Everyone has various point of view on various matters. And these points of views are affected by each and everyone's background, environment, education, religious, exposure, culture, and many more. No matter how much you would like to make some people see things from your own perspective, they will never and it's not their fault.
As much as I don't like engaging in arguments, it still happens. In moments like that, I don't walk away and let the next person take the victory; I would ensure to communicate my side of the story just like I would listen to the next person communicate theirs and see where the whole issue started.
As for this, I would rather walk away then to forcefully make a point that would eventually waste my time, cost me my voice and strength and over all, still not result in the other person admitting that I am right. As time goes on, time will softly tell who is right, and it time doesn't, then so be it.