New(ish) Foundations for Theories of Entailment

In this chapter, I present a systematic study of theories of entailment as developed in relevant modal logics of varying strength. I consider several relevant normal modal logics and the properties of entailment, as defined as necessitated relevant implication, in these systems. Fine-Urquhart style semantics are provided for these relevant modal logics and proofs of soundness and completeness are given. The semantics is used to prove admissibility results for certain weak relevant modal theories. Connections between some of these relevant modal logics and preexisting theories of entailment are pointed out.