The CAUSALGRAPH Procedure
Causal Graphs and Potential Outcomes
The CAUSALGRAPH procedure emphasizes the graphical or structural model framework of causality as developed by Pearl, among others (Pearl 2009b). This is in contrast with the Neyman-Rubin potential outcomes framework (Neyman, Dabrowska, and Speed 1990; Rubin 1980, 1990). Although the notation differs, these two frameworks are equivalent in the sense that any theorem that can be proved in one framework can also be proved in the other framework (Galles and Pearl 1998; Elwert 2013). For an extended discussion, see Pearl (2009b, chap. 7) and Pearl (2012).
For a single outcome variable Y and a single treatment variable X, the potential outcome Y(x) is a random variable that describes the possible values of the outcome variable for an experimental unit that is associated with the treatment X=x. The identity
establishes a map between the potential outcomes framework and the structural model framework (Pearl 2009b).
The do-operator is meant to emphasize the interpretation of a causal effect as the effect of an action or intervention (Pearl 2009b). That is, the quantity 
reflects the distribution of the outcome variable that would result from the hypothetical act of intervening and imposing the condition X=x (Elwert 2013).
The correspondence between potential outcomes and the do-operator, together with the stratification estimator, indicates another important link between the potential outcomes framework and the structural model framework (Elwert 2013). In particular, the conditional ignorability of treatment assignment (in the potential outcomes framework) is exactly equivalent to the criterion for the existence of an adjustment set. That is, 
if and only if Z satisfies the adjustment criterion (Shpitser, VanderWeele, and Robins 2010). For more information about the adjustment criterion, see the section Identification by Adjustment. Significantly, the adjustment criterion involves only observed quantities, whereas conditional ignorability requires reasoning about counterfactuals that might be unobserved (Elwert 2013).