 This study supports the conclusion that Hempel’s conception of scientific explanations as involving the subsumption of singular events by means of covering laws was well-founded, even though his commitment to an extensional methodology inhibited him from embracing a more adequate account of natural laws. The symmetry thesis turns out to require qualification, not only with respect to predictions for events that occur only with low probability, but also for retrodictions derived by modus tollens. The link that most accurately embodies the relationship between Hempel’s work on explanation and decision-and-inference, as we have seen, appears to be reflected by the Principal Principle, which formalizes the recognition that subjective (or personal) probabilities, as degrees of belief, should have the same values as corresponding objective probabilities, when they are known. The values of propensities as properties of laws, no doubt, should be given precedence over the values of frequencies, insofar as laws, unlike frequencies, cannot be violated and cannot be changed and thus provide the more reliable guide.