Information Technology Reference
In-Depth Information
Preliminaries
The most fundamental and generally accepted principle is that of
Constant
Exchangeability, Ex
,whichsaysthatif
ʸ
(
a
1
,...,a
m
) is a sentence of
L
and
a
1
,...,a
m
is any other choice of distinct constant symbols from amongst the
a
1
,a
2
,...
then
ʸ
(
a
1
,...,a
m
)and
ʸ
(
a
1
,...,a
m
) should have the same probabil-
ity. We shall assume Ex. We will also need the
Principle of Regularity, Reg
,which
requires any consistent quantifier free sentence of
L
to have non-zero probability.
For the purpose of this article we will restrict our attention to languages
with finitely many unary predicate symbols
R
1
,...,R
p
, finitely many binary
relation symbols
Q
1
,...,Q
q
and no relation symbols of higher arities. Sentences
ʘ
(
a
1
,...,a
m
)oftheform
m
p
q
i
=1
±
i
=1
±
R
i
(
a
j
)
∧
Q
i
(
a
j
,a
l
)
(1)
j
=1
j,l
∈{
1
,...,m
}
2
where
Q
i
(
a
j
,a
l
),
are called
state descriptions for a
1
,...,a
m
. Note that state descriptions for
a
1
,...,a
m
are mutually exclusive and exhaustive (their disjunction is a tau-
tology).
Any probability function
w
is uniquely determined by its values on state
descriptions and in many situations it suces to think of probability func-
tions as functions defined on state descriptions and such that probabilities of
state descriptions for
a
1
,...,a
m
sum to 1, and probabilities of state descriptions
for
a
1
,...,a
m
+1
which extend a given state description
ʘ
(
a
1
,...,a
m
)sumto
w
(
ʘ
(
a
1
,...,a
m
)), see [5]. We shall use this in the present paper.
In the unary context, that is, when
q
= 0 and the language consists merely of
the unary predicates
R
1
,...,R
p
, a state description for
a
1
,...,a
m
is any sentence
±
R
i
(
a
j
) denotes one of
R
i
(
a
j
),
¬
R
i
(
a
j
) and similarly for
±
j
=1
i
=1
±
R
i
(
a
j
). The formulae
i
=1
±
R
i
(
x
) are called atoms and denoted
ʱ
1
(
x
)
,...,ʱ
2
p
(
x
). Unary state descriptions are usually written as
j
=1
ʱ
h
j
(
a
j
),
where
h
j
∈{
1
,...,
2
p
.
Johnson's Su
cientness Postulate, JSP.
w
ʱ
i
(
a
m
+1
)
}
|
j
=1
ʱ
h
j
(
a
j
)
de-
pends only on m and m
i
,wherem
i
is the number of times that i appears amongst
the h
j
.
The classical result discussed in the Introduction tells us that as long as the
language has at least two predicates (
p
2), the only probability functions
satisfying Reg, Ex and JSP are the Carnap's
c
ʻ
functions (0
<ʻ
≥
≤∞
) defined
as follows
2
⊛
⊞
⊠
=
m
i
+
ʻ
2
−p
m
+
ʻ
m
⊝
ʱ
i
(
a
m
+1
)
c
ʻ
|
ʱ
h
j
(
a
j
)
j
=1
where
m
i
is as above.
2
Note that the definition does yield the values of
c
ʻ
on all state descriptions and gives
a probability function.
