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$\gamma $-ADMISSIBILITY IN FIRST-ORDER RELEVANT LOGICS: PROOF USING NORMAL MODELS IN THE MARES–GOLDBLATT SETTING
- Part of
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- Journal:
- The Review of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 12 December 2024, pp. 1-22
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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.
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 
and 
admit 
