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in just (2). Therefore, we have M*
X
P
M. Furthermore, since M*
¬
M, we have
M*
X
P
M.
Let T be a set of beliefs, and let P be a predicate occurs in T. During the
extension process, we should seek formula ϕ
P
such that for any model M of
T
∧
ϕ
P
there is no model M* of T which satisfies
M
*
X
P
M
The formula T
∧
ϕ
P
which satisfies such a principle of minimization is called
circumscription of P on T.
Let P* be a predicate constant which has the same number of variables of that
of P. Then, it can be demonstrated that any model of the following formula is a
minimal model of P on T:
(
∀
x P
*
(x)
ŗ
P(x))
∧
¬(
∀
x P(x)
ŗ
P
*
(x))
∧
T(P
*
)
Therefore, any model of the following formula is a minimal model of P on T:
¬ ((
∀
x P
*
(x))
ŗ
P(x))
∧
¬(
∀
x P(x)
ŗ
P
*
(x))
∧
T(P
*
))
As a result, the following is a circumscription formula of P on T:
∀
P
*
¬((
∀
x P
*
(x)
ŗ
P(x))
∧
¬(
∀
x P(x)
ŗ
P
*
(x))
∧
T(P
*
))
ϕ
P
=
Definition 2.19
A formula
ϕ
is entailed by the predicate circumscription of P in
A, written as T
ż
P
ϕ
or CIRC(T, P)
żϕ
, if and only if
ϕ
is true with respect to all
the
X
P
-minimal model of P.
The predicate circumscription CIRC(T,P) of P in T is defined as:
∀
P
*
¬((
∀
x)(P
*
(x)
ŗ
P(x))
∧
¬(
∀
x)(P(x)
CIRC(TP) = T
∧
ŗ
P
*
(x))
∧
T(P
*
))
(2.7)
It can also be rewrited as:
CIRC(TP) = T
∧
∀
P
*
((T(P
*
)
∧
(
∀
x)(P
*
(x)
ŗ
P(x)))
ŗ
(
∀
x)(P(x)
ŗ
P
*
(x)))
(2.8)
Since it is a formula of high-order logic, we can rewrite it as:
ϕ
P
=
∀
P
*
((T(P
*
)
∧
(
∀
x)(P
*
(x)
ŗ
P(x)))
ŗ
(
∀
x)(P(x)
ŗ
P
*
(x)))
(2.9)
∀
x (P* (x)
ŗ
P(X)), then
∀
x
It states that if there is a P* such that T(P*) and
(P(x)
ŗ
P* (x)) can be deduced as a conclusion.
