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Extending Carnap's Continuum
to Binary Relations
Alena Vencovska
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
alena.vencovska@manchester.ac.uk
Abstract.
We investigate a binary generalization of Carnap's Contin-
uum of Inductive Methods based on a version of Johnson's Sucientness
Postulate for polyadic atoms and determine the probability functions
that satisfy it.
Introduction
The problem of drawing conclusions inductively has puzzled philosophers for
centuries: how to use the available evidence to support a hypothesis to a certain
degree, thus extending the classical deductive reasoning to allow us to reach less-
than-certain but probable conclusions. A substantial contribution to this topic
was made by logical positivism as represented by Rudolf Carnap and others
during the earlier parts of the 20th century. They developed a formal framework
in which rational assignments of probabilities to sentences could be studied,
aiming to capture our reasoning about the world which was assumed to combine
(just) elementary experiences and pure logic.
Carnap's ambitious program has been subsequently largely seen as a failure
on the grounds that the framework cannot be made to correspond to the way
we interpret the world and reason about it (see [1], [2]). Still, the advent of
artificial reasoning agents justifies a re-examination of the purely formal and
uninterpreted aspect of Carnap's proposal, a mode of reasoning which Carnap
himself referred to as
Pure
Inductive Logic.
1
While Carnap experimented with various formal frameworks to investigate
this logic the most transparent seems to be when all statements are expressed
in first order logic with a language involving countably many individuals and
finitely many predicate or relation symbols, and principles are adopted for as-
signing belief values to these statements in a rational, logical fashion indepen-
dently of any intended interpretation.
There are good arguments for identifying belief with subjective probability,
and for identifying belief values based on some evidence with conditional proba-
bilities. So Pure Inductive Logic works with probability functions. More formally,
Research Grant R117181.
1
For Carnap's approach, see for example [3], [4]. For more recent developments, see
for example [5], [6]. [6] also contains an extensive bibliography of related works.
