Nikhil Pal Singh wrote about an "exit" - a "representation of some credible alternative." This is an important idea, but I think it also applies to persuasive argument. Okay, do you remember that argument you had with that asshole who sat behind you in Spanish class or some person on the bus that you wished had just left you alone? Now, you had a really sound argument. You whipped out all of the right statistics and facts to prove their thesis wrong. And, the person you were arguing with was classically educated enough to understand what you were saying and the implications of your argument. But, still, they didn't agree with you. They didn't admit that they were wrong. Why? Because the point of this conversation was probably not to have an intellectual debate of ideas but to prove that the other person was wrong. You did this: congratulations. But nobody likes to admit that they were wrong. Instead, this person that you were talking with is now quickly backtracking to the land of "who are you to tell me this?" and something about an eternal deity permitting or not permitting whatever it is.
If the point of a discussion is to prove that you are right (and, therefore, that the other person is wrong), then, even if you "win," there will be no consensus between you. If, instead, you present your facts amicably and acknowledge some of the other person's points as valid but show them gently why they are incorrect, then perhaps this person might agree with you. In essence, you have provided this person with an exit.
I think a lot of political issues could be resolved if people were less keen on winning and more focused on change. To do this, the party with the weaker facts must be let down gently and not made out to be an idiot. As Singh says, "It isn't legislation. It's presentation."