Claim
State their case first, in a form they accept
Before you argue, say back what you think their position is, and ask whether you have it right. Then improve it: take the strongest version rather than the weakest, which is usually called steelmanning.
Two things happen. The other person stops defending and starts correcting, which is a completely different posture, and you frequently discover you were arguing with something they do not hold. It also costs you nothing, because if their position is bad, it will be more obviously bad in its strongest form, and you will not be able to be accused of attacking a caricature.
Claim
Find the smallest real disagreement
Big arguments are usually a stack of smaller ones, and the stack is what makes them unwinnable. Work down to the specific claim where you actually differ: a number, a mechanism, a value.
That is the only place a conversation can move. If the disagreement turns out to be about a fact, one of you can go and look. If it is about a mechanism, you can both describe how you think it would play out, which is day 5. If it is about a value, name it and stop, because you have found the floor, and pretending to keep arguing about evidence after that point is what turns a discussion into a fight.
Action
Agree an exit before you need one
With the person you most often argue with, agree a rule while you are both calm. Either of you can say pass on any subject, once, and the other drops it immediately with no comment and no scoring.
It sounds childish and it works, for a mechanical reason: it gives one party the ability to end the exchange unilaterally, which removes the need for either to win in order for it to stop. Try it with one person this week. The subject also becomes less interesting rather than more forbidden, which is the opposite of what everyone expects.
Example
Two colleagues who worked well together stopped having lunch after a long argument in 2023 about a national referendum.
One of them wrote three sentences a fortnight later: I do not want to agree with you about this and I do not want to stop having lunch with you, can we leave it. They had lunch the following week. Neither has changed his mind. What was recovered was not the argument but about forty lunches a year, and the man who sent the message says the only difficult part was accepting that he was not going to get the last word.
Caution
Not everything should be smoothed over
This day has a limit and it needs saying clearly. Some positions are not a difference of weighting: when a view treats a group of people as less than fully human, keeping the peace is a choice to be comfortable at their expense.
There is no technique that makes that comfortable, and a man who uses conversational method to avoid ever taking a position has not achieved balance. He has achieved cover. The rule I would defend: use these moves for disagreements about policy, mechanism and value, and do not use them to be pleasant about contempt. Knowing which one you are in is the judgment, and it is yours.
Claim
Ask what would change their mind, and answer it yourself
One question, used sparingly, reorganizes a conversation: what would have to be true for you to change your position.
It only works if you answer it first, about yourself, honestly and out loud. Used as a weapon it is insufferable and everybody can tell. Used symmetrically it does two things: it distinguishes the people in a discussion from the people in a fight, and it gives you both a concrete thing to go and check. If neither of you can answer it, you have discovered that this is an identity conversation, and the honest move is to notice and choose something else to talk about.
Aside
Against the middle as a default
One heuristic I want to reject explicitly, because it sounds like moderation and is not. The idea that the truth is reliably in the middle of two positions is no better than picking a side: it makes your view a function of where other people stand, which means anyone can move you by staking out a more extreme position.
On many questions one side is straightforwardly right on the facts. On others both are wrong in the same direction. Splitting the difference is a way of avoiding the work of the previous five days while appearing judicious, and it is worth noticing that it is also the most socially comfortable position available, which is usually a warning rather than a recommendation.