A Note on the Relevance of Semilattice Relevance Logic

Abstract

A propositional logic has the variable sharing property if φ →ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R⁺_u) and its extension with an involution negation (R^¬_u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed.