(This is once again me taking what old material from my personal blog and reposting it here with some revisions.)

Graham's hierarchy of disagreement is pretty well known and fairly useful, but I think the upper levels have some issues. (Are the lower levels necessarily the best they could be? Maybe not, but they also don't particularly matter.) I suggest a few changes.

First, and most minorly, let's remove DH6 "refuting the central point" as separate from "refutation". I think it should just always be implicit that whether you're counterarguing or refuting, it should be on the central point. If it's not, well... I guess we should add a second change -- let's add a DH3.5, above "contradiction" but below "counterargument", which is "arguing with a peripheral point". (So I guess really we haven't actually removed "refuting the central point" as separate, so much as we've stuffed both the original "counterargument" and "refutation" into DH3.5, but...) Don't waste your time arguing with peripheral points; spot them the point if necessary (you can always un-spot it later) and get on to the central point.

Now, one could notionally separate this out further -- counterarguing against a peripheral point, vs refuting a peripheral point -- but really, if you're on a peripheral point, it doesn't matter too much which you're doing. Get on the central point first. (If you miss it accidentally, hopefully your interlocutor is helpful and tells you you're off the point, rather than getting dragged into a peripheral argument! If you're entirely unsure what the central point is, some sort of double-crux or similar exercise may be necessary...)

Next, let's get rid of the strict hierarchy and put "counterargument" and "refutation" parallel to one another, rather than having the one on top of the other. Counterargument and refutation do different things; neither alone is superior to the other. But there is something that goes on top of both and that we should add, and that's counterargument combined with refutation. If you only counterargue, you just leave things in a state of "well we both have arguments for our side". But if you only refute, you just leave things in a state of "your argument is wrong, but who knows what's correct?" If you want to really do a proper job of arguing, you need to both refute and counterargue; until then it's incomplete!

(Yeah, none of this fits great into a Bayesian framework; I guess really the Bayesian framework is what you pull out when you're in one of the above inconclusive states!)

Finally, let's add a DH4.5 -- well, it would be a DH4.5 if we hadn't just moved DH4 and DH5 to be parallel to one another... now it sits below DH5 and above DH3 -- which is nonconstructive refutation. This is where you point out that there must be something wrong with an argument, without actually finding the hole in it. For instance, showing that an argument proves too much is an example of this: "Your argument that God exists can't be right, because the same argument would prove that a perfect island exists" (as the classic example goes). This is definitely not a counterargument (it doesn't do anything to show that God doesn't exist, it only addresses the particular argument), and it's clearly better than simple contradiction, but it's only kind of a refutation -- you still have to find the actual hole. I think it's worth separating this out.

So, in summary, my revised hierharchy:

• Refutation + counterargument
• Refutation
• Nonconstructive refutation
• (parallel to the above two, not actually below them) Counterargument
• Arguing a peripheral point
• Contradiction
• Lower levels (not important here)

(Doesn't counteragument + nonconstructive refutation belong on there somewhere? Yeah, technically, I suppose, but you can fill it in.)