Science approximates that method. Most scientists don't explicitly assign their hypotheses prior probabilities and then use Bayesian updating on the results of their experiments, but their brains have to assign different levels of credence to different hypotheses (to determine which ones get the attention) and adjust those credence levels as new data comes in, and the larger scientific community performs a similar process to determine consensus; Bayes-structure is implicit in there even if you're not actually using probability math.

http://en.wikipedia.org/wiki/Scientific_method

I think your third idea is right: science cannot be used to justify its own fundamental principles. For one thing, that argument would be very circular.