Let's assume there exist two groups of people. The members of group A think that some improved product is better because it tastes great. The members of group B, on the other hand, believe that the same product is better because it's now less filling. Keeping in mind that the product is better because it both tastes great, and is less filling. Which group is mistaken? Which group would benefit from changing their minds?

Their ideas don't really contradict. But they think they do because, apparently, both sides have the mistaken view that a product has to be better for exactly one reason.

My example was obviously a gross simplification, but not all disagreement stems from contradiction. Worse, sometimes when people agree on many points they are in inreconcilable disagreement with each other on others. If we were to consider that both groups have held their position for many years, it should be immediately apparent that it would be impossible for either to change their mind. Time calcifies opinion.

You are using the word "impossible" to mean "very hard", and I am not.

You're right, though I think I would say "extremely difficult, tending toward impossible". People do not change their minds easily, especially on subjects they feel a strong emotional attachment to.