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Theorem 1.
(i) If conditions 1.1, 1.2 and 1.3a hold, then
pref
(
a
,
b
)
(resp.
eqpref
(
a
,
b
)
)
is a sceptically justified conclusion of the argumentation framework iff
a
is strictly
(resp. equally) preferred over
b
according to the lexicographic preference ordering.
(ii) If conditions 1.1, 1.2 and 1.3b hold, then
pref
(
a
,
b
)
(resp.
eqpref
(
a
,
b
)
) is a scepti-
cally justified conclusion of the argumentation framework iff
a
is strictly (resp. equally)
preferred over
b
according to the ceteris paribus preference ordering.
Proof.
We prove the theorem for strict preference. The same line of argument can be
followed for equal preference.
(i)
: Suppose
a
is strictly lexicographically preferred over
b
. This means that there
is an importance level on which
a
satisfies more interests (say,
P
1
,...,
P
n
)than
b
(say,
P
1
,...,
P
m
,
n
>
m
), and on all more important levels,
a
and
b
satisfy an equal number
of interests. In this case, we can construct the following arguments, where the first two
arguments are subarguments of the third (note that these arguments can also be built if
m
is equal to 0, by using the empty set count).
⇐
P
1
(
a
)
...
P
n
(
a
)
I
(
P
1
)
...
I
(
P
n
)
P
1
≈ ...≈
P
n
sat
(
a
,
[
P
1
]
,
n
)
P
1
(
b
)
...
P
m
(
b
)
I
(
P
1
)
...
I
(
P
m
)
P
1
≈ ...≈
P
m
sat
(
b
,
[
P
1
]
,
m
)
sat
(
a
,
[
P
1
]
,
n
)
sat
(
b
,
[
P
1
]
,
m
)
P
1
≈
P
1
n
>
m
pref
(
a
,
b
)
We will now try to defeat this argument. Premises of the type
P
(
a
)
are justified by
condition 1.2. Premises of the type
I
(
P
)
and
P
1
≈
P
2
cannot be defeated (conditions
1.1 and 1.3a). There are three inferences we can try to undercut (the last inference of
the argument and the last inferences of two subarguments). For the first count, this can
only be done if there is another
P
j
such that
I
(
P
j
)
and
P
j
≈
and
P
j
(
a
)
is the case. However,
P
1
...
P
n
encompass all interests that
a
satisfies on this level,
so count undercut is not possible. The same argument holds for the other count. At this
point it is useful to note that these two counts are the only ones that are undefeated. Any
lesser count will be undercut by the count undercutter that takes all of
P
1
...
P
n
(resp.
P
1
...
P
m
) into account. Such an undercutter has no defeaters, so any non-maximal count
is not justified. The undercutter of
prefinf
(
a
,
b
,
[
P
1
])
is based on two counts. We have
seen that any non-maximal count will be undercut. If the maximal counts are used,
we have
n
=
m
for undercutter arguments that use
Q
P
and
P
j
∈{
P
1
,...,
P
n
}
P
, since we have that on all
more important levels than
[
P
1
]
,
a
and
b
satisfy an equal number of interests. So the
undercutter inference rule cannot be applied since
n
=
m
is not true. For that reason,
a rebutting argument with conclusion
pref
(
b
,
a
)
will not be justified. This means that
for every possible type of defeat, either the defeat is inapplicable or the defeater is itself
defeated by undefeated arguments. This means that the argument is sceptically justified.
⇒
: Suppose that
a
is not strictly lexicographically preferred over
b
. This means that
for all importance levels
[
P
]
, either
a
does not satisfy more interests than
b
on that
