I'd say that the incoherent speaker is arguing at DH(-1). DH0 would be an improvement. You would be counterarguing at DH(No) - argument by pointing out conversational emptiness.
(edited to clarify that it is the person who makes the incoherent argument who is arguing badly, and the person arguing against that who is doing something entirely outside the hierarchy.
Other DH(No) arguments-that-are-about-non-argument include "We aren't actually arguing about the same thing" and "let me take some time to do more reading before I reply.")
- Gandhi
Now that most communication is remote rather than face-to-face, people are comfortable disagreeing more often. How, then, can we disagree well? If the goal is intellectual progress, those who disagree should aim not for name-calling but for honest counterargument.
To be more specific, we might use a disagreement hierarchy. Below is the hierarchy proposed by Paul Graham (with DH7 added by Black Belt Bayesian).1
DH0: Name-Calling. The lowest form of disagreement, this ranges from "u r fag!!!" to "He’s just a troll" to "The author is a self-important dilettante."
DH1: Ad Hominem. An ad hominem ('against the man') argument won’t refute the original claim, but it might at least be relevant. If a senator says we should raise the salary of senators, you might reply: "Of course he’d say that; he’s a senator." That might be relevant, but it doesn’t refute the original claim: "If there’s something wrong with the senator’s argument, you should say what it is; and if there isn’t, what difference does it make that he’s a senator?"
DH2: Responding to Tone. At this level we actually respond to the writing rather than the writer, but we're responding to tone rather than substance. For example: "It’s terrible how flippantly the author dimisses theology."
DH3: Contradiction. Graham writes: "In this stage we finally get responses to what was said, rather than how or by whom. The lowest form of response to an argument is simply to state the opposing case, with little or no supporting evidence." For example: "It’s terrible how flippantly the author dismisses theology. Theology is a legitimate inquiry into truth."
DH4: Counterargument. Finally, a form of disagreement that might persuade! Counterargument is "contradiction plus reasoning and/or evidence." Still, counterargument is often directed at a minor point, or turns out to be an example of two people talking past each other, as in the parable about a tree falling in the forest.
DH5: Refutation. In refutation, you quote (or paraphrase) a precise claim or argument by the author and explain why the claim is false, or why the argument doesn’t work. With refutation, you're sure to engage exactly what the author said, and offer a direct counterargument with evidence and reason.
DH6: Refuting the Central Point. Graham writes: "The force of a refutation depends on what you refute. The most powerful form of disagreement is to refute someone’s central point." A refutation of the central point may look like this: "The author’s central point appears to be X. For example, he writes 'blah blah blah.' He also writes 'blah blah.' But this is wrong, because (1) argument one, (2) argument two, and (3) argument three."
DH7: Improve the Argument, then Refute Its Central Point. Black Belt Bayesian writes: "If you’re interested in being on the right side of disputes, you will refute your opponents' arguments. But if you're interested in producing truth, you will fix your opponents' arguments for them. To win, you must fight not only the creature you encounter; you [also] must fight the most horrible thing that can be constructed from its corpse."2 Also see: The Least Convenient Possible World.
Having names for biases and fallacies can help us notice and correct them, and having labels for different kinds of disagreement can help us zoom in on the parts of a disagreement that matter.
Let me illustrate by labeling excerpts from Alvin Plantinga's critical review of Richard Dawkins' The God Delusion.
DH1, Ad Hominem:
DH2, Responding to Tone:
DH4, Counterargument:
DH6, Refuting the Central Point:
Of course, even a DH6 or DH7 disagreement can still be wrong. But at the very least, these labels can help us highlight the parts of a disagreement that matter for getting at the truth.
Also see: Causes of Disagreements.
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1 This article is an update to my earlier post on CSA.
2 Sometimes the term "steel man" is used to refer to a position's or argument's improved form. A straw man is a misrepresentation of someone's position or argument that is easy to defeat: a "steel man" is an improvement of someone's position or argument that is harder to defeat than their originally stated position or argument.
3 For an example of DH7 in action, see Wielenberg (2009). Wielenberg, an atheist, tries to fix the deficiencies of Dawkins' central argument for atheism, and then shows that even this improved argument does not succeed.