Just ask what they want from the argument.
For instance, you might be upset that your spouse forgot your birthday, but instead you get upset about dirty dishes, an open window, or a cluttered nightstand. To the other person, it looks like you’re just suddenly upset by clutter, which can be confusing if that’s not your regular M.O. When your spouse doesn’t understand why you’re upset, that’s likely to make you even more upset, and the whole situation can spiral out of control.
Instead, respond to an argument with a calm and respectful request asking what the other person is trying to accomplish with the fight.
Hopefully, that will facilitate him or her either telling you what the issue is, or looking at the situation and realizing what they really want out of it is a hug and snack, or maybe they just want an apology for some wrong that you did them you don’t even realize. By asking, you can get to the root of the problem a little quicker, and hopefully avoid the whole epic battle portion of the conflict. Or at the very least you’re offering to listen to that person’s concerns, which is often the point of a fight, to begin with.
Once you hear what an issue is, you can respond (again, calmly and respectfully) and address their concerns. Or, even better, you can just respond with “I hear you,” because you did, in fact, hear them. Maybe you still disagree, but in some cases, that’s still ok. Often just acknowledging that you hear and understand the other person’s point of view might be enough to end the conflict (or at least open up a productive dialogue about it) and preserve your friendship/marriage/job in the process.