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.

Let be an argumentation system, we have:

**Conflict-free set**- A set
is
conflict-free if and only if
such
that
.
**Collective defence**- Consider , . collectively defends if and only if , if such that . defends all its elements if and only if , collectively defends .

[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).

**Admissible semantics (admissible set)**- A set
is admissible if
and only if is conflict-free and defends all its elements.
**Preferred semantics (preferred extension)**- A set
is a
preferred extension if and only if is maximal for set
inclusion among the admissible sets.
**Stable semantics (stable extension)**- A set is a stable extension if and only if is conflict-free and attacks each argument which does not belong to ( , such that ).

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 is included in a preferred extension of .
- There always exists at least one preferred extension of .
- If 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.

Marie-Christine Lagasquie 2005-02-04