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let
L
be a language with finitely many predicate or relation symbols (without
equality) and with countably many constant symbols
a
1
,a
2
,a
3
,...
.Wesaythat
a function
w
which assigns real numbers between 0 and 1 to sentences of
L
is a
probability function if for any sentences
ʸ,ˆ
and
∃
xˈ
(
x
) the following conditions
hold:
•
If
ʸ
is logically valid then
w
(
ʸ
)=1.
•
If
ʸ
and
ˆ
are mutually exclusive then
w
(
ʸ
∨
ˆ
)=
w
(
ʸ
)+
w
(
ˆ
).
ˈ
(
a
n
))
.
The conditional probability of
ʸ
given
ˆ
is defined when
w
(
ˆ
)
•
w
(
∃
xˈ
(
x
)) = lim
nₒ∞
w
(
ˈ
(
a
1
)
∨
ˈ
(
a
2
)
∨
...
∨
= 0 as the ratio
w
(
ʸ∧ˆ
)
w
(
ˆ
)
.
For any given language as above there are many probability functions, but
some of them are better for inductive reasoning then others: for example, it is
desirable that the conditional probability of a 'new' individual having some prop-
erties increases (or at least does not decrease) on the basis of another individual
found having these properties. Similarly, it is desirable that to start with
w
does
not code any unintended information about the individual constants, predicates
or relations; any particular information about them should be supplied by the
evidence on a case by case basis.
Accordingly, a number of arguably rational principles have been formulated
that can be imposed on
w
. Not all of them are compatible and the question
for Pure Inductive Logic is which combination(s) of these principles should be
chosen, what are the probability functions satisfying them and what inferences
they authorize.
Carnap and others in the 20th century studied the situation where there were
only unary predicate symbols and no relation symbols of higher arities in the
language. They independently identified a principle subsequently known as
John-
son's Su
cientness Postulate
, see [7], which yields a particular one-parameter
family of probability functions
c
ʻ
for positive numbers
ʻ
(real or
): the
Carnap
Continuum of Inductive Methods
. Their result is remarkable for its power and
elegance, apparently reducing the choice of a rational probability function down
to the choice of a single parameter
ʻ
, and it has long been seen as the corner
stone of
unary
Pure Inductive Logic. The recent development of Pure Inductive
Logic for languages containing relation symbols of higher arities therefore raises
the major question of whether, and if so how, this continuum can, or should be,
extended to these larger languages.
One possible answer to this is given in [8] where it is shown that for any
positive
ʻ
the family of probability functions
c
ʻ
as we vary the unary language
L
has a natural continuation also to not necessarily unary
L
which preserves the
key property of
Spectrum Exchangeability
(for details see [8]).
In this paper however we shall propose an alternative extension of Carnap
Continuum (to binary languages) based on satisfying a natural generalization of
the original Johnson's Su
cientness Postulate. In this sense then we would claim
that it comes closer to capturing Johnson's and Carnap's original intuitions and
insights.
∞
