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3.3 Arguments
In order to turn the decision framework presented in the previous section into a concrete
argumentation framework, we need first to define the notion of argument. Since we want
that our AF not only suggests some decisions but also provides an intelligible explanation of
them, we adopt a tree-like structure of arguments. We adopt here the tree-like structure for
arguments proposed in (Vreeswijk, 1997) and we extend it with presumptions on the missing
information.
Informally, an argument is a deduction for a conclusion from a set of presumptions
represented as a tree, with conclusion at the root and presumptions at the leaves. Nodes in
this tree are connected by the inference rules, with sentences matching the head of an inference
rule connected as parent nodes to sentences matching the body of the inference rule as children
nodes. The leaves are either presumptions or the special extra-logical symbol
, standing for
an empty set of premises. Formally:
Definition 10
(Structured argument)
.
Let
DF
=
DL
P
I
T
P
RV
be a decision
framework. A
structured argument
built upon
DF
is composed by a conclusion, some premises, some
presumptions, and some sentences. These elements are abbreviated by the corresponding prefixes (e.g.
conc
stands for conclusion). A structured argument
¯
Acanbe:
1. a
hypothetical argument
built upon an unconditional ground statement. If L is either a decision
literal or an presumable belief literal (or its strong/weak negation), then the argument built upon a
ground instance of this presumable literal is defined as follows:
conc
(
,
sm
,
,
,
,
A
)=
L,
A
premise
(
)=∅
,
A
psm
(
)=
{
}
L
,
A
sent
(
)=
{
}
L
.
or
2. a
built argument
built upon a rule such that all the literals in the body are the conclusion of
arguments.
3
), then the
trivial
argument A built upon this fact is
2.1) If f is a fact in
T
(i.e.
body
(
f
)=
defined as follows:
A
(
)=
(
f
)
,
conc
head
A
premise
(
)=
{}
,
A
psm
(
)=
∅
,
A
sent
(
)=
{
head
(
f
)
}
.
2.2) If r is a rule in
T
with
body
(
r
)=
{
L
1
,...,
L
j
,
∼
L
j
+
1
,...,
∼
L
n
}
and there is a collection
A
1
,...,
A
n
of structured arguments
{
}
such that, for each strong literal L
i
∈
body
(
r
)
,L
i
=
A
i
A
i
conc
(
)
with i
≤
j and for each weak literal
∼
L
i
∈
body
(
r
)
,
∼
L
i
=
conc
(
)
with i
>
j,
we define the
tree
argument A built upon the rule r and the set
A
1
,...,
A
n
{
}
of structured
arguments as follows:
conc
(
A
)=
head
(
r
)
,
A
premise
(
)=
body
(
r
)
,
A
A
i
psm
(
)
=
psm
(
)
,
A
i
∈{
A
1
,...,
A
n
}
A
A
i
)
sent
(
)
=
body
(
r
)
∪{
head
(
r
)
}∪
sent
(
.
A
i
∈{
A
1
,...,
A
n
}
3
denotes the unconditionally true statement.
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