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•
S
is
admissible
(denoted
adm
aaf
(
S
)
) iff
S
is conflict-free and
S
attacks every argument
a
such
that
a
attacks some arguments in
S
;
•
S
is
preferred
iff
S
is maximally admissible (with respect to set inclusion);
•
S
is
complete
iff
S
is admissible and
S
contains all arguments
a
such that
S
attacks all attacks
against
a
;
•
S
is
grounded
iff
S
is minimally complete (with respect to set inclusion);
•
S
is
ideal
iff
S
is admissible and it is contained in every preferred set.
These declarative model-theoretic
semantics
of the AAF capture various degrees of
justification, ranging from very permissive conditions, called
credulous
, to restrictive
requirements, called
sceptical
. The semantics of an admissible (or preferred) set of arguments
is credulous, in that it sanctions a set of arguments as acceptable if it can successfully
dispute every arguments against it, without disputing itself. However, there might be
several conflicting admissible sets. That is the reason why various sceptical semantics have
been proposed for the AAF, notably the grounded semantics and the sceptically preferred
semantics, whereby an argument is accepted if it is a member of all maximally admissible
(preferred) sets of arguments. The ideal semantics was not present in (Dung, 1995), but it has
been proposed recently (Dung et al., 2007) as a less sceptical alternative than the grounded
semantics but it is, in general, more sceptical than the sceptically preferred semantics.
Example 1
(AAF)
.
In order to illustrate the previous notions,
let us consider the abstract
argumentation framework
aaf
=
A
,
attacks
where:
•
A
=
{
a
,
b
,
c
,
d
}
;
•
.
The following graph represents this AAF, whereby the fact that “x
attacks
y” is depicted by a directed
arrow from x to y:
attacks
=
{
(
a
,
a
)
,
(
a
,
b
)
,
(
b
,
a
)
,
(
c
,
d
)
,
(
d
,
c
)
}
abc d
We can notice that:
•
{}
is grounded;
•
{
b
,
d
}
and
{
b
,
c
}
are preferred.
•
{
b
}
is the maximal ideal set.
As previously mentionned, Dung's seminal calculus of opposition deals neither with the
nature of arguments nor with the semantics of the attacks relation.
Unlike the abstract argumentation, assumption-based argumentation considers neither the
arguments nor the attack relations as primitives. Arguments are built by reasoning backwards
from conclusions to assumptions given a set of inference rules. Moreover, the attack relation is
defined in terms of a notion of “contrary”(Bondarenko et al., 1993; Dung et al., 2007). Actually,
assumption-based argumentation frameworks (ABFs, for short) are concrete instances of
AAFs built upon deductive systems.
The abstract view of argumentation does not deal with the problem of finding arguments and
attacks amongst them. Typically, arguments are built by joining rules, and attacks arise from
conflicts amongst such arguments.
Definition 3
(DS)
.
A deductive system is a pair
(
L
,
R
)
where
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