A Note on the Relevance of Semilattice Relevance Logic

  • Yale Weiss The Graduate Center, CUNY

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.

Published
2019-10-15
How to Cite
WEISS, Yale. A Note on the Relevance of Semilattice Relevance Logic. The Australasian Journal of Logic, [S.l.], v. 16, n. 6, p. 177-185, oct. 2019. ISSN 1448-5052. Available at: <https://ojs.victoria.ac.nz/ajl/article/view/5416>. Date accessed: 18 nov. 2019. doi: https://doi.org/10.26686/ajl.v16i6.5416.