$a'$ for elements in $A$. For any $V_i \in V$, define the potential outcome $V_i(\pi,a,a')$ by setting $A$ to $a$ for the purposes of paths in $\pi$, and to $a'$ for the purposes of proper causal paths from $A$ to $Y$ not in $\pi$. Formally, the definition is as follows, for any $V_i \in V$, $V_i(\pi, a, a') \equiv a \text{ if }V_i \in A$, otherwise [$$\begin{aligned}
    \label{eqn:pse}
    V_i(\pi, a, a') \equiv & V_i \Big( \big\{ V_j(\pi, a, a') \mid V_j \in \operatorname{pa}_{\cal G}^{\pi}(V_i) \big\}, \nonumber \\ 
    & \hspace{2cm}  \big\{ V_j(a') \mid V_j \in \operatorname{pa}_{\cal G}^{\overline{\pi}}(V_i) \big\} \Big)\end{aligned}$$ ]{}where $V_j(a') \equiv a'$ if $V_j \in A$, and given by recursive substitution otherwise, $\operatorname{pa}_{\cal G}^{\pi}(V_i)$ is the set of parents of $V_i$ along an edge which is a part of a path in $\pi$, and $\operatorname{pa}_{\cal G}^{\overline{\pi}}(V_i)$ is the set of all other parents of $V_i$.

A counterfactual $V_i(\pi, a, a')$ is said to be *edge inconsistent* if counterfactuals of the form $V_j(a_k, \ldots)$ and $V_j(a_k', \ldots)$ occur in $V_i(\pi, a, a')$, otherwise it is said to be *edge consistent*. It is well known that a joint distribution $p(V(\pi, a, a'))$ containing an edge-inconsistent counterfactual $V_i(\pi, a, a')$ is not identified 