popper-logic-scientific-discovery

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some structural components of a theory of experience
Thus (2), i.e. the question When is n sufficiently long? has been
reduced to (1′), i.e. the question When is ε small? (and vice versa).
But this means that all three questions could be answered if only we
could decide what particular value of ε is to be neglected as ‘negligibly
small’. Now our methodological rule amounts to the decision to
neglect small values of ε; but we shall hardly be prepared to commit
ourselves for ever to a definite value of ε.
If we put our question to a physicist, that is, if we ask him what ε he
is prepared to neglect—0.001, or 0.000001, or . . . ? he will presumably
answer that ε does not interest him at all; that he has chosen not ε
but n; and that he has chosen n in such a way as to make the correlation
between n and ∆p largely independent of any changes of the value ε which we
might choose to make.
The physicist’s answer is justified, because of the mathematical peculiarities
of the Bernoullian distribution: it is possible to determine for
every n the functional dependence between ε and ∆p.* 4 An examination
of this function shows that for every (‘large’) n there exists a
characteristic value of ∆p such that in the neighbourhood of this value
∆p is highly insensitive to changes of ε. This insensitiveness increases
with increasing n. If we take an n of an order of magnitude which we
should expect in the case of extreme mass-phenomena, then, in the
neighbourhood of its characteristic value, ∆p is so highly insensitive
to changes of ε that ∆p hardly changes at all even if the order of
* 4 The remarks that follow in this paragraph (and some of the discussions later in this
section) are, I now believe, clarified and superseded by the considerations in appendix
*ix; see especially points 8 ff of my Third Note. With the help of the methods there used,
it can be shown that almost all possible statistical samples of large size n will strongly
undermine a given probabilistic hypothesis, that is to say give it a high negative degree of
corroboration; and we may decide to interpret this as refutation or falsification. Of the
remaining samples, most will support the hypothesis, that is to say, give it a high positive
degree of corroboration. Comparatively few samples of large size n will give a probabilistic
hypothesis an undecisive degree of corroboration (whether positive or negative).
Thus we can expect to be able to refute a probabilistic hypothesis, in the sense here
indicated; and we can expect this perhaps even more confidently than in the case of a
non-probabilistic hypothesis. The methodological rule or decision to regard (for a large
n) a negative degree of corroboration as a falsification is, of course, a specific case of the
methodological rule or decision discussed in the present section—that of neglecting
certain extreme improbabilities.