Jaynes' book PT:LoS has a good chapter on this, where he derives Bayes' theorem from simple assumptions (use of numbers to represent plausibility, consistency between paths that compute the same value, continuity, and agreement with common sense qualitative reasoning). The assumptions are sound.

Note that the validity of Bayes' theorem is a separate question from the validity of any particular set of prior probabilities, which is on much shakier ground.