Information Technology Reference
In-Depth Information
and hence (2) implies
ʳ
[
k,c,h,d
]
(
a
j
,a
l
) just when it implies
ʳ
[
c,k,d,h
]
(
a
l
,a
j
). Since
we wish the signature to record the numbers of (unordered) pairs of constants
with certain behaviour and the decision to write (2) using
ordered
pairs
a
j
,a
l
with
j<l
is merely a matter of convention,
ʳ
[
k,c,h,d
]
and
ʳ
[
c,k,d,h
]
should play
the same role. Note that
n
[
k,c,h,d
]
=
n
[
c,k,d,h
]
and that the sum of the
m
k
is
m
.
We remark that
n
uniquely determines
m
.
Furthermore we define
n
k,c
to be the number of
j, l
∈
A
such that (3) holds
1
,...,
2
q
for some
h, d
∈{
}
.Wehave
n
k,c
=
h,d∈{
1
,...,
2
q
k,k
=
h,d∈{
1
,...,
2
n
[
k,c,h,d
]
(
k
=
c
)
,
n
[
k,k,h,d
]
.
(5)
q
}
}
h≤d
It may seem that requiring a probability function to give state descriptions
with the same signature equal probability is equivalent to Ex. However, this is
not the case: the following principle is strictly stronger than Ex:
Binary Exchangeability, BEx.
For a state description ʘ
(
a
1
,...,a
m
)
of L
the probability w
(
ʘ
)
depends only on the signature of ʘ.
Even so many probability functions do satisfy BEx, and there is a represen-
tation theorem for them similar to the de Finnetti representation theorem for
probability function satisfying Ex, see [9]. We shall employ
4
the following result
from [9]:
Theorem 1.
Let w be a probability function and assume that w satisfies BEx.
Let ʔ
(
a
1
,...,a
m
)
be a partial state description as in (2), s, t, r, g
∈{
1
,...,m
}
,
s<t, r<g,
r, g
∈
/
A and ʳ a binary atom such that ʔ
∧
ʳ
(
a
r
,a
g
)
∧
ʳ
(
a
s
,a
t
)
is consistent. Then
w
(
ʳ
(
a
r
,a
g
)
|
ʔ
)
≤
w
(
ʳ
(
a
s
,a
t
)
|
ʔ
∧
ʳ
(
a
r
,a
g
))
.
(6)
An Example.
Let
p
=
q
=1so
L
=
{
R,Q
}
where
R
is unary and
Q
is binary.
We have
ʲ
1
(
x
)=
R
(
x
)
∧
Q
(
x, x
)
ʴ
1
(
x, y
)=
Q
(
x, y
)
ʲ
2
(
x
)=
R
(
x
)
∧¬
Q
(
x, x
)
ʴ
2
(
x, y
)=
¬
Q
(
x, y
)
Q
(
x, x
)
ʲ
4
(
x
)=
¬R
(
x
)
∧¬Q
(
x, x
)
and the binary atom
ʳ
[2
,
3
,
1
,
2]
(
x, y
) is the formula
ʲ
3
(
x
)=
¬
R
(
x
)
∧
R
(
x
)
∧¬
Q
(
x, x
)
∧¬
R
(
y
)
∧
Q
(
y,y
)
∧
Q
(
x, y
)
∧¬
Q
(
y,x
)
.
4
We remark that this theorem from a forthcoming paper is not essential for the result
presented here (Theorem 2) in the sense that only a special case of it is needed in
the proof and it could be added to the assumptions. Indeed the original Johnson's
proof in [7] for the unary case introduces a corresponding postulate although it was
subsequently shown to be unnecessary. To be precise, we could replace the usage
of Theorem 1 by
assuming
that (6) from Theorem 1 holds when
ʔ
(
a
1
,a
2
,a
3
)=
ʲ
k
(
a
1
)
∧ ʲ
c
(
a
2
)
∧ ʲ
k
(
a
3
)(where1
≤ k ≤ c ≤
2
p
+
q
).