Agreed, I don't think people really debate emotionally charged issues from first principles. I think it's an error to think of this as a matter of logical axioms. That's not how people (usually) debate.
Its obvious that logic isn't a good model for how people form and change opinions. (I just feel that it isn't)
So... what's the state of the art?
It's been out of favor since Derrida, but I think Structuralism is due for another pass in the limelight. An extreme subjectivism about reality, shaped by the stimuli you are exposed to. These external stimuli are what define your axioms, and from there, you use the same (fuzzy, bad, and prone to inducing contradctions) logic as everyone else. It provides a compelling model for why people have difficulty reconciling their opinions; they're not even talking about the same things, just using the same words, the same signs.