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to be counted, otherwise we would conclude incorrect preferences. To ensure that only
maximal counts are used, we provide an inference scheme to construct arguments that
undercut non-maximal counts (inference scheme 6). An argument of this type says that
any count which is not maximal is not applicable. Inference scheme 7 says that an out-
come
a
is preferred over an outcome
b
if the number of interests of a certain importance
level that
a
satisfies is higher than the number of interests on that same level that
b
satis-
fies. Inference scheme 8 undercuts scheme 7 if there is a more important level than that
of
P
on which
a
and
b
do not satisfy the same number of interests. Finally, inference
schemes 9 and 10 do the same as 7 and 8, but for equal preference.
Defeat.
The most common type of defeat is rebuttal. An argument rebuts another argu-
ment if its conclusion contradicts conclusion of the other argument. Conclusions con-
tradict each other if one is the negation of the other, or if they are preference or impor-
tance statements that are incompatible (e.g.
pref
(
a
,
b
)
and
pref
(
b
,
a
)
,or
pref
α
(
a
,
b
)
α
α
and
eqpref
(
a
,
b
)
). Defeat by rebuttal is mutual. Another type of defeat is undercut. An
undercutter is an argument for the inapplicability of an inference used in another argu-
ment. Undercut works only one way. Defeat is defined recursively, which means that
rebuttal can attack an argument on all its premises and (intermediate) conclusions, and
undercut can attack it on all its inferences.
α
Definition 5. (Defeat)
An argument
A defeats
an argument
B
(
A
B
)if
conc
(
A
)
and
conc
(
B
)
are contradictory (
rebuttal
), or
conc
(
A
)=
'
inf
(
B
)
is inapplicable' (
under-
cut
), or
A
defeats a subargument of
B
.
→
Let us return to the example. With the information from the knowledge base, the argu-
ments
A
and
B
in Table 5 can be formed.
A
advocates a preference for
c
, based on the
interest wealth.
B
advocates a preference for
f
, based on the interest family. Without
an ordering on these interests, no decision between these arguments can be made. But
if
wealth
M
family
is known, argument
C
can be made, which undercuts
B
. Similarly,
with
family
M
wealth
, argument
D
can be made, which undercuts
A
.
Validity.
If some conditions in the input knowledge base (KB) hold, it can be
shown that the proposed argumentation framework models ceteris paribus and lexi-
cographic preference. In the following, we consider a single audience and leave out the
subscript
α
.
Condition 1.
Let
C
be a set of interests to be used as criteria, with importance order
.
(1) For all
P
,'
I
(
P
)
'isinKBiff
P
∈C
.
(2) For all
P
∈C
,
a
,'
P
(
a
)
' is a conclusion of a sceptically justified argument iff
a
satisfies
P
.
(3) The relative importance among interests is
(a) a total preorder,
(b) the identity relation,
and for all
P
,
Q
∈C
,'
P
Q
'isinKBiff
P
Q
,and'
P
≈
Q
'isinKBiff
P
≈
Q
.
