What the person arguing with you is really trying to do is protect their own sense of worth in one way or another so meet them half-way. Agree with what they are trying to say. By agreeing you will have given the person what they wanted enabling you to calmly express your own point of view without having to insist or prove anything. When you take this approach you not only go with the flow; you implement outer intention. The results will be beyond compare and far superior to anything you could have achieved via sophisticated intellectual contrivances.