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•
L
is a formal language consisting of countably many sentences, and
•
R
is a countable set of inference rules of the form r:
α
←
α
1
,...,
α
n
where
α
∈L
, called the
head
of the rule (denoted
head
(
r
)
),
α
1
,...,
α
n
∈L
, called the
body
(denoted
body
(
r
)
), and n
≥
0
.
If
n
). A deductive
system does not distinguish between domain-independent axioms/rules, which belong to
the specification of the logic, and domain-dependent axioms/rules, which represents a
background theory.
Due to the abductive nature of the practical reasoning, we define and construct arguments
by reasoning backwards. Therefore, arguments do not include irrelevant information such as
sentences not used to derive a conclusion.
=
0, then the inference rule represents an axiom (written simply as
α
(
L
R
)
Definition 4
(Deduction)
.
Given a deductive system
,
and a selection function f , a (backward)
based on a set of premises P is a sequence of sets S
1
,...,
S
m
,whereS
1
=
deduction of a conclusion
α
{α}
=
{
}
≤
<
,S
m
P
, and for every
1
i
m, where
σ
is the sentence occurrence in S
i
selected by f :
is not in P then S
i
+
1
=
S
i
−{σ}∪
σ ←
1.
if
σ
S for some inference rule of the form
S in the set of
R
inference rules
;
is in P then S
i
+
1
=
S
i
.
Deductions are the basis for the construction of arguments in assumption-based
argumentation. In order to obtain an argument from a backward deduction, we restrict the
premises to those ones that are taken for granted (called
assumptions
). Moreover, we need to
specify when one sentence
contraries
an assumptions to specify when one argument attacks
another. In this respect, an ABF considers a deductive system augmented by a non-empty
set of assumptions and a (total) mapping from assumptions to their contraries. In order to
perform decision making, we consider the generalisation of the original assumption-based
argumentation framework and the computational mechanisms, whereby multiple contraries
are allowed (Gartner & Toni, 2007).
2.
if
σ
Definition 5
(ABF)
.
An
assumption-based argumentation framework
is a tuple
abf
=
L
,
R
,
A
sm
,
C
on
where:
•
(
L
,
R
)
is a deductive system;
•
A
sm
⊆L
is a non-empty set of
assumptions
.Ifx
∈A
sm, then there is no inference rule in
R
such that x is the head of this rule;
2
L
is a (total) mapping from assumptions into set of sentences in
•
C
on:
A
sm
→
L
,i.e. their
contraries
.
In the remainder of the paper, we restrict ourselves to finite deduction systems, i.e.
with finite languages and finite set of rules. For simplicity, we restrict ourselves to flat
frameworks (Bondarenko et al., 1993), i.e. whose assumptions do not occur as conclusions
of inference rules, such as logic programming or the argumentation framework proposed in
this paper.
In the assumption-based approach, the set of assumptions supporting a conclusion
encapsulates the essence of the argument.
Definition 6
(Argument)
.
An
argument
for a conclusion is a deduction of that conclusion whose
premises are all assumptions. We denote an argument
a
for a conclusion
α
supported by a set of
assumptions
A
simply as
a
:
A
α
.
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