Jaegwon Kim is often viewed as having proposed a dilemma for Donald Davidson: if when Davidson appealed to psychophysical supervenience in “Mental Events,” his appeal was to weak supervenience, then he failed to state how mental properties depend on physical properties; and if his appeal was instead to strong supervenience, then he was appealing to a thesis that is incompatible with anomalous monism. I examine this dilemma in detail, pointing out that it is actually a dilemma for the doctrine of anomalous monism itself. I ultimately argue that it is a false dilemma. But I then state a related dilemma that poses a serious, unresolved prima facie problem for anomalous monism.