Then the counterfactual conditional $~$\\mathbb P(\\mathbf x | \\operatorname{do}(X\_j\=x\_j))$~$ is calculated via:

This "do" notation may seem mysterious, as it is not part of standard probability theory. However, as Pearl shows in Section 3.2.2 ("Interventions as Variables") of *Causality*, second edition, all of this can be treated as a notational convenience, as a causal model can be reduced to a certain kind of PGM with variables for interventions, and the "do" notation (or lack of a "do") can be considered to be a statement about the value of an intervention variable.

See also this Powerpoint deck.