RELEVANCE LOGICS AND RELATION ALGEBRAS

Abstract

Relevance logicsare known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds ofdynamic semantics(i.e., proper relation algebras and relevant families of relations). We prove severalsoundnessresults here. We also prove thecompletenessof a certain positive fragment ofRas well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define anew tableauxformalization and newsequent calculi(with the single cut rule admissible) for various relevance logics.