## Towards a gradual individual acceptability

The individual acceptability is based on the comparison of an argument with its attackers.

The first proposal has been to select an argument if and only if it does not have any attacker (see [12]).

This has later been extended by [1] where, using a preference relation between arguments (an intrinsic valuation), an argument is accepted if and only if it is preferred to each of its attackers.

Following this proposal, we propose the same mechanism but with the interaction-based valuation.

Given a gradual valuation, the preordering induced by can be directly used in order to compare, from the acceptability point of view, an argument and its attackers28. This defines a new class of acceptable arguments: well-defended arguments.

Definition 20 (Well-defended argument)   Consider , is well-defended (for ) if and only if such that , .

Thus, we capture the idea that an argument will be better accepted if it is at least as good as its direct attackers (or incomparable with them in the case of a partial ordering). The set of well-defended arguments will depend on the valuation used.

Using this new notion, the set of the arguments is partitioned in three classes:

• the first class contains the arguments which are not attacked,
• the second class contains the arguments which are attacked but are well-defended,
• the third class contains the other arguments (attacked and not well-defended).

Note that the set of the well-defended arguments corresponds to the union of the two first classes. A further refinement uses the gradual valuation inside each of the classes as in Section 4.2.

In Example 9 presented in Section 4.2, the well-defended arguments are:

• , , , and ( is incomparable with but better than ) for the valuation with tuples,
• though with the valuation of [4] the well-defended arguments are , , , and ( is better than ).

Note also that, as in the semantics of [9], Definition 20 considers the attackers one by one. It is not suitable for a valuation which handles the direct attack'' as a whole (as the valuation of [4] - see the counterexamples presented in Section 4.4).

Marie-Christine Lagasquie 2005-02-04