Arrows Pointing at Arrows: Arrow Logic, Relevance Logic, and Relation Algebras

Abstract

Richard Routley and Robert K. Meyer introduced a ternary relational semantics for various relevance logics in the early 1970s. Johan van Benthem and Yde Venema introduced “arrow logic” in the early 1990s and about the same time I showed how a variation of the Routley–Meyer semantics could be used to provide an interpretation of Tarski’s axioms for relation algebras. In this paper I explore the relationships between the van Benthem–Venema semantics for arrow logics, and the Routley–Meyer semantics for relevance logic, and conclude with a comparison between van Benthem’s version of the semantics for arrow logic aimed at relation algebras, and my own version of the Routley–Meyer semantics which I used to give a representation of relation algebras (but at a type level higher than Tarski’s original intended interpretation of an element as a relation, for me it is a set of relations). In the process I show how van Benthem’s semantics for arrow logic can be just slightly tweaked (just one additional constraint) so as to give a representation of relation algebras.