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 $A$ and $B$ which are exi-accepted and such that $A {\mathcal{R}}B$ or $B {\mathcal{R}}A$?

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 $S$.

Definition 19 (Cleanly-accepted argument)   Consider $A \in {\mathcal{A}}$, $A$ is cleanly-accepted if and only if $A$ belongs to at least one extension of $S$ and $\forall B \in {\mathcal{A}}$ such that $B {\mathcal{R}}A$, $B$ does not belong to any extension of $S$.

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

Property 13   Consider $A \in {\mathcal{A}}$ and a semantics $S$ such that each extension for $S$ is conflict-free. If $A$ is uni-accepted then $A$ is cleanly-accepted. The converse is false.

The notion of cleanly-accepted argument refines the class of the exi-accepted arguments. For a semantics $S$ and an argument $A$, we have the following states:

• $A$ can be uni-accepted, if $A$ belongs to all the extensions for $S$ (so, it will also be cleanly-accepted);
• or $A$ 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 $A$ can be only-exi-accepted, if $A$ is not cleanly-accepted, but $A$ is exi-accepted;
• or $A$ is not-accepted if $A$ does not belong to any extension for $S$.

Example 9   Consider the following argumentation system.

There are two preferred extensions $\{D, C_2, A, G\}$ and $\{D, C_2, E, G, I\}$. So, for the preferred semantics, the acceptability levels are the following: $D$, $C_2$ and $G$ are uni-accepted, $I$ is cleanly-accepted but not uni-accepted, $A$ and $E$ are only-exi-accepted, $B$, $C_1$, $F$, $H$ and $J$ 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 $v$ be a gradual valuation and let $\succeq$ be the associated preordering (partial or complete) on ${\mathcal{A}}$. 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:

$$D, C_2 \succ I \succ E \succ G \succ J \succ C_1,F \succ A \succ H \succ B$$

With the global valuation with tuples presented in Section 3.2, we obtain the following comparisons:

$$D, C_2 \succ G \succ B \succ F, C_1$$

$$D, C_2 \succ A \succ E$$

$$D, C_2 \succ H \succ E$$

$$D, C_2 \succ I$$

$$D, C_2 \succ J$$

So, all the arguments belonging to a cycle are incomparable with $G$, $B$, $F$, $C_1$ 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, $E \succ G$ despite the fact that $G$ is uni-accepted and $E$ is only-exi-accepted.
• With the valuation with tuples and under the preferred semantics, $H \succ E$ despite the fact that $E$ is only-exi-accepted and $H$ 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).