In real life or even on the Internet, there are many occurrences when you can't agree with the person standing next to you and want to move on from the conversation without agreeing or surrendering. I used to say this:
"Let's agree to disagree."
However, this sounds a bit rude and sometime the opposition side still continues after it (especially on SE discussions). So how can I convey "Let's agree to disagree" in a more polite and effective way?
Don't actually disagree (any more) and wind up that thread of conversation in favor of something else.
There could two (or more) possibilities as to whether the person wants to keep talking.
He may think you do not understand his point well (or well enough). If you haven't already, simply say you understand. Or take all doubt away, and summarize what he is trying to say. You can do this without indicating at all whether you agree with it or not. You are simply stating or proving you understand their point.
Secondly, they may think that continued discussion might change your mind. You can manage this by not being the second person in the conversation. Stay silent on this thread of conversation (or wind it down). You can change the subject. You may elect to simply excuse yourself because you have something to do (like go back to work, go get a drink, go to the bathroom).
In my opinion, the key is to tactfully conclude the exchange or simply to stop being one of the parties to the conversation.
State the underlying disagreement and suggest a next action.
Usually, when you can't agree on a small thing, it's due to some more fundamental disagreement. For example, you may disagree on core values like how much risk is acceptable or whether religious doctrine should be used as evidence. You may have had different experiences, where something that worked for you in the past didn't work for them. You might just be having too much trouble communicating your viewpoints and can't find a way to fix it.
Whatever the underlying issue seems to be, point that out. Then, suggest what you want to do instead of continuing the discussion. Maybe you can just stop the conversation and move on. Sometimes, however, you want to make a decision and will need to make a compromise, or you have an unmet request that sparked the discussion. Make a suggestion for the next action you'll take and make sure it doesn't depend upon resolving the disagreement.
Some examples of what you might say in various situations:
How about something like, "This is a bigger issue than we're going to resolve today; let's move on to [something, blessedly anything else]". I've used this before and was happy with results.
Another thing to consider is whether you are trying to close the discussion for all time, or just for now. When you get down to it, most of these kind of disagreements are built on top of emotionally-held axioms which are not really amenable to debate. We've gracefully shut down many pointless debates by pointing out when my colloquator [1] and I actually uncover which of these axioms we hold differently. That's when you shake hands, tip your hat, and move on.
[1] Spell check doesn't think that "colloquator" is a word, but dammit it should be
Even if they cannot get your agreement, people want to be heard. I suggest you tell them that first: "I hear you and I can see this is important to you". Then you can acknowledge your disagreement if you like but I would move the conversation toward why it is important to them, which actually might be a great place to find common ground.
Take for example guns. The other person wants to own guns and you think they should not be kept in the house. When you find out their why you might see that they are concerned for their families safety just like you are. Now you can agree on a deeper value and the desire to own guns or not is more relatable. You might see that this disagreement is really not as fundamental as you thought, since there is a deeper level..something more fundamental and human.
And you move the conversation towards connection, or a possible connection, by going deeper into the reasons a person has a particular opinion or point of view.
