A few comments above I wrote,

My impression is that the conversation ends because at least one side isn't willing to argue about initial assumptions. (Indeed, if there was an argument for it, it wouldn't be an assumption.) People will just say things like, 'it is self-evident that ....' or, 'oh, if you don't even believe that then there's nothing to talk about!'.

I was presenting this as an example of a discussion failure mode. I don't suppose that it is necessary. But I also hadn't known about those other ways of arguing about initial assumptions -- utility, parsimony, etc. Those sound like very productive places to begin.

So we agree it's not a conversation halter unless one person won't proceed past, 'my assumptions are self-evident!'.