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 loosing any information about them. We then present an in-depth proof-theoretical study of the introduced rules by focusing, in particular, on the question whether grounding operators can be considered as logical operators and whether balanced rules for grounding operators can be defined.