Often in a conversation we are proven wrong. We could be wrong in our facts. Or we could be wrong (different) in our opinion. When wrong in facts, humility is the best approach. Accept your mistake. Unfortunately many people argue their false correctness because they find it a bit difficult to accept their statistical defeat. It only proves a point to their stubbornness, and rigidity. However, often you may have a different opinion. In such a case, you can stand by your opinion while allowing the other person to maintain his right to his opinion.
We often find the worth of a man how he reacts when he is proven wrong.