I could never quite articulate why I felt that way, but now I think I can. And I have Dale Carnegie to thank for that:
Nine times out of ten, an argument ends with each of the contestants more firmly convinced than ever that he is absolutely right.
You can't win an argument. You can't because if you lose it, you lose it; and if you win it, you lose it. Why? Well, supposed you triumph over the other man and shoot his argument full of holes... Then what? You will feel fine. But what about him? You have made him feel inferior. You have hurt his pride. He will resent your triumph. And-
A man convinced against his will
Is of the same opinion still.
Ben Franklin illustrates it with this trade-off:
If you argue and rankle and contradict, you may achieve a victory sometimes; but it will be an empty victory because you will never get your opponent's good will.
So figure it out for yourself. Which would you rather have: an academic, theatrical victory or a person's good will? You can seldom have both.
Lincoln chimes in with what to do instead of arguing:
No man who is resolved to make the most of himself can spare time for personal contention, still less can he afford to take the consequences, including the vitiation of his temper and the loss of self control. Yield to larger things to which you show no more than equal rights, and yield to lesser ones though clearly your own. Better give your path to a dog, than be bitten by him in contesting for the right. Not even killing the dog will cure the bite.
As Dale says, will proving someone wrong make him want to agree with you? Of course not! You just made him hurt and angry. In other words, you've evoked emotions. Negative ones. And once those are in play, all the reason and logic in the world won't do any good.