This ended up being shared around quite a few places, spawning interesting and largely constructive discussions as it went. It also attracted not a small amount of scorn for the person who the letter was addressed to. This got us into a conversation about the social effects of scorn and ridicule, their effects on persuasiveness, and how to invest your energy in arguing well. My comments follow - I'd be interested to hear yours.
It took me a while to get to the point where I realised that making a persuasive argument is about more than laying out facts (as I saw them) and the (i.e. my) line of reasoning that links them. That whole theory-of-minds-of-other-people thing was admittedly slow to develop, in the way it often is for geeks.
I find it helps to think of it in terms of working out where their thinking is at, and trying to lay out a clear path for them that acknowledges their values. You may not be able to bring them all of the way all at once, but you can mark a trail that will be easier to follow in future.
Boiled down into two fancy words: memetic aikido.
The good thing about doing things that way with an audience is that, like the article above, you have the opportunity for collateral education, rather than collateral damage.
A direct, hard push doesn't usually get good results. It just tends to harden opinions on both sides of the argument. The "no, that's ridiculous..." or "well actually..." often makes someone's firmly-seated opinions dig in further. It might make you look strong to any sympathetic onlookers and those who respect a performance of strength, but it's a dead loss to everyone else.
Shame has little constructive potential if someone doesn't already care about what you think of them - and even then, laying it on too thick becomes a burden rather than an encouragement to be better.
That, in essence, is what I think we should be looking for if we're going to invest in arguments: the encouragement to be better. If we're not doing that in some way, isn't everyone better off if we use that energy somewhere else?