What this and your other comment make clear is that once you start talking about the probability that X is true, especially in the context of hypothesis testing, you've moved (usually unwittingly) into a Bayesian framing, and you better make your priors explicit.
What this and your other comment make clear is that once you start talking about the probability that X is true, especially in the context of hypothesis testing, you've moved (usually unwittingly) into a Bayesian framing, and you better make your priors explicit.