Collective acceptability of [9]

In the framework of collective acceptability, we have to consider the acceptability of a set of arguments. This acceptability is defined with respect to some properties and the sets which satisfy these properties are called acceptable sets or extensions. An argument will be said acceptable if and only if it belongs to an extension.

Definition 17 (Basic properties of extensions following [9])  
Let ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$ be an argumentation system, we have:

Conflict-free set

A set $E \subseteq {\mathcal{A}}$ is conflict-free if and only if $\not \exists A, B \in E$ such that $A {\mathcal{R}}B$.

Collective defence

Consider $E \subseteq {\mathcal{A}}$, $A \in {\mathcal{A}}$. $E$ collectively defends $A$ if and only if $\forall B \in {\mathcal{A}}$, if $B {\mathcal{R}}A, \exists C \in E$ such that $C {\mathcal{R}}B$. $E$ defends all its elements if and only if $\forall A \in E$, $E$ collectively defends $A$.

[9] defines several semantics for collective acceptability: mainly, the admissible semantics, the preferred semantics and the stable semantics (with corresponding extensions: the admissible sets, the preferred extensions and the stable extensions).

Definition 18 (Some semantics and extensions following [9])   Let ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$ be an argumentation system.

Admissible semantics (admissible set)

A set $E \subseteq {\mathcal{A}}$ is admissible if and only if $E$ is conflict-free and $E$ defends all its elements.

Preferred semantics (preferred extension)

A set $E \subseteq {\mathcal{A}}$ is a preferred extension if and only if $E$ is maximal for set inclusion among the admissible sets.

Stable semantics (stable extension)

A set $E \subseteq {\mathcal{A}}$ is a stable extension if and only if $E$ is conflict-free and $E$ attacks each argument which does not belong to $E$ ( $\forall A \in {\mathcal{A}}\setminus E$, $\exists B \in E$ such that $B {\mathcal{R}}A$).

Note that in all the above definitions, each attacker of a given argument is considered separately (the ``direct attack'' as a whole is not considered). [9] proves that:

• Any admissible set of ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$ is included in a preferred extension of ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$.
• There always exists at least one preferred extension of ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$.
• If ${{<}{\mathcal{A}},{\mathcal{R}}{>}}$ is well-founded then there is only one preferred extension which is also the only stable extension.
• Any stable extension is also a preferred extension (the converse is false).
• There is not always a stable extension.

Property 12   The set of leaves (i.e. $\{A \vert {\mathcal{R}}{^-}(A) = \varnothing\}$) is included in every preferred extension and in every stable extension.