One Variable Relevant Logics are S5ish

Abstract Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5 ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula A A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L. 5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt 15 semantics for quantified L are transformed into ternary (plus two binary) relational semantics for S5 -like extensions of L (for a general presentation, see Seki 26, 27). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete 11.