How do you solve Hempel's Dilemma?

Here.  How you solve it is important  because it affects the strength of the argument from past explanatory successes, which Danaher dealt with first. The reason for this we need to give an account of what it is for something to be a success for physicalistic explanations. It would have been thought, prior to the discovery of quantum mechanics, that physicalistic explanation means deterministic explanation, but that got abandoned when quantum mechanics came along. So, not everything turned out to be physicalistically explicable if pre-quantum expectations are presupposed. Similarly, the discovery of a beginning of the universe would have been thought to have been an explanatory failure for physicalistic explanation from the point of view of the people prior to the discovery of Big Bang cosmology. What this suggests is that we ought to be rather cautious as to what exactly we are claiming when we say that the mind-body problem will have a physicalistic explanation. What might turn out to be the case is everything that the dualist thinks is true of the mind really is true, but since neuroscience discovers this, the "soul" gets built into physics, and physicalism "triumphs" after all.