$\gamma $-ADMISSIBILITY IN FIRST-ORDER RELEVANT LOGICS: PROOF USING NORMAL MODELS IN THE MARES–GOLDBLATT SETTING

Abstract For relevant logics, the admissibility of the rule of proof $\gamma $ has played a significant historical role in the development of relevant logics. For first-order logics, however, there have been only a handful of $\gamma $ -admissibility proofs for a select few logics. Here we show that, for each logic L of a wide range of propositional relevant logics for which excluded middle is valid (with fusion and the Ackermann truth constant), the first-order extensions QL and LQ admit $\gamma $ . Specifically, these are particular “conventionally normal” extensions of the logic $\mathbf {G}^{g,d}$ , which is the least propositional relevant logic (with the usual relational semantics) that admits $\gamma $ by the method of normal models. We also note the circumstances in which our results apply to logics without fusion and the Ackermann truth constant.