Archive for Mathematical Logic manuscript No. (will be inserted by the editor) Leibniz filters and the strong version of a protoalgebraic logic Josep Maria Font, Ramon Jansana Department of Logic, History and Philosophy of Science University of Barcelona, E-08071 Barcelona, Spain e-mail: font@mat.ub.es , jansana@mat.ub.es 31st May 1999 Abstract A filter of a sentential logic S is Leibniz when it is the smal- lest one among all the S -filters on the same algebra having the same Leib- niz 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 theor- ems. Topics studied include an enhanced Correspondence Theorem, char- acterizations of the weak algebraizability of S + and of the explicit defin- ability 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 Lukasiewicz’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. Key words protoalgebraic logic – Leibniz filter – strong version – al- gebraizable logic – Leibniz operator – modal logics – quantum logics – many-valued logics – abstract algebraic logic – transfer theorem Mathematics Subject Classification: Primary: 03G99. Secondary: 03B22, 03B45, 03B50, 03G12.