It definitely depends on the situation. If I feel someone's life (or lives) may be at stake, I will argue my point until I am proven wrong beyond a shadow of a doubt. Otherwise, it's not really worth arguing. Being 'right' is really overrated...if that's what will bring the other person happiness, then I'll confess to wrongs whether real or imagined. On the other hand, if I believe the key to the other person's true eternal happiness is learning how to deal with being wrong, I will again argue my point endlessly, whether it be baseless or not.
You learn from your mistakes...how do you think I got so smart?