popper-logic-scientific-discovery

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some structural components of a theory of experience
how special assumptions may be combined with an equal-chance
hypothesis, leading in each case to different definitions of the reference
sequences and the primary properties for which equal distribution is
assumed.
The following example may perhaps illustrate the fact that frequency
assumptions are indispensable even when we may be inclined to do
without them.
Imagine a waterfall. We may discern some odd kind of regularity:
the size of the currents composing the fall varies; and from time to
time a splash is thrown off from the main stream; yet throughout all
such variations a certain regularity is apparent which strongly suggests
a statistical effect. Disregarding some unsolved problems of
hydrodynamics (concerning the formation of vortices, etc.) we can,
in principle, predict the path of any volume of water—say a group
of molecules—with any desired degree of precision, if sufficiently
precise initial conditions are given. Thus we may assume that it
would be possible to foretell of any molecule, far above the waterfall,
at which point it will pass over the edge, where it will reach
bottom, etc. In this way the path of any number of particles may, in
principle, be calculated; and given sufficient initial conditions we
should be able, in principle, to deduce any one of the individual
statistical fluctuations of the waterfall. But only this or that individual
fluctuation could be so obtained, not the recurring statistical regularities
we have described, still less the general statistical distribution as
such. In order to explain these we need statistical estimates—at
least the assumption that certain initial conditions will again and
again recur for many different groups of particles (which amounts
to a universal statement). We obtain a statistical result if, and only if,
we make such specific statistical assumptions—for example, assumptions
concerning the frequency distribution of recurring initial
conditions.
71 FORMALLY SINGULAR PROBABILITY STATEMENTS
I call a probability statement ‘formally singular’ when it ascribes a
probability to a single occurrence, or to a single element of a certain