Research published last year and brought to my attention by Mind Hacks shows how this effect might help you convince people they are wrong.
Mind Hacks summarizes the work:
One group was asked to give their opinion and then provide reasons for why they held that view. This group got the opportunity to put their side of the issue, in the same way anyone in an argument or debate has a chance to argue their case.
Those in the second group did something subtly different. Rather than provide reasons, they were asked to explain how the policy they were advocating would work. They were asked to trace, step by step, from start to finish, the causal path from the policy to the effects it was supposed to have.
The results were clear. People who provided reasons remained as convinced of their positions as they had been before the experiment. Those who were asked to provide explanations softened their views, and reported a correspondingly larger drop in how they rated their understanding of the issues.