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Abstract
We introduce a general approach for representing and reasoning with argumentation-based systems. In our framework arguments are represented by Gentzen-style sequents, attacks (conflicts) between arguments are represented by sequent elimination rules, and deductions are made according to Dung-style skeptical or credulous semantics. This framework accommodates different languages and logics in which arguments may be represented, allows for a flexible and simple way of expressing and identifying arguments, supports a variety of attack relations (including those that reflect relevance or quantitative considerations), and is faithful to standard methods of drawing conclusions by argumentation frameworks. Altogether, we show that argumentation theory may benefit from incorporating proof theoretical techniques and that different non-classical formalisms may be used for backing up intended argumentation semantics.
Conflict of interest disclosure statement
No potential conflict of interest was reported by the authors.
Funding
Christian Straßer's research was supported by the Alexander von Humboldt Foundation and the German Ministry for Education and Research.
Notes
1. Clearly, the argument A may be split into two BH-arguments, and
, but in general such rewriting requires further processing and might cause a blowup in the number of arguments.
2. Another argument that is sometimes pleaded for set-inclusion minimisation is that it reduces the number of attacks. Again, it is disputable whether set-inclusion minimisation is the right principle for assuring this property, since, for instance, the singletons and
, supporting (e.g. in classical logic) the claim
, are incomparable w.r.t. set-inclusion (and moreover they even do not share any atomic formula), but it is obvious that as n becomes larger
becomes more exposed to attacks than
.
3. The availability of an implication connective is not required – see Remark 4.3.
4. Following the usual convention, we shall omit set-brackets from the left-hand sides of sequents.
5. Obviously, for the definition of an -argument it does not matter which of the calculi that are sound and complete for
is chosen.
6. Following the usual conventions, we use commas in a sequent for denoting the union operation, and omit curly brackets of singletons (i.e. we write ψ instead of ).
7. Note that is cautiously
-reflexive:
for a consistent formula
.
8. Gorogiannis & Hunter (Citation2011) consider a stricter implication between attacks, in which .
9. To reduce the amount of notations, we use the same names for the rules with and without the implication connective. This will not cause ambiguity in what follows.
10. Note that ‘-Spec’ or ‘
-SpecNeg’ variants would not be sensible, since permissions with incompatible content do not conflict in any intuitive sense.
11. Note that this condition does not imply a subset-minimality of Γ, but rather assures that Γ does not contain information that is irrelevant (in the sense of Item (a)) for its conclusion.
12. For instance, when and
, we have that
.
13. On the other hand, relevant attack rules have some desirable properties that are not necessarily shared by other rules, like being invariant with respect to irrelevant information – see Lemma 5.15.
14. See also the conclusion of this paper for a short discussion on implementing preferences in sequents.
15. Recall that by the definition of , this implies that
.
16. Similar entailment relations may of-course be defined for other semantics of abstract argumentation frameworks, such as semi-stable semantics (Caminada, Citation2006), ideal semantics (Dung, Mancarella, & Toni, Citation2007), eager semantics (Caminada, Citation2007), and so forth.
17. It is important to note that is attacked by
-derivable arguments (such as
), but none of them is in
.
18. In case that is infinite, compactness of
should be assumed along the proof.
19. Note that the condition is indeed required here. For instance, in an argumentation framework based on and Undercut it holds that
. (Indeed, according to any semantics considered here
is undefended, since it is attacked by
, and the latter is not attacked by any other sequent since its left-hand side is empty).
20. Recall that by Notation 5.4 this means that the proposition holds for every entailment of the form considered in Definition 5.3, where
and
are as defined in the proposition,
is any of the standard argumentation semantics considered in this paper, and
.
21. Other techniques for generating arguments are considered, for example, in Besnard, Grégoire, Piette, and Raddaoui (Citation2010) and Efstathiou & Hunter (Citation2011).