Leibniz filters and the strong version of a protoalgebraic logic
by Josep Maria Font and Ramon Jansana (University of Barcelona)
Archive for Mathematical Logic 40 (2001) 437--465
Abstract
A filter of a sentential logic S is Leibniz when it is the least one among all the
S-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic 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 the strong version of S and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from S to its strong version. For finitely equivalential logics better results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Lukasiewicz's many-valued logics, finding that the existence of a weak and a strong version of a logic correspond to well-known situations in the literature, such as the local and the global consequences for normal modal logics.