Why do long, uninspiring, and seemingly-childish debates sometimes emerge even in a community like LessWrong?  And what can we do about them?  The key is to recognize the potentially harsh environmental effect of an audience, and use a dying debate to fertilize a more sheltered private conversation.

Let me start by saying that LessWrong generally makes excellent use of public debate, and naming two things I don't believe are solely responsible for fruitless debates here: rationalization biases and self-preservation¹.  When your super-important debate grows into a thorny mess, the usual aversion to say various forms of "just drop it" are about signaling that:

1. you're not skilled enough to continue arguing, so you'd look bad,
2. the other person isn't worth your time, in which case they'd be publicly insulted and compelled to continue with at least one self-defense comment, extending the conflict, or
3. the other person is right, which would risk spreading what appear to be falsehoods.

"Stop the wrongness", the last concern, is in my opinion the most perisistent here simply because it is the least misguided.  It's practically the name of the site.  Many LessWrong users seem to share a sincere, often altruistic desire to share truth, abolish falsehood, and overcome conflict.  Public debate is a selection mechanism generally used very effectively here to grow and harvest good arguments.  But we can still benefit from diffusing the weed-like quibbling that sometimes shows up in the harsh environment of debate, and for that you need a response that avoids the problematic signals above.  So try this:

"I'm worried that debating this more here won't be useful to others, but I want to keep working on it with you, so I'm responding via private message.  Let's post on it again once we either agree or better organize our disagreement.  Hopefully at least one of us will learn and refine a new argument from this conversation."

Take a moment to see how this carefully avoids (1)-(3).  Then you can try changing the tone of the private message to be more collaborative than competitive; the change in medium will help mark the transition.  This way you'll each be less afraid of having been wrong and more concerned with learning to be right, so rationalization bias will also be diminished.  As well, much social drama can disintegrate without the pressure of the audience environment (I imagine this might contribute to couples fighting more after they have children, though this is just anecdotal speculation).  Despite being perhaps obvious, these effects are not to be underestimated!

But hang on...  if you're convinced someone is very wrong, is it okay to leave such a debate hanging midstream in public?  Why doesn't "stop the wrongness" trump our social concerns and compel us to flog away at our respective puddles of horsemeat?

The usual necessary condition for you to wind up in a pointless online debate is that you're very convinced your co-poster is wrong, but isn't obviously wrong enough to attract negative comment Karma.  So you keep posting in an attempt to "clear up" the issue for everyone else, or at the very least dilute the false content.  But somewhere along the line, you might end up with something that looks to you like:

You (1 point):  Very smart/correct response to the parent discussion.
Other (1 point):  Convincing but wrong/vague/irrelevant comment that somehow got upvoted.
You (0 points):  Correction, or return to parent discussion.
Other (0 points):  More trickily wrong/vague/irrelevant stuff.
…

If you've read these from the outside, it often looks more like

User A (1 point):  «something that mildly interests you»
User B (1 point):  «tolerable response»
User A (0 points):  «quibbling you don't care to read»
User B (0 points):  «yup, definitely don't care»
…

By now, or with very little practice, you should be able to tell from the inside when a conversation is entering a public failure spiral.  When that happens, no matter how smart and right and well-intentioned your responses might be, your co-poster's responses are still going to show up at around the same density.  And since the average non-negative-scoring comment on LessWrong is pretty correct, if your assessment is accurate, then your opponent's wrong comments are going to drag down the average rightness more than you're going to raise it.  And you're certainly not helping the situation if you're wrong.  And, on LessWrong in particular, the Recent Comments feed just diverts more attention to the trickily wrong ideas the more you argue against them.

So, in the unfortuate event² that a debate starts going awry (which thankfully is a relatively rare occurrence here), the task is to stop the spiral without looking bad, and not making the other person look bad is a necessary ingredient (else they'll continue the spiral in self-defense).  The "switch to private message" signal I suggest above is a pretty self-sufficient way to do that.  Though it stands pretty well on its own, it becomes more effective the more we send and interpret it consistently as a community, and the more we follow through with posting conclusions to the resulting private debates.

Thus, above all, "switch to private" requests must be neither presented nor interpreted as an insult to either debater.  Not everyone is as courageous as cousin_it to publicly change their minds.  We must praise the effort of a debater who aims to work through a contentious argument privately.  We must consider "switch to private" as a sign of respect for the other debater: it shows value for his/her interaction.  We must not judge "winners" or "losers" of a debate as a function of who requests privacy first.  Then perhaps awkward social battles can become collaborations as routinely as mulched weeds can become crops.

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Footnotes

¹ Regarding rationalization biases, most posters here know the difference between trying to learn what right is and trying to change what right is in an attempt to escape confirming past errors.  Regarding classic self-preservation, in this case it yields a desire to signal intelligence at the cost of deceiving the audience.  This is pretty misguided, not just because it's usually immoral, but because the LessWrong audience isn't easily deceived...  especially by a long and uninteresting-looking debate.  I'm sure these factors are present too, but the last one most needed addressing.

² I can't praise LessWrong enough for the fact that its debates tend to be much more fruitful than elsewhere.  I very much don't want to stifle the open and productive arguments that go on here; only to encourage a way out of the undesirable ones.