I've been thinking about something like this.

If there's two people arguing, there's a 50% chance a given one is right. There is generally no reason to believe the the correct one happens to be you.

That only really adds a bit of evidence against you, which doesn't seem like much. That said, if the other person realizes this, and doesn't change their mind, their evidence was one bit stronger than you previously thought.

Furthermore, if they realize that you know that and haven't changed your mind, and they still don't change their mind, that adds another bit.

Etc.

If both people understand this, they will either immediately figure that the one who's more sure is probably right, or just not be sure.