraic logic

A filter of a sentential logic Y is Leibniz when it is the smallest one amo ng all the Y-filters on the same algebra having the same Leibniz congruence . This paper studies these filters and the sentential logic Y+ defined by t he class of all Y-matrices whose filter is Leibniz, which is called the str ong version of Y, in the context of protoalgebraic logics with theorems. To pics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of Y+ and of the explicit definability of Leib niz filters, and several theorems of transfer of metalogical properties fro m Y to Y+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logic s, quantum logics and Lukasiewiez's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corre sponds to well-known situations in the literature, such as the local and th e global consequences for normal modal logics; while in others these constr uctions give an independent interest to the study of other lesser-known log ics, such as the lattice-based many-valued logics.