The problem is a general one, but the believer/non-believer instance is the most obvious one. When presenting a case, each person proceeds from certain assumptions - when these assumptions are different and irreconcilable, you have a problem. I find this happens when I'm debating cannabis law reform (some operate on a different set of assumptions concerning the effects than I do), political theory (libertarians have some funny assumptions going on, especially if they're objectivists), as well as theology (I met a creationist the other day who had the assumption 'humans are exempt from certain biological effects that other organisms may be subject to'. The case in point was evolution).

The only way to solve these arguments is to either:

  1. Reconcile the assumptions - show that the two differnet sets are not incommensurable after all. If the two sets of assumptions can be shown to be in line with one another, then the conclusion reached from them should be acceptable to both parties. (this assumes that both parties follow a similarly logical path from the assumptions. big if.)
  2. Change one set of assumptions. YMMV with this one, depending on what other assumptions the person has, like 'observable facts take precedence over what the Bible/my parents/the tentacle from Xibalbi told me' or whatever.
If the two parties still operate on different assumptions, your best bet is to smile and withdraw, and look over the assumptions of your own that have been called into question.
themusic: The point is that attempting to argue with people by making claims they do not agree with is not going to work. fullstop.