If I am certain that p but another person is certain that ~p, does the bare fact of his certainty give me reason to doubt my own?

This is actually a fascinating question. I think the answer must surely depend on the circumstances. Why, precisely, do you feel so certain that p is true? There are a couple of possible answers to this. It could be that p just strikes you as so obvious that it has never really occurred to you to question it at all. Simply in order for us to be able to live our lives and get around in the world, we need to believe innumerable things, far too many for us ever to conduct a thorough investigation into the evidence for each and every one. It's perfectly normal for us to feel no doubt whatsoever about many things, for no other reason than that we've heard them repeated over and over without ever encountering a dissenting voice. And this shouldn't be regarded as an epistemic failing: but, unfortunately, such beliefs do sometimes turn out to be false. A case which is often presented to illustrate this fact is that, back in the Middle Ages, the predominant opinion was that the Earth was flat; a...