  The do-calculus is a well-known deductive system for deriving connections
between interventional and observed distributions, and has been proven complete
for a number of important identifiability problems in causal inference.
Nevertheless, as it is currently defined, the do-calculus is inapplicable to
causal problems that involve complex nested counterfactuals which cannot be
expressed in terms of the "do" operator. Such problems include analyses of
path-specific effects and dynamic treatment regimes. In this paper we present
the potential outcome calculus (po-calculus), a natural generalization of
do-calculus for arbitrary potential outcomes. We thereby provide a bridge
between identification approaches which have their origins in artificial
intelligence and statistics, respectively. We use po-calculus to give a
complete identification algorithm for conditional path-specific effects with
applications to problems in mediation analysis and algorithmic fairness.
