Of course, I could totally agree with what you are saying, and just be arguing to defend the opposing point. If so, I may be totally disproving your point, which would invalidate my position too. So if I do agree with you and my possibly-attempted counterattack works, I would be confusing myself thoroughly – If You and I both think that it is impossible to argue something you don’t believe in, and I successfully convince you that it is possible to argue something I don’t believe in, I have then argued something I don’t believe in, thereby disproving my point. In which case I suppose I would believe that it is possible, causing that I argued for something I did hold of, which makes me lose any legitimate source to my claim that it is possible.
Oh, I don’t know…