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jsteinhardt comments on Original Research on Less Wrong - Less Wrong
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No frequentist says this. They don't believe in P(H|E). That's the explicit basis of the whole philosophy. People who talk about the probability of a hypothesis given the evidence are Bayesians, full stop.
Statistical significance is, albeit in a strange and distorted way, supposed to be about P(E|null hypothesis), and so, yes, two experiments with a p-value of 0.05 should add up to somewhere in the vicinity of p < 0.0025, because it's about likelihoods, which do multiply, and not posteriors.
While some frequentist methods do use likelihoods, the mapping from likelihood to p-value is non-linear, so multiplying them would still be a mistake, at least as far as I can tell.