popper-logic-scientific-discovery

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new appendices
behaviour of E and C. Consequently, no paradox arises; and we may
quite naturally measure the weight of the evidence e with respect to the hypothesis h
by either E(h, e), or C(h, e), or else—keeping more closely to Keynes’
idea—by the absolute values of either of these functions.
7. If, as in our case, h is a statistical hypothesis, and e the report of
the results of statistical tests of h, then C(h, e) is a measure of the degree
to which these tests corroborate h, exactly as in the case of a nonstatistical
hypothesis.
It should be mentioned, however, that as opposed to the case of a
non-statistical hypothesis, it might sometimes be quite easy to estimate
the numerical values of E(h, e) and even of C(h, e), if h is a statistical
hypothesis. 6 (In 8 I will briefly indicate how such numerical calculations
might proceed in simple cases, including, of course, the case of
h = ‘p(a, b) = 1’.)
The expression
(4)
P(e, h) − P(e)
is crucial for the functions E(h, e) and C(h, e); indeed, these functions
are nothing but two different ways of ‘normalizing’ the expression (4);
they thus increase and decrease with (4). This means that in order to
find a good test-statement e—one which, if true, is highly favourable to
h—we must construct a statistical report e such that (i) e makes P(e,
h)—which is Fisher’s ‘likelihood’ of h given e—large, i.e. nearly equal
to 1, and such that (ii) e makes P(e) small, i.e. nearly equal to o. Having
constructed a test statement e of this kind, we must submit e itself to
empirical tests. (That is to say, we must try to find evidence refuting e.)
importance. In this Journal, 1954, 5, 324, I suggested that we define
C(x, y, z) = (P(y, xz) − P(y, z))/(P(y, xz) − P(xy, z) + P(y, z)).
From this we obtain C(x, y) by assuming z (the ‘background knowledge’) to be tautological,
or non-existent (if this manner of expression is preferred).
6 It is quite likely that in numerically calculable cases, the logarithmic functions suggested
by Hamblin and Good (see my ‘Second Note’) will turn out to be improvements
upon the functions which I originally suggested. Moreover, it should be noted that from
a numerical point of view (but not from the theoretical point of view underlying our
desiderata) my functions and the ‘degree of factual support’ of Kemeny and Oppenheim
will in most cases lead to similar results.