popper-logic-scientific-discovery

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new appendices
either if δ is comparatively large (roughly, if δ ≈ 1/2) or—in case δ is
small—if n, the sample size, is a large number. We therefore find that
P(e, h) − P(e), and thus our functions E and C, can only be large if δ is
small and n large; or in other words, if e is a statistical report asserting a good fit
in a large sample.
Thus the test-statement e will be the better the greater its precision
(which will be inverse to 2δ) and consequently its refutability or content,
and the larger the sample size n, that is to say, the statistical
material required for testing e. And the test-statement e so constructed
may then be confronted with the results of actual observations.
We see that accumulating statistical evidence will, if favourable,
increase E and C. Accordingly, E or C may be taken as measures of the
weight of the evidence in favour of h; or else, their absolute values may
be taken as measuring the weight of the evidence with respect to h.
8. Since the numerical value of P(e, h) can be determined with the
help of the binomial theorem (or of Laplace’s integral), and since
especially for a small δ we can, by (6), put P(e) equal to 2δ, it is
possible to calculate P(e, h)—P(e) numerically, and also E.
Moreover, we can calculate for any given n a value δ = P(e)/2 for
which P(e, h)—P(e) would become a maximum. (For n = 1,000,000,
we obtain δ = 0.0018.) Similarly, we can calculate another value, of
δ = P(e)/2, for which E would become a maximum. (For the same n,
we obtain δ = 0.00135, and E(h, e) = 0.9946).
For a universal law h such that h = ‘P(a, b) = 1’ which has passed
n severe tests, all of them with the result a, we obtain, first, C(h, e) =
E(h, e), in view of P(h) = 0; and further, evaluating P(e) with the help
of the Laplacean distribution and d = o, we obtain C(h, e) = n/
(n + 2) = 1 − (2/(n + 2)). It should be remembered, however, that
non-statistical scientific theories have as a rule a form totally different
from that of the h here described; moreover, if they are forced into this
form, then any instance a, and therefore the ‘evidence’ e, would
become essentially non-observational.* 3
* 3 One might, however, speak of the degree of corroboration of a theory with respect to a
field of application, in the sense of appendices i and *viii; and one might then use the method
of calculation here described. But since this method ignores the fine-structure of content
and probability, it is very crude, as far as non-statistical theories are concerned. Thus in
these cases, we may rely upon the comparative method explained in footnote 7 to the