 Whether Pearl provides the correct causal explanation of Simpson’s Paradox remains a topic of continued debate (Armistead 2014 see also  Section 4).  What should not, however, be controversial is that recent causal modeling techniques enable one to systematically distinguish between causal and probabilistic claims in a much more general and precise way than had previously been possible. While Cartwright required that all causes of \(E\) be included in the background context, for the sake of eliminating confounding it is only necessary to hold fixed common causes (and other variables needed to block back-door paths). Theorists of probabilistic causality were to some extent aware that one did not need to hold fixed all causes of the effect in order to eliminate confounding, but they lacked a general account of which variable sets are sufficient for identifying the effect. Simpson’s Paradox was especially threatening, since there was no way to provide general conditions under which an apparent positive causal relationship in a population would disappear entirely upon partitioning. Using Pearl’s framework, it is trivial to show that as long as one does not condition on mediators, if a probabilistic expression identifies an average positive effect between \(X\) and \(Y\) in a population, intervening on \(X\) must raise \(Y\)’s effect in at least some subpopulations (Weinberger 2015).