In 2008, Paul Graham wrote How To Disagree Better, ranking arguments on a scale from name-calling to explicitly refuting the other person's central point.

And that's why, ever since 2008, Internet arguments have generally been civil and productive.

Graham's hierarchy is useful for its intended purpose, but it isn't really a hierarchy of disagreements. It's a hierarchy of types of response, within a disagreement. Sometimes things are refutations of other people's points, but the points should never have been made at all, and refuting them doesn't help. Sometimes it's unclear how the argument even connects to the sorts of things that in principle could be proven or refuted.

If we were to classify disagreements themselves – talk about what people are doing when they're even having an argument – I think it would look something like this:

Most people are either meta-debating – debating whether some parties in the debate are violating norms – or they're just shaming, trying to push one side of the debate outside the bounds of respectability.

If you can get past that level, you end up discussing facts (blue column on the left) and/or philosophizing about how the argument has to fit together before one side is "right" or "wrong" (red column on the right). Either of these can be anywhere from throwing out a one-line claim and adding "Checkmate, atheists" at the end of it, to cooperating with the other person to try to figure out exactly what considerations are relevant and which sources best resolve them.