Under a given semantics, and following Dung, the acceptability of an argument depends on its membership to an extension under this semantics. We consider three possible cases25:
However, these three levels seem insufficient. For example, what should be concluded in the case of two arguments and which are exi-accepted and such that or ?
So, we introduce a new definition which takes into account the situation of the argument w.r.t. its attackers. This refines the class of the exi-accepted arguments under a given semantics .
Thus, we capture the idea that an argument will be better accepted, if its attackers are not-accepted.
The notion of cleanly-accepted argument refines the class of the exi-accepted arguments. For a semantics and an argument , we have the following states:
Note that, in all the cases where there is only one extension, the first three levels of acceptability coincide26. This is the case:
Looking more closely, we can prove the following result (proof in Appendix A):
Then, using a result issued from [10,11] and reused in  which shows that, when there is no odd cycle, all the preferred extensions are stable27, we apply Property 14 and we obtain the following consequence:
Finally, the exploitation of the gradual interaction-based valuations (see Section 3) allows us to define new levels of collective acceptability.
Let be a gradual valuation and let be the associated preordering (partial or complete) on . This preordering can be used inside each acceptability level (for example, the level of the exi-accepted arguments) in order to identify arguments which are better accepted than others.
Example 9 (continuation) Two different
gradual valuations are applied on the same graph:
With the instance of the generic valuation proposed in  (see Section 3.1), we obtain the following comparisons:
With the global valuation with tuples presented in Section 3.2, we obtain the following comparisons:
If we apply the preordering induced by a valuation without respecting the acceptability levels defined in this section, counter-intuitive situations may happen. In Example 9, we obtain:
These counter-intuitive situations illustrate the difference between the acceptability definition and the valuation definitions (even if both use the interaction between arguments, they do not use it in the same way).
Marie-Christine Lagasquie 2005-02-04