One Variable Relevant Logics are S5ish

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].