Here’s a handy visual aid for levels of discussion.
The peak of the pyramid, unfortunately, does not seem to be useful. An end state, perhaps? In any case, whether or not your opponent’s point has been successfully refuted is likely to be a subjective judgement call on the basis of the audience. Very rarely, it seems, will anyone engage in a discussion with someone and actually convince them to “switch sides,” so to speak. Although it does happen!
Therefore, when engaging in serious discussion, the next two steps are the useful ones: refutation, and contradiction. Not every argument will have a mistake to refute, I suspect, so contradiction is the real heart of a proper serious discussion.
Not exactly something you can do on Twitter, what with the 140 character word lengths. Twitter really only gives you space for contradiction, unless whatever subject you’re discussing lends itself to image-as-evidence. (Useful, the image uploading. Especially for presenting evidence.)
The bottom three levels of the pyramid are not part of argumentation at all. Tone policing, as indicated in the graphic, does not address the argument at all.
Now, since tone policing is one of the primary weapons of the intellectually dishonest, I have a severe allergy to seeing it used. If you must address the tone in which something is written, either do so in a humorous (rather than moralizing) way, or take your concerns private rather than public. The only case in which I would consider it appropriate to publicly “tone police” is to disavow harassment, or if the person doing the “policing” has some legitimate authority to do so. Someone with a fanbase can certainly request specific behavior out of that fanbase. (In which case, I wouldn’t really consider it “tone policing” – the offensive part of tone policing is that the person doing it often abrogates to herself an authority that she does not legitimately possess.)
Ad hominem is one of the other primary weapons of the intellectually dishonest: DISQUALIFY. Most often you will see this combined with the last bit: insult. As GamerGate has found out, the opposition has only one card: disqualify with insults. Misogynists, harassers, trolls – nobody needs to address the arguments of such horrible people! After all, they don’t have any.
Which is one reason to investigate disqualification claims very carefully, especially when dealing with generalized accusations against entire groups.