I think this is an interesting question. If the arguer is cherry-picking evidence, we should ignore that to a large degree. We are often even justified in updating in the opposite direction of a motivated argument. In the pure mathematical case, it doesn't matter anymore, so long as we are prepared to check the proof thoroughly. It seems to break down very quickly for any other situation, though.
In principle, the Bayesian answer is that we need to account for the filtering process when updating on filtered evidence. This collides with logical uncertainty when "evidence" includes logical/mathematical arguments. But, there is a largely seperate question of what we should do in practice when we encounter motivated arguments. It would be nice to have more tools for dealing with this!
Yes, this in an interesting issue. One unusual (at least, I have not seen anyone advocate it seriously elsewhere) perspective is that mentioned by Tyler Cowen here. The gist is that in Bayesian terms, the fact that someone thought an issue was important enough to lie about is evidence that their claim is correct.
Abram Demski and Grognor
Much of rationality is pattern-matching. An article on lesswrong might point out a thing to look for. Noticing this thing changes your reasoning in some way. This essay is a list of things to look for. These things are all associated, but the reader should take care not to lump them together. Each dichotomy is distinct, and although the brain will tend to abstract them into some sort of yin/yang correlated mush, in reality they have a more complicated structure; some things may be similar, but if possible, try to focus on the complex interrelationships.