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popper-logic-scientific-discovery

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422<br />

new appendices<br />

length, to take the simplest example. We make the assumption (favourable<br />

to our opponents) that we are given some <strong>logic</strong>ally necessary finite<br />

lower and upper limits, I and u, to its values. Assume that we are given a<br />

distribution function for the <strong>logic</strong>al probability of this property; for<br />

example, a generalized equidistribution function between l and u. We<br />

may discover that an empirically desirable change of our theories leads<br />

to a non-linear correction of the measure of our physical property<br />

(based, say, on the Paris metre). Then ‘<strong>logic</strong>al probability’ has to be<br />

corrected also; which shows that its metric depends upon our empirical<br />

knowledge, and that it cannot be defined a priori, in purely <strong>logic</strong>al<br />

terms. In other words, the metric of the ‘<strong>logic</strong>al probability’ of a measurable<br />

property would depend upon the metric of the measurable<br />

property itself; and since this latter is liable to correction on the basis<br />

of empirical theories, there can be no purely ‘<strong>logic</strong>al’ measure of<br />

probability.<br />

These difficulties can be largely, but not entirely, overcome by making<br />

use of our ‘background knowledge’ z. But they establish, I think,<br />

the significance of the topo<strong>logic</strong>al approach to the problem of both<br />

degree of confirmation and <strong>logic</strong>al probability.* 2<br />

But, even if we were to discard all metric considerations, we should<br />

still adhere, I believe, to the concept of probability, as defined, implicitly,<br />

by the usual axiom systems for probability. These retain their full<br />

significance, exactly as pure metrical geometry retains its significance<br />

even though we may not be able to define a yardstick in terms of pure<br />

(metrical) geometry. This is especially important in view of the need to<br />

identify <strong>logic</strong>al independence with probabilistic independence (special multiplication<br />

theorem). If we assume a language such as Kemeny’s (which, however,<br />

breaks down for continuous properties) or a language with relative-<br />

* 2 I now believe that I have got over these difficulties, as far as a system S (in the sense of<br />

appendix *iv) is concerned whose elements are probability statements; that is to say, as far as<br />

the <strong>logic</strong>al metric of the probability of probability statements is concerned or, in other words,<br />

the <strong>logic</strong>al metric of secondary probabilities. The method of the solution is described in my<br />

Third Note, points 7 ff.; see especially point *13.<br />

As far as primary properties are concerned, I believe that the difficulties described here<br />

in the text are in no way exaggerated. (Of course, z may help, by pointing out, or<br />

assuming, that we are confronted, in a certain case, with a finite set of symmetrical<br />

or equal possibilities.)

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