Leibniz filters and the strong version of a protoalgebraic logic |
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Authors: | Josep Maria Font Ramon Jansana |
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Institution: | (1) Department of Logic, History and Philosophy of Science, University of Barcelona, 08071 Barcelona, Spain. e-mail: font@mat.ub.es; jansana@mat.ub.es, ES |
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Abstract: | A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-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 ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of
protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak
algebraizability of ?+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to
?+. 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.
Received: 30 October 1998 /?Published online: 15 June 2001 |
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Keywords: | Mathematics Subject Classification (2000): Primary: 03G99 Secondary: 03B22 03B45 03B50 03G12 |
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