A Representation of Relation Algebras Using Routley-Meyer Frames

Abstract

A representation of relation algebras is given by modifying the Routley-Meyer semantics for relevance logic. This semantics is similar to the Kripke semantics for intutionistic logic, but its frames use a ternary accessibility relation instead of a binary one. The representation is foreshadowed in the work of Lyndon as well as by the representation of Boolean algebras with operators by Jonsson and Tarski, but the aim here is to make it explicit and to connect it to the related representations of algebras arising in the study of relevance logic as well as other “substructural logics,” e.g., linear logic. A philosophical interpretation is given of the representation, showing that an element of a relation algebra can be understood as a set of relations, rather than as the intended interpretation as a single relation. It is shown how a Routley-Meyer frame can be represented so its states are relations, and an interpretation is provided in terms of a “relational database”. Connections are mentioned to recent work by Jon Barwise on “information channels” between “sites” and by J. M. Dunn and R. K. Meyer on a ternary frame semantics for combinatory logic.