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Note of the full generalized models of the extensions of a logic

Albuquerque H., Font J. M., Jansana R.

Reports on Mathematical Logic

2017

Vol. 52

57--68

In this short note we show that the full generalized models of any extension of a logic can be determined from the full generalized models of the base logic in a simple way. The result is a consequence of two central theorems of the theory of full generalized models of sentential logics. As applications we investigate when the full generalized models of an extension can also be full generalized models of the base logic, and we prove that each Suszko lter of a logic determines a Suszko lter of each of its extensions, also in a simple way.

On semilattice-based logics with an algebraizable assertional companion

Font J. M.

Reports on Mathematical Logic

2011

Vol. 46

109-132

This paper studies some properties of the so-called semilattice-based logics (which are defined in a standard way us- ing only the order relation from a variety of algebras that have a semilattice reduct with maximum) under the assumption that its companion assertional logic (defined from the same variety of algebras using the top element as representing truth) is algebraizable. This describes a very common situation, and the conclusion of the paper is that these semilattice-based logics exhibit some of the good behaviour of protoalgebraic logics, without being necessarily so. The main result is that all these logics have enough Leibniz filters, a fact previously known in the literature to occur only for protoalgebraic logics. Another significant result is that the two companion logics coincide if and only if one of them en- joys the characteristic property of the other, that is, if and only if the semilattice-based logic is algebraizable, and if and only if its assertional companion is selfextensional. When these condi- tions are met, then the (unique) logic is finitely, regularly and strongly algebraizable and fully Fregean; this places it at some of the highest ranks in both the Leibniz hierarchy and the Frege hierarchy.

In memory of Wim Blok

Font J. M., Wroński A.

Reports on Mathematical Logic

2006

Nr 41,spec. iss.

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