As soon as the oracle is trying to make predictions that are affected by what the oracle says, the problem she has to solve shifts from “estimate the probabilities” to “choose what information to give, so as to produce consistent results given how that information will affect what happens”. In some cases there might not be anything she can say that yields consistent results. Exactly where (if at all) that becomes impossible depends on the details of the situation.
As soon as the oracle is trying to make predictions that are affected by what the oracle says, the problem she has to solve shifts from “estimate the probabilities” to “choose what information to give, so as to produce consistent results given how that information will affect what happens”. In some cases there might not be anything she can say that yields consistent results. Exactly where (if at all) that becomes impossible depends on the details of the situation.