I think this fits in the politics section most of all since it mostly considers debates, which are a big part of politics. Since I have arrived on this grand forum, I have frequently found situations where I would have to refer on the apparent irresolvability of debates, that one cannot logically argue another out of his or her point. A current professor of epistemology, Markus Lammenranta, has considered this issue from the view point of skepticism. He points out the modes given by an philosopher Agrippa who was of the Academic Skepticism vein (not really the Pyrrhonic as labeled although the Pyrrhonic Skeptics did use the development of the Academic Skeptics). In his paper he goes through traditional academic style arguments for skepticism and show they generally do not hold, until he does a specific argument for Agrippa's arguments. You can find his paper here.
http://www.absolutewrestling.net/youtube/lammenranta.pdf
I'll leave this paper up for a little while, but eventually I will delete it. I found it as an preliminary article for the book
Oxford Handbook of Skepticism. It is quite a good book on modern interpretations of Skepticism. Anyway the relevant part of the paper begins on Page 8. I will do a little summary below. So Lammenranta identifies Agrippa's modes as follows.
Suppose we have a question to which there are just two possible answers
p and ~p. Then the mode of Disagreement works as follows:
(1) S1 believes that p.
(2) S2 believes that ~p.
(3) At most, one of them is right.
(4) The disagreement between S1 and S2 is irresolvable.
(5) We should suspend judgment about p.
Markus Lammenranta goes on to defend counters to parts 3 and 4. So if one counters (4) and says it is resolvable, a reason r would be given, and one could simply disagree with r, and send the debate into infinite regress (or circularity if one goes back to a reason already used).
One could disagree with part 3 and say both of them are right in their own way as everything is relative (called Relativism). But as Lammenranta points out below it is faulty.
The traditional objection is that relativism is self-referentially incoherent: When the relativist says that relativism is true, she may mean that relativism is absolutely true or that relativism is relatively true. In the former case, she contradicts her own view that truth is relative. In the latter case, she acknowledges that relativism is true only for the relativist. It is not true for the absolutist. This is dialectically inefficient. She cannot argue the absolutist out of his view.
According to Lammenranta we cannot solve this disagreement without begging the question and accepting our belief that way. In terms of arguments, begging the question is where an argument takes for granted which it sets out to prove. Such an argument could be the apple is red because it is red or this belief is valid because the belief in itself is valid (although they can be more complex). As he points out if one simply accepts begging the question arguments as acceptable, then anything and everything can be defended as truth. Now he goes into a lot more that we may all read and discuss, but maybe this is enough for now to get things rolling
.