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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 ).
 
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