Semantics for a Basic Relevant Logic with Intensional Conjunction and Disjunction (and some of its Extensions)

This paper proposes a new relevant logic B+⊓⊔, which is obtained by adding two binary connectives, intensional conjunction ⊓ and intensional disjunction ⊔, to Meyer–Routley minimal positive relevant logic B+, where ⊓ and ⊔ are weaker than fusion ◦ and fission +, respectively. We give Kripke-style semantics for B+⊓⊔, with →, ⊓ and ⊔ modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B+⊓⊔, including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained.