The proper way to rescue the probability-raising idea is with the do-operator: we can say that X causes Y if P(Y | do(X)) > P(Y). Since intervention is a rung-two concept, this definition can capture the causal interpretation of probability raising, and it can also be made operational through causal diagrams. In other words, if we have a causal diagram and data on hand and a researcher asks whether P(Y | do(X)) > P(Y), we can answer his question coherently and algorithmically and thus decide if X is a cause of Y in the probability-raising sense.
