Beth Definability in the Logic KR

TL;DR

This paper proves that the Beth Definability Property fails for the logic KR by demonstrating non-surjective epimorphisms in related algebraic structures, extending previous results to a broader class of relation algebras.

Contribution

It establishes the failure of Beth Definability in the logic KR through algebraic methods, specifically using modular lattices and relation algebras.

Findings

Beth Definability fails for KR

Epimorphisms are not always surjective in related algebras

The result extends to Boolean monoids and related structures

Abstract

The Beth Definability Property holds for an algebraizable logic if and only if every epimorphism in the corresponding category of algebras is surjective. Using this technique, Urquhart in 1999 showed that the Beth Definability Property fails for a wide class of relevant logics, including T, E, and R. However, the counterexample for those logics does not extend to the algebraic counterpart of the super relevant logic KR, the so-called Boolean monoids. Following a suggestion of Urquhart, we use modular lattices constructed by Freese to show that epimorphisms need not be surjective in a wide class of relation algebras. This class includes the Boolean monoids, and thus the Beth Definability Property fails for KR.