The goal of an argument should be ending up on the same side of the problem.

Winning an argument and solving the thing you were arguing about are two different projects. The first ends with somebody conceding and remembering it; the second ends with the two of you facing the problem instead of each other. Most fights go wrong because the goal quietly switched in the first thirty seconds.

Ask yourself mid-argument what you are now trying to get. If the honest answer is to be declared right, say the other thing out loud: that you want to sort this out together. Naming the shared problem usually drops the volume by itself.