Christopher Hill and Joseph Levine have argued that the conceivabilitiesinvolved in anti-materialist arguments are defeated as evidence ofpossibility. Their strategy assumes the following principle: theconceivability of a state of affairs S constitutes evidence for thepossibility of S only if the possibility of S is the bestexplanation of the conceivability of S. So if there is a betterexplanation of the conceivability of S than its possibility, then theconceivability of S is thereby defeated as evidence of possibility. Hilland Levine proceed to offer alternative explanations of theseconceivabilities, concluding that these conceivabilities are therebydefeated as evidence. However, this strategy fails because theirexplanations generalize to all conceivability judgments concerningphenomenal states. Consequently, one could defend absolutely any theoryof phenomenal states against conceivability arguments in just this way.This result conflicts with too many of our common sense beliefs aboutthe evidential value of conceivability with respect to phenomenalstates. The general moral is that the application of such principles ofexplanatory defeat is neither simple nor straightforward.