 This last point is a very general, very important phenomenon. Like the axioms of first-order logic, the axioms of probability are quite weak (Howson and Urbach 1993; Christensen 2004). Unless \(H\) is a tautology or contradiction, the axioms only tell us that its probability is somewhere between \(0\) and 1. If we can express \(H\) as a disjunction of two logically incompatible sub-hypotheses, \(H_1\) and \(H_2\), and we know the probabilities of these sub-hypotheses, then the third axiom lets us compute \(p(H) = p(H_1)+p(H_2)\). But this just pushes things back a step, since the axioms by themselves only tell us that \(p(H_1)\) and \(p(H_2)\) must themselves lie between \(0\) and 1.