Much philosophical thinking requires a solid grasp of the difference between what's true and what's demonstrable. This doesn't always come naturally to students. (For example, many seem to confuse the questions of whether an argument is sound, and whether it can be shown to be sound.) In such cases, what are the best pedagogical methods an instructor could use to help struggling students acquire the skill of working with this distinction in a philosophically mature way?
I guess that will depend on the nature of their resistance. It could stem from either:
(i) difficulty grasping the objective metaphysical concepts (truth, facts, etc.) as distinct from the more subjective epistemic ones (proof, demonstration, etc.); or
(ii) skepticism as to whether the former could ever matter without the latter.
The first problem seems like it should be easy enough to overcome with a few examples: we can't tell whether there are an odd number of stars in the sky, but there's presumably some fact of the matter nonetheless.
The second is trickier. I guess one way to bring out the importance is to note the important difference between "not certain" and "certainly not". That is, it's one thing to be unsure whether an argument is sound or not (because you can't decisively determine the truth or falsity of the premises), but it's quite another to claim that it is definitely unsound. But we need the metaphysics-epistemology distinction to make sense of this difference. If there's no difference between truth and proof, then one's inability to prove a premise would allow one, absurdly, to infer that it must be false.
A more complicated example is the epistemic externalist's response to radical skepticism. More generally: everyone agrees that if I'm actually deceived by an evil demon then I thereby lack knowledge. So the controversy instead concerns the epistemic status of my beliefs in those cases in which they are true. That is, to properly assess radical skepticism, one must consider a scenario in which the subject's beliefs are true, but where this fact is not absolutely transparent to the subject (since, from her perspective, she can't rule out the possibility of the evil demon). We may find -- especially if we accept an externalist theory of knowledge -- that the subject can count as having knowledge in this scenario after all. That suffices to refute the radical skeptic's claim that knowledge is impossible -- that we lack knowledge even if our beliefs happen to be true. But note that we never could have gotten this far if we didn't take care to distinguish the epistemic and metaphysical aspects of the problem, i.e. the difference between what's "not certain" and what's simply "not".
So that's one example of how it can be philosophically valuable to abstract away to a kind of "God's eye view" whereby we can consider what's true in a scenario independently from the question of what anybody in the scenario can conclusively demonstrate to be true. Can anyone suggest some other (perhaps better) examples of this philosophical method?