Appeal to the Nobler Motives
I can see that you have a mind for fairness, so I ask that you consider my concerns carefully. I trust that, in the end, I can count on you to make the right decision. You are, of course, free to make up your own mind.
As Dale Carnegie rightly pointed out, people are not wired to believe they have made poor decisions. Although we are often proven wrong, we never think we are. After all, if we thought that we would have made a different decision in the first place.
The problem is, having once convinced ourselves of our correctness, it takes quite a shock for us to reject our initial point of view. Consider this video describing the McGurk effect. Even after we have presented incontrovertible evidence that we are wrong, our brain supplies an illusion that conforms what we are perceiving to our previous belief. In fact, we create a hallucination to allow the new fact to conform with our previous sense of reality.
Having seen the McGurk effect in action, it is worth pointing out that this is not a flaw that only applies to you (or to other people.) It is part of our wiring as humans. There is no point getting frustrated at someone who has taken a different point of view and won't listen to any other. In fact, telling them they are wrong will only cause them to look into their memory for evidence to the contrary, and push them farther away. They were made to be like that, and so were you. It is only through effort that we can overcome this natural tendency.
Carnegie tells us to "appeal to the nobler motives." Instead of demanding that they listen, compliment them on their willingness to consider other points of view, and ask if they will consider yours. In Pre-suasion, Dr. Robert Cialdini suggests that we not "so much tell people what to think as what to think about."
So what are these nobler motives to which we should appeal? In The Righteous Mind, Jonathan Haidt identifies six fundamental ideas that seem common to most cultures: care, fairness, liberty, loyalty, authority and sanctity. How each culture defines these fundamental ideas may vary, but the ideas themselves are shared. To find out why a person is motivated in a way that doesn't make sense to us, it is worth considering how their point of view may reflect one of these ideas. Then, by identifying how we relate to that same idea, we may be able to find common ground and perhaps even influence their perspective.
Even with this knowledge, however, it is important to be genuinely interested in the other person and learn which ideas they value most. There is no sense in appealing to sanctity if our debate partner values it much less than he does fairness. Still, an appeal to a fairly shallow belief in sanctity is nonetheless likely to have a better outcome than directly contradicting a deeply held belief in one's own rectitude.
Have you had success appealing to the nobler motives? Tell me about it in the comment section.