It seems to me that there are chiefly four ways of responding to a remark that challenges one’s position:
Reactive: give in to your natural impulses and fight back. This is a most common response. When people say something you do not agree with, the natural instinct is to argue, to correct them, or even to fight them. However stupid and irrational a reactive response may be, it is the most common approach to human interactions. In general, such responses make the situation worse because the people you confront will rarely be willing to change their position.
Tactical: apply a tried and tested procedure to handle a particular type of situation. This approach denies base impulses by substituting them for a rational framework to handle the interaction. A good example of this is the training phone support operators undergo. However argumentative and confrontational the customer may be, there are ways to respond that will help calm them down and bring the conversation closer to resolving the issue at hand.
Strategic: ignore the challenge and instead use the opportunity to promote your own ideas. Very few people use this kind of response, but a good example would be that of a typical politician answering questions from the media or the public. Blunt questions or challenges are typically sidestepped and turned into a platform for the speaker’s own agenda – e.g. “good question, but the real issue here is…”. The intended audience of such a response is typically not the person they are talking to but rather everyone else that happens to be listening.
Moral: answer the challenge by earnestly helping out the person who made it. Confrontational people aren’t the sort of people that are successful and happy with their lives. People are argumentative or mean because of a deep dissatisfaction with life, because their spirit is wounded in some way. They react to this inner pain with violent outbursts aimed at regaining control, but instead of being successful, those outbursts only end up damaging the situation further, leading to a dreary cycle. Instead of reacting to their mistakes, or worse, exploiting them for your own gain, see if you can help the individual by giving them a kindness they cannot ignore. A moral response goes beyond a mere tactical handling of the situation. In essence, it is a strategic response, but the goal of the strategy is intertwined with one’s philosophy, directing it towards a long-term good rather than a short-term profit. Additionally, unlike the typical strategic response, the audience is the person who initiated the conversation rather than someone else. Very few people can operate on this level, and even fewer consistently, but nevertheless, it seems to me the only one worth striving for at all times. In all cases, aim to help, but if you can’t do that then do not make the situation worse with a petty response. Make things better, or do not engage.
If they’ve made a mistake, correct them gently and show them where they went wrong. If you can’t do that, then the blame lies with you. Or no one.