Summary of `Popper's Contributions to the Theory of Probability' by ...

Summary of `Popper's Contributions to the Theory of Probability' by David Miller
The rst draft of this paper, planned for The Cambridge Companion to Popper (forthcoming),
was written in 2006. Many changes have been made since. Further criticisms and suggestions
for improvement will be valued, even if it is impossible to incorporate them all before publication.
0 The body of the paper deals with four distinct aspects of Popper's philosophical and logical
work in probability theory and its applications: the version of the frequency interpretation
developed in Chapter VI of Logik der Forschung in 1934; the propensity interpretation,
rst proposed in the 1950s in The Postscript, and rened and extended in the next 40
years; the axiomatization of probability, which was begun in 1938, and has its most
detailed development in appendices ∗iv and ∗v of The Logic of Scientic Discovery; and the
application of the theory of probability to methodological problems, which culminated
in the work of Popper & Miller in 1983 and 1987 on the impossibility of inductive support.
1 Popper was in agreement with the outlines of von Mises's frequency (or statistical) theory
of probability, which was based on the axioms of convergence and of randomness, but highlighted
two problems that Mises did not deal with satisfactorily. One was the fundamental
problem of the theory of chance, the problem of explaining `[t]he seemingly paradoxical inference
from the unpredictability and irregularity of singular events to the applicability of
the rules of the probability calculus': how does order arise out of disorder? His ingenious
response was to strengthen the axiom of randomness in such a way that a form of the
axiom of convergence became deducible. He was thus able to establish Bernoulli's theorem
without any explicit assumption of convergence. The second problem was the problem of
decidability, the problem that probability hypotheses are not empirically falsiable: no observable
sequence of Heads and Tails formally refutes a hypothesis such as p(Heads) = 1/2;
a run of a million Heads is not impossible. Nor can any nite sequence refute a hypothesis
of randomness in Mises's sense. By Popper's criterion of demarcation, the unfalsiability of
hypotheses of probability excludes them from empirical science. His solution was to specify
a class of nite random sequences with relative frequency 1/2, for which convergence
is provable, and only later to consider innite random sequences. He then invited us to
understand the hypothesis p(Heads) = 1/2 to imply that from the beginning the sequence
of Heads and Tails is (approximately) as random as possible. A methodological rule xes
the degree of approximation at which hypotheses are deemed refuted. But here, Popper
claimed, probability hypotheses do not dier signicantly from other universal hypotheses.
2 A third problem with interpreting physical probabilities as frequencies, at which Mises
merely shrugged his shoulders, is the problem of the single case. By the mid-1950s Popper
was convinced that an interpretation that permitted genuine single-case probabilities would
dissolve many of the unsolved paradoxes of quantum theory. In his rst published exposition
he wrote that the propensity interpretation `diers from the purely statistical or
frequency interpretation only in this that it considers the probability as a characteristic
property of the experimental arrangement rather than a property of a sequence'. This
summary tells less than half the full story. The truth is that there are three sizable steps
on the road from the frequency interpretation to the full-blown propensity interpretation
of Popper's later writings. The rst step abandons the frequency interpretation's emphasis
on collectives, and makes probabilities manifestations of dispositional properties of experimental
statistical arrangements. This step allows the frequency theory to give an account
of probabilities for some singular events. The second step, the crucial one, still relates
probabilities to experimental arrangements, but understands them as propensities to produce
one outcome or another in each single case, rather than as dispositions to produce
frequencies. The resort to relative frequencies is relegated to the (not at all unimportant)

2
level of empirical testing. Popper did not always succeed in keeping properly distinct these
two ways of grounding probabilities in physical dispositions. The third step, which came
later, releases the single-case probabilities or propensities from reliance on anything akin to
experimental arrangements, and begins to make intelligible the attribution of probabilities
not only to repeated events but to unique events (such as the outbreak of World War 2).
Popper emphasized that the propensity interpretation was supposed to a new physical (or
metaphysical) hypothesis, and he painted in contrast to the classical mechanistic world
view a metaphysical picture of `a world of propensities'. It is its decidedly metaphysical
avour that has led many critics (wrongly) to reject the third step in Popper's development.
3 Popper regarded as one of his principal achievements in the philosophy of science the
unmasking of the traditional claim (in the work of Laplace, of Keynes, and of Carnap, for
example) that the degree to which a scientic hypothesis H is conrmed by evidence E is
appropriately measured by p(H, E), where p is a function that satises the axioms of the
calculus of probability. But what are these axioms? Being unaware of the axiomatization
published by Kolmogorov in 1933 Popper set out to axiomatize the theory of probability
in the most general way possible. His eorts led to a strikingly original set of axioms
B, somewhat similar to, but more powerful than, axioms oered by de Finetti. Popper's
system takes as primitive a set S on which are dened a binary operation⁀ (shortened
to concatenation), a singulary operation ′ , and a two-place function p that takes values
in the real numbers. The main features of the system B are these: (a) there are no
explicit assumptions about the algebraic structure of ⟨S,⁀ , ′ ⟩; that it is reducible to a
Boolean algebra is a consequence of the probability axioms alone; (b) the axioms of B are
autonomously independent, which means that none of them is a disguised formulation of
some Boolean law; (c) the term p(a, c) is well dened, even for terms that turn out to
stand for zero elements (contradictions) in the Boolean algebra; (d) there are no explicit
assumptions concerning the range of the function p, yet it is a theorem that 0 ≤ p(a, c) ≤ 1.
4 Popper distinguished in Logik der Forschung between objective (physical) interpretations
of probability and subjective interpretations. Among the latter he classed the logical interpretation,
according to which the probability function p measures the 'logical proximity' of
statements. Like many authors he shied away from the idea (due originally to von Kries,
and embraced by Carnap in 1950) that the values of the logical probability function can
all be determined a priori. He argued, nevertheless, that something positive could be said,
for example (a) that the logical probability of a statement varies inversely with its content,
and (b) that the logical probability of a universal hypothesis must be zero. From (a), which
is true, he concluded that science prefers improbable hypotheses to probable ones, even
after severe testing, and from (b), which is not true, that Carnap's programme to dene
the degree of conrmation of a hypothesis by its logical probability cannot succeed. He
pointed out something that in its time was scandalous, but is now generally accepted, that
if we dene the relation of conrmation of H by E by the inequality s(H, E) > 0 (where
s(H, E), the degree of support that H enjoys on E, is dened as equal to p(H, E)−p(H)), then
we cannot sensibly also call p(H, E) the degree of conrmation of H by E. In 1983 Popper
& Miller purported to demonstrate that although a hypothesis H (for which p(H) > 0) can
be supported by evidence E, in the sense that s(H, E) = p(H, E) − p(H) > 0, such support
is wholly due to the existence of a deductive relation between H and E (namely, that their
contents overlap), and disappears when they are, in the appropriate sense, deductively
disconnected. They argued, not very originally, that that part of the content of H that
properly goes beyond the content of E, called the excess content E(H, E) of H over E, should
be identied with with the conditional E → H; and, more originally, that E(H, E) is never
supported by E: indeed, s(E(H, E), E) ≤ 0. In other words, probabilistic support is not
inductive support. This argument has met with a great deal of critical reaction, especially
with regard to the identication of the excess function E(H, E) with the conditional E → H.