Sometimes you're right, and you know it.  Unfortunately, we live in a world where those around us may not know what we know.  Worse than that (by an order of magnitude), others around us may intellectually know they are wrong - or you are right - and fight you regardless; out of spite, or fear, or anger, or whatever.

At the end of the day, all of the logic and data in the world won't always win others over to your way of thinking, especially if they didn't come to their current set of conclusions using facts and logic to begin with.

When something like this happens in your world, you have two choices.  First, you can dig in deeper, fighting with more logic, facts, figures, and eloquent arguments.  Or, instead, you can work to understand the alternate perspective and change your communication approach to suit your audience.  The latter may take more time, energy, and patience than you would like to expend, but is ultimately more effective in the long run.

One of the keys to cultivating solid relationships is the ability to identify these situations and intentionally adjust your approach when you think it will lead to a better result.  This practice takes a lifetime (and then some) to master, but is worth the journey to improve on over time.

Next time you feel the other side digging in, don't respond in kind.  Instead, adjust and look to address a potential deeper issue.