ON SEMILATTICE-BASED LOGICS WITH AN ALGEBRAIZABLE ASSERTIONAL COMPANION

Abstract

This paper studies some properties of the so-called semilattice-based logics (which are dened in a standard way using only the order relation from a variety of algebras that have a semilattice reduct with maximum) under the assumption that its companion assertional logic (dened 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 lters, a fact previously known in the literature to occur only for protoalgebraic logics. Another signicant result is that the two companion logics coincide if and only if one of them enjoys 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 conditions are met, then the (unique) logic is nitely, 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. Let K be a class of algebras of the same (arbitrary) similarity type. The class K is said to be semilattice-based when each algebra in K has a semilattice reduct uniformly dened