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.
翻译:本文研究了贴有标签和嵌套的计算法之间的关系,以得出理论直觉逻辑,一阶直觉逻辑与非恒定域的关系,以及一阶直觉逻辑与恒定域的关系。 该文件表明,Fitting的嵌套计算法自然产生于其对应的标有标签的计算法,适用于上述每种逻辑,即消除有标签的衍生物的结构规则。这两类系统之间的翻译对应法被利用来表明,嵌套的计算法继承了与其关联的标有标签的计算法的校对理论特性,如完整性、规则的可逆性和可接受性。 由于标有标签的计算法很容易通过逻辑的语义学获得,本文中介绍的方法可以被视为一种方法,根据逻辑的语义直接衍生出具有有利特性的精细的标有标签的计算法(将嵌套的计算法作为碎片)。