Abstract

This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

中文翻译：

---

直觉逻辑的嵌套计算与语义系统之间的对应关系

摘要

本文针对命题直觉逻辑，具有非恒定域的一阶直觉逻辑和具有恒定域的一阶直觉逻辑，研究了标记和嵌套计算之间的关系。结果表明，通过消除标记派生中的结构规则，Fitting的嵌套结石自然来自于它们对应的标记结石（对于上述每种逻辑）。两种类型系统之间的翻译对应关系被利用来表明嵌套的结石从其关联的带标签的结石继承了证明理论特性，例如完整性，规则的可逆性和切割可采性。由于可以通过逻辑的语义轻松获得标记的结石，