An argumentation framework  is a graph of nodes called arguments, and edges called attacks. If arguments are propositions, and “p1 attacks p2” reads “if you believe p1 then you shouldn’t believe p2”, then an argumentation framework looks like something you can use to represent the relationship between arguments and counterarguments in, say, a debate.
I liked argumentation frameworks. I used them in various past academic publications – see this, for example. I used them with students when I taught decision analysis and decision theories. I no longer see much value in argumentation frameworks.
The concept of “acceptable argument”, or intuitively speaking, of an argument that you would be justified in believing, gets a precise meaning in an argumentation framework. An argument is acceptable if and only if all arguments attacking it are not acceptable. This simple condition leads to a procedure for evaluating the acceptability of any argument: every source node in the graph is an acceptable argument, and you then compute acceptability of the rest; cycles make things more complicated, in that any one given argumentation framework with at least one cycle will have multiple sets of acceptability assignments on all arguments. In a sense, acceptability is unstable: there is no rule that says which of the alternative acceptability assignments is better than the others.
For example, say that you have an argumentation framework as follows. A1, A2, etc. are arguments, –> are attack relations: A1 –> A2, A2 –> A1.
So there are two arguments, and they attack each other. Here, you have two alternative assignments of acceptability: A1 is acceptable and A2 isn’t; or A2 is acceptable and A1 isn’t.
This clearly isn’t great. You might resolve it by adding another relation, such as preference, and say that you prefer that A1 is acceptable, which solves it. The problem, however, is that by adding a preference relation, you no longer have an argumentation framework, but more than that.
Anyway, if we stick to the original concept of argumentation framework, then it is simply necessary to accept that some instances thereof will have multiple alternative acceptability assignments, when they they have cycles.
Keeping that caveat in mind, note that if you think of an argumentation framework as representing a debate, where arguments are being offered for and against a position (itself an argument in the framework), the winner of the debate can always be the one that has the last word.
This is easy to see: I say A1, you say A2 –> A1, I then say A3 –> A2, and so on. If we stopped when you said A2 –> A1, you would have won; but if I was allowed to take the last step, and offer A3 –> A2, I would win.
It follows that the one with more resources wins. There is an economic angle to acceptability.
The important nuance, of course, is that acceptability is not equivalent to truth. An acceptable argument is not necessarily true, and vice versa. Acceptability is a value an argument obtains in the context of an argumentation framework; so if the argumentation framework leads to some argument A being acceptable, there is no reason we cannot have A represent a proposition which is false. For example, if there is a debate on a question, and one of the participants is given the final word before an audience, that person can ensure that their arguments compute as acceptable; and this happens only because they have the last word.
Not only is acceptability not equivalent to truth, but establishing one or the other is fundamentally different. Acceptability is cheap, it suffices to have the last word. Truth, on the other hand, is a different matter: it takes evidence, and often an accumulation of it. As I wrote in the note on evidence, there is a lot to do before we can claim that A is evidence for B, and therefore, that A supports B (as opposed to attacks, but you can turn it around, and ask what conditions should hold for some C to be evidence against B?).
This means that argumentation frameworks is a deficient concept. It simply means that it cannot be taken seriously when talking about truth. What can be done to make acceptability more relevant in a given case, is to increase the cost of attack: this can be done by introducing criteria that are required before an attack relation can be claimed from some argument A to another argument B. A simple approach that raises costs considerably, is to ask that A should be evidence against B as a necessary and sufficient condition for A –> B. Then again, this is no longer an argumentation framework, but one augmented with much more complicated rules. What’s twisted with this approach, is that you can end up with acceptable arguments that have no evidence against them, but have neither evidence for them; so you need more than the attack relation alone – there needs to be a support relation as well.
- Dung, Phan Minh. “On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.” Artificial intelligence 77.2 (1995): 321-357.