I think they are impossible. Morality can say "no option is right" all it wants, but we still must pick an option, unless the universe segfaults and time freezes upon encountering a dilemma. Whichever decision procedure we use to make that choice (flip a coin?) can count as part of morality.
I take it for granted that faced with a dilemma we must do something, so long as doing nothing counts as doing something. But the question is whether or not there is always a morally right answer. In cases where there isn't, I suppose we can just pick randomly, but that doesn't mean we've therefore made the right moral decision.
Are we ever damned if we do, and damned if we don't?
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