Argumentation can get tiresome. While the Socratic ideal would have us stay stoically emotionally uninvested in our points of view, and would keep us from resorting to ad hominem attacks, the ideal is a bit, well... idealistic. Sometimes, disagreement about facts just leads to charged conflict between people.
The best response to this, and here I really don't think I'm guilty of overambitious optimism, is to identify and clarify what the point at issue really is, and to concertedly stay conscious of the fact that people are not their individual beliefs, and that it is alright to have been wrong about facts. If people can manage that, they can have disagreements without conflict. Or, they can achieve the same effect if they simply accept that conflict is not necessarily bad so long as it ends and has a reasonable expectation of getting somewhere.
But most people are not so high-minded. Much of the time, people simply want to bury the hatchet (which is perfectly well admirable, except that a disagreement is really a lot more like a land mine than a hatchet in that regard).
So they resort to the old chestnut of "Why don't we agree to disagree." (I neglect to use a question mark because it is never said with that inflection. It is not really a question, it is a demand, and a cravenly presented one at that. Impolite, really.)
But it's rather anticlimactic, isn't it? I am reminded of a joke: "There are two types of people: those who don't need closure."
In fact, the nature of disagreements, if they are genuine disagreements is that we disagree whether we assent to it or not. To disagree, at least two people must have at least two beliefs, and those beliefs must be incompatible.
Things are incompatible by virtue of their containing a contradiction. Contradictions are endemic to the concepts themselves. If I happen to believe that all school buses are red, and you believe that they are all yellow, our agreeing to disagree affects the facts that we have incompatible views not at all.
Not to be overly literal. I know that people who say this don't think that it really settles the discussion. They want merely to avoid conflict. But it is the height of bad manners to not only interrupt somebody, but also to interrupt them to tell them that they can not and may not go on expressing themselves, lest they offend the holy rules of social grace and the satisfy the insane need of the un-argumentative to "keep everyone happy".
Who, anyway, is the graceless and conflicted one in the situation, he who happens to have an opinion which is conceptually incompatible with someone else's, or he who would rather censor a discussion than be challenged?
The dissonance that goes with disagreement is a tension, sure. But it is certainly better to relieve a tension than to ignore it.
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