Grounding operators: transitivity and trees, logicality and balance

We formally investigate immediate and mediate grounding operators from an inferential perspective. We discuss the differences in behaviour displayed by several grounding operators and consider a general distinction between grounding and logical operators. Without fixing a particular notion of grounding or grounding relation, we present inferential rules that define, once a base grounding calculus has been fixed, three grounding operators: an operator for immediate grounding, one for mediate grounding – corresponding to the transitive closure of the immediate grounding one – and a grounding tree operator, which enables us to internalise chains of immediate grounding claims without losing any information about them. We then present an in-depth proof-theoretical study of the introduced rules by focussing, in particular, on the question whether grounding operators can be considered as logical operators and whether balanced rules for grounding operators can be defined.

本研究从推理视角出发，对直接奠基（immediate grounding）与间接奠基（mediate grounding）算子展开形式化探究。我们探讨了多种奠基算子所展现出的行为差异，并对奠基算子与逻辑算子之间的一般性差异进行了辨析。在未限定特定奠基概念或奠基关系的前提下，我们给出了一套推理规则：在确立基础奠基演算（grounding calculus）的基础上，该规则可定义三类奠基算子：直接奠基算子、对应直接奠基算子传递闭包的间接奠基算子，以及可将直接奠基断言链内化且不丢失任何相关信息的奠基树算子。随后，我们对所提出的规则展开了深入的证明论研究，尤其聚焦于两个核心问题：其一，奠基算子能否被视作逻辑算子；其二，能否为奠基算子定义平衡性的推理规则。