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Let E be an extension for <D, W>. The result can be demonstrated as follows. On
the one hand, if W is inconsistent, then the extension E is also inconsistent
since W⊆E. On the other hand, if E is inconsistent, then any default rule of D
can be applied since any formula can be deduced from E; therefore,
according to Theorem 2.3, we will get the result that E=T
h
(W). So, W is also
inconsistent.
(2) If a closed default theory has an inconsistent extension then this is the unique
extension for this default theory.
In the case that there are more then one extension for a default theory, some
conclusions on the relationship between these extensions have been summed
up also:
(3) If E and F are extensions for a closed normal default theory and if E⊆F, then
E=F.
(4) Suppose ∆
1
=<D
1
,W
1
> and ∆
2
=<D
2
,W
2
> are two different default theories,
and that W
1
⊆W
2
. Suppose further that extensions of ∆
2
is consistent. Then
extensions of ∆
1
is also consistent.
Definition 2.11
A default rule is normal iff it has the following form:
A
:
MB
(2.3)
B
where A and B are any formulas. A default theory
∆
=<D,W> is normal iff every
default rule of D is normal.
Normal default theories hold the following properties:
(1) Every closed normal default theory has an extension.
(2) Suppose E and F are distinct extensions for a closed normal default theory,
then E∪F must be inconsistent.
(3) Suppose ∆=<D,W> is a closed normal default theory, and that D'⊆D.
Suppose further that E'1 and E'2 are distinct extensions of <D',W>. Then ∆
