Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish 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 \(\mathcal {A}\) 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].

在这里，我展示了几个一阶相关逻辑的单变量片段对应于底层命题相关逻辑的某些S5 ish 扩展。特别是，给定模态语言和单变量语言之间相当标准的翻译以及置换命题相关逻辑L ，单变量片段的公式\(\mathcal {A}\)是LQ ( QL ) 的定理，当且仅当平移是L5定理（L.5）。证明是模型理论的。在一个方向上，基于 Mares-Goldblatt [15] 量化L语义的语义被转换为L的类似S5扩展的三元（加两个二元）关系语义（有关一般介绍，请参见 Seki [26, 27]） 。另一方面，对基于L 的完整一阶相关逻辑进行评估，将其转化为L的合适S5扩展的模型。我还讨论了这项工作与寻找恒定域的完整公理化、非通用框架三元关系语义的关系，其中RQ是不完整的 [11]。