Arguing something like this we sometimes confuse the need to win with the need to come to an understanding (note I didn't say agreement).
If you present your case in the following format:
"I win, you lose" -- in other words you say in plain english "I'm right, you're wrong" that automatically sends someone on the defensive! I've been guilty of this from time to time but I'm just observing here
Perhaps try to simply state your points of view and see what is in common WITHOUT trying to say something someone else believes is outright wrong. Just say how you differ and why.
Otherwise it's just a fight.. no one wants that.. I hope?