A suggested answer to this problem (one I personally disagree with) is to use a basic system of fuzzy logic to accelerate the process of coming to an agreement. The idea behind this method was that you would admit that you probably were't 100% correct, and so give your arguments a value of (0.95) true, and the oppositions an appropriate value. Then, working through slowly and calmly, re-assign values as the conversation goes on (so that both sides can hear the progress numerically), until, probably, both sides are at 0.5:0.5, by which time, I'd imagine, an agreement would have been reached.

This solution to one of the problems with logic would probably only be prevalent among mathematicians. Most people wouldn't admit that it's sometimes easier with numbers.