An argument doesn't have to end in agreement for there to be a resolution.
Sometimes, no matter what is said or how it is said, you just continue to see things differently from the other person. That's OK. If you were able to get clear in your position and the other person was able to get clear in his/her position, sometimes that is the best that you can do.
Yes, it would feel better if you could agree. Yes, a true agreement would take up much less mental energy than continuing to disagree (and trying to figure out a way to make the other person agree with you). But sometimes it just isn't possible.
Sometimes the best you can do is to agree to disagree and move on from there.