Different levels of collective acceptability

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:

• the argument can be uni-accepted, when it belongs to all the extensions of this semantics,
• or the argument can be exi-accepted, when it belongs to at least one extension of this semantics,
• or the argument can be not-accepted when it does not belong to any extension of this semantics.

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 .

Definition 19 (Cleanly-accepted argument)   Consider , is cleanly-accepted if and only if belongs to at least one extension of and such that , does not belong to any extension of .

Thus, we capture the idea that an argument will be better accepted, if its attackers are not-accepted.

Property 13   Consider and a semantics such that each extension for is conflict-free. If is uni-accepted then is cleanly-accepted. The converse is false.

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:

• can be uni-accepted, if belongs to all the extensions for (so, it will also be cleanly-accepted);
• or can be cleanly-accepted (so, it is by definition also exi-accepted); note that it is possible that the argument is also uni-accepted;
• or can be only-exi-accepted, if is not cleanly-accepted, but is exi-accepted;
• or is not-accepted if does not belong to any extension for .

Example 9   Consider the following argumentation system.

 There are two preferred extensions and . So, for the preferred semantics, the acceptability levels are the following: , and are uni-accepted, is cleanly-accepted but not uni-accepted, and are only-exi-accepted, , , , and are not-accepted.

Note that, in all the cases where there is only one extension, the first three levels of acceptability coincide26. This is the case:

• Under the preferred semantics, when there is no even cycle (see [8]).
• Under the basic semantics (another semantics proposed by Dung - see [9,8] - which is not presented here and which has only one extension).

Looking more closely, we can prove the following result (proof in Appendix A):

Property 14   Under the stable semantics, the class of the uni-accepted arguments coincides with the class of the cleanly-accepted arguments.

Then, using a result issued from [10,11] and reused in [8] 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:

Consequence 1   Under the preferred semantics, when there is no odd cycle, the class of the uni-accepted arguments coincides with the class of the cleanly-accepted arguments.

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 [4] (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:

So, all the arguments belonging to a cycle are incomparable with , , , and, even between them, there are few comparison results.

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:

• With the valuation of [4] and under the preferred semantics, despite the fact that is uni-accepted and is only-exi-accepted.
• With the valuation with tuples and under the preferred semantics, despite the fact that is only-exi-accepted and is not-accepted.

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