popper-logic-scientific-discovery

398
new appendices
(−)
d(a 1)p(a 2).
This result shows that our theorem (1) is violated. But an assignment
of the kind described which leads to this result is unavoidable if we
wish to avoid assigning the same probability—that is, zero—to all
theories. Consequently our theorem (1) entails the assignment of zero
probabilities to all theories.
Wrinch and Jeffreys themselves arrived at a very different result.
They believed that the possibility of empirical knowledge required the
possibility of raising the probability of a law by accumulating evidence
in its favour. From this they concluded that (2) must be false, and
further, that a legitimate method must exist of assigning non-zero
probabilities to an infinite sequence of explanatory theories. Thus
Wrinch and Jeffreys drew very strong positive conclusions from the
‘transcendental’ argument (as I called it in the preceding appendix). 5
Believing, as they did, that an increase in probability means an increase
in knowledge (so that obtaining a high probability becomes an aim of
science), they did not consider the possibility that we may learn from
experience more and more about universal laws without ever increasing their probability;
that we may test and corroborate some of them better and better,
thereby increasing their degree of corroboration without altering their probability
whose value remains zero.
Jeffreys and Wrinch never described the sequence of theories, and
the assignment of probability values, in a sufficiently clear way. Their
main idea, called the ‘simplicity postulate’, 6 was that the theories
should be so ordered that their complexity, or number of parameters,
increases, while the probabilities which they assign to them decrease;
this, incidentally, would mean that any two theories of the sequence
would violate our theorem (1). But this way of ordering cannot be
carried through, as Jeffreys himself noticed. For there may be theories
5 Cf. note 3 to appendix *vii.
6 In his Theory of Probability, § 3.0, Jeffreys says of the ‘simplicity postulate’ that it ‘is not . . .
a separate postulate but an immediate application of rule 5’. But all that rule 5 contains,
by way of reference to rule 4 (both rules are formulated in § 1.1) is a very vague form of
the ‘transcendental’ principle. Thus it does not affect our argument.