Some people like to argue. You can almost count on them taking the other side of anything you say. Of course, it can be easy to manipulate those who are predictable in this way, like you might use reverse psychology on a stubborn child.
One of the more frustrating things of my life, since I love ideas, is that it often isn't possible to win a debate. I wish God would swoop in sometimes and pronounce a winner. You know, the end of Job type thing where God comes in and says, "Where were you when I plucked Leviathan out with a fishhook?"
You can be completely convinced of something and the majority think you're crazy or evil... and of course you may be. Those who are wrongly convinced or paranoid or schizophrenic may be completely convinced that they are seeing the rest of the world as it is. At the same time, the average intelligence of the world is, well, average. So it is likely in many circumstances that as far as the truth is concerned, "narrow is the way and few there be who find it."
It is immensely frustrating that God doesn't show up to say, "And the winner is..." We are left to how convinced we remain after an argument and, perhaps more importantly, when you count up all the people standing by, who do they think is right. We've all voted for the person who lost the election. So we all probably agree that the majority isn't always right.
Even at the Society of Biblical Literature, where people of the highest biblical knowledge are debating, I often hear papers that I think are complete and utter nonsense. What world are they from? Having a large church means nothing as far as the truth is concerned. Crowds flock to rock concerts and all sorts of things, including Hitler rallies.
So why argue? There's no point in arguing unless it leads to some positive consequence. "You can't win an argument," so if the person you are arguing with is the only person involved and you're not going to have any effect on him or her, then just walk away.
Sometimes you can have a good effect on the people listening to the debate. That's a reason to push back. Rarely does putting people in their place work and, perhaps more often, it backfires.
Basically, don't argue because you think you're right. As logical as you might think you are, that's illogical. Argue if you think you might actually change something for the better. If not, just walk away.