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Definition 15
(Semantics)
.
Let
DF
=
DL
,
P
sm
,
I
,
T
,
P
,
RV
be a decision framework and
AF
=
be our argumentation framework for decision making. For S
A
(
DF
)
,
defeats
⊆A
(
DF
)
a set of
structured arguments, we say that:
•
¯
Sis
subjectively conflict-free
iff
A
,
B
S it is not the case that A defeats B;
∀
∈
¯
Sis
subjectively admissible
(s-admissible for short), denoted
sadm
AF
(
, iff S is subjectively
conflict-free and S defeats every argument
¯
Asuchthat A defeats some argument in S;
S
)
•
We restrict ourselves to the subjective admissibility, but the other Dung's extension-based
semantics (cf Definition 2) can be easily adapted.
Formally, given a structured argument A, let
A
A
dec
(
)=
{
(
)
∈
psm
(
)
|
}
D
a
D
is a decision predicate
be the set of decisions supported by the structured argument A.
The decisions are
suggested
to reach a goal if they are supported by a structured argument
concluding this goal and this argument is a member of an s-admissible set of arguments.
Definition 16
(Credulous decisions)
.
Let
DF
=
DL
P
I
T
P
RV
,
sm
,
,
,
,
be a decision
∈G
be a goal and
D
⊆D
framework, g
be a set of decisions. The decisions
D
credulously argue
for
g iff there exists an argument A in a s-admissible set of arguments such that
conc
(
A
)=
gand
A
dec
(
)=
D
.Wedenote
val
c
(
D
)
G
the set of goals in
for which the set of decisions
D
credulously
argues.
It is worth noticing that the decisions which credulously argue for a goal cannot contain
mutual exclusive alternatives for the same decision predicate. This is due to the fact that a
s-admissible set of arguments is subjectively conflict-free.
If we consider the structured arguments
¯
AandB supporting the decisions
D
(
a
)
and
D
(
b
)
respectively where
a
and
b
are mutually exclusive alternatives, we have
D
(
a
)
I
D
(
b
)
and
and so, either A
defeats
¯
BorB
defeats
A or both of them depending on
D
(
a
)
I
D
(
b
)
the strength of these arguments.
Proposition 1
(Mutual exclusive alternatives)
.
Let
DF
=
DL
P
I
T
P
RV
∈G
be a goal and
AF
=
,
sm
,
,
,
,
be a decision framework, g
be the argumentation framework for decision making built upon
DF
.If
¯
Sbe
a s-admissible set of arguments such that, for some A
A
(
DF
)
,
defeats
S, g
A
A
∈
=
conc
(
)
(
)
∈
psm
(
)
and D
a
,then
A
(
)
∈
psm
(
)
=
D
b
iff a
b.
However, notice that mutual exclusive decisions can be suggested for the same goal through
different s-admissible sets of arguments. This case reflects the credulous nature of our
semantics.
Definition 17
(Skeptical decisions)
.
Let
DF
=
DL
be a goal and
D
,
D
⊆D
be two sets
of decisions. The set
D
of decisions
skeptically argue for
g iff for all s-admissible set of arguments S
such that for some arguments
¯
AinS
conc
(
,
P
sm
,
I
,
T
,
P
,
RV
be a decision framework, g
∈G
A
A
)=
g, then
dec
(
)=
D
.Wedenote
val
s
(
D
)
the set of
goals in
for which the set of decisions
D
skeptically argues. The decisions
D
is
skeptically preferred
to the decisions
D
iff
val
s
(
D
)
P
val
s
(
D
)
G
.
Due to the uncertainties, some decisions satisfy goals for sure if they skeptically argue for
them, or some decisions can possibly satisfy goals if they credulously argue for them. While
the first case is required for convincing a risk-averse agent, the second case is enough to
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