On Semilattice-based Logics with an Algebraizable Assertional Companion

Josep Maria Font

Titles

Abstract

This paper studies some properties of the so-called semilattice-based logics (which are defined 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 (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 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 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.

Keywords

semilattice-based logics, assertional logic, frege hierarchy

References

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Information

Information: Reports on Mathematical Logic, 2011, Number 46, pp. 109 - 132

DOI: https://doi.org/10.4467/20842589RM.11.007.0285

Article type: Original article

Titles:

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On Semilattice-based Logics with an Algebraizable Assertional Companion

English:

On Semilattice-based Logics with an Algebraizable Assertional Companion

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Josep Maria Font

University of Barcelona, Barcelona, Spain

Published at: 15.12.2011

Article status: Open

Licence: None

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Josep Maria Font (Author) - 100%

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