I actually tried pretty hard to teach an intelligent person off the street to argue productively with this post. (Agonized over the wording a fair amount, started with simple techniques and then discussed advanced ones, made no assumptions about prior knowledge, etc.) So, share this prior to your argument?

Clearly, your argument's productivity will be bounded if one participant is totally intransigent. So the best you can do in that case is collect evidence from the intransigent participant and move on.