Two theorems on invariance and causality

In much recent work, invariance under intervention has become a hallmark of the correctness of a causal-law claim. Despite its importance this thesis generally is either simply assumed or is supported by very general arguments with heavy reliance on examples, and crucial notions involved are characterized only loosely. Yet for both philosophical analysis and practicing science, it is important to get clear about whether invariance under intervention is or is not necessary or sufficient for which kinds of causal claims. Furthermore, we need to know what counts as an intervention and what invariance is. In this paper I offer explicit definitions of two different kinds for the notions intervention, invariance, and causal correctness. Then, given some natural and relatively uncontroversial assumptions, I prove two distinct sets of theorems showing that invariance is indeed a mark of causality when the concepts are appropriately interpreted.