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# K. Popper - Logic scientific discovery

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logico-subjective theory, of which Keynes3 is the principal exponent, treats the probability relation as a special kind of logical relationship between two statements. The two extreme cases of this probability relation are derivability and contradiction: a statement q ‘gives’,4 it is said, to another statement p the probability 1 if p follows from q. In case p and q contradict each other the probability given by q to p is zero. Between these extremes lie other probability relations which, roughly speaking, may be inter- preted in the following way: The numerical probability of a statement p (given q) is the greater the less its content goes beyond what is already contained in that statement q upon which the probability of p depends (and which ‘gives’ to p a probability). The kinship between this and the psychologistic theory may be seen from the fact that Keynes deﬁnes probability as the ‘degree of rational belief’. By this he means the amount of trust it is proper to accord to a statement p in the light of the information or knowledge which we get from that statement q which ‘gives’ probability to p. A third interpretation, the objective interpretation, treats every numerical *1 The reasons why I count the logical interpretation as a variant of the subjective inter- pretation are more fully discussed in chapter *ii of the Postscript, where the subjective interpretation is criticized in detail. Cf. also appendix *ix. 2 Waismann, Logische Analyse des Wahrscheinlichkeitsbegriﬀs, Erkenntnis 1, 1930, p. 237: ‘Prob- ability so deﬁned is then, as it were, a measure of the logical proximity, the deductive connection between the two statements’. Cf. also Wittgenstein, op. cit., proposition 5.15 ﬀ. 3 J. M. Keynes, A Treatise on Probability, 1921, pp. 95 ﬀ. 4 Wittgenstein, op. cit., proposition 5.152: ‘If p follows from q, the proposition q gives to the proposition p the probability 1. The certainty of logical conclusion is a limiting case of probability.’ some structural components of a theory of experience136 probability statement as a statement about the relative frequency with which an event of a certain kind occurs within a sequence of occurrences.5 According to this interpretation, the statement ‘The probability of the next throw with this die being a ﬁve equals 1/6’ is not really an assertion about the next throw; rather, it is an assertion about a whole class of throws of which the next throw is merely an element. The state- ment in question says no more than that the relative frequency of ﬁves, within this class of throws, equals 1/6. According to this view, numerical probability statements are only admissible if we can give a frequency interpretation of them. Those prob- ability statements for which a frequency interpretation cannot be given, and especially the non-numerical probability statements, are usually shunned by the frequency theorists. In the following pages I shall attempt to construct anew the theory of probability as a (modiﬁed) frequency theory. Thus I declare my faith in an objective interpretation; chieﬂy because I believe that only an objective theory can explain the application of the probability calculus within empirical science. Admittedly, the subjective theory is able to give a consistent solution to the problem of how to decide probability state- ments; and it is, in general, faced by fewer logical diﬃculties than is the objective theory. But its solution is that probability statements are non- empirical; that they are tautologies. And this solution turns out to be utterly unacceptable when we remember the use which physics makes of the theory of probability. (I reject that variant of the subjective theory which holds that objective frequency statements should be derived from subjective assumptions—perhaps using Bernoulli’s the- orem as a ‘bridge’:6 I regard this programme for logical reasons as unrealizable.) 5 For the older frequency theory cf. the critique of Keynes, op. cit., pp. 95 ﬀ., where special reference is made to Venn’s The Logic of Chance. For Whitehead’s view cf. section 80 (note 2). Chief representatives of the new frequency theory are: R. von Mises (cf. note 1 to section 50), Dörge, Kamke, Reichenbach and Tornier. *A new objective interpretation, very closely related to the frequency theory, but diﬀering from it even in its mathemat- ical formalism, is the propensity interpretation, introduced in sections *53 ﬀ. of my Postscript. 6 Keynes’s greatest error; cf. section 62, below, especially note 3. *I have not changed my view on this point even though I now believe that Bernoulli’s theorem may serve as a ‘bridge’ within an objective theory—as a bridge from propensities to statistics. See also appendix *ix and sections *55 to *57 of my Postscript. probability 137 49 THE FUNDAMENTAL PROBLEM OF THE THEORY OF CHANCE The most important application of the theory of probability is to what we may call ‘chance-like’ or ‘random’ events, or occurrences. These seem to be characterized by a peculiar kind of incalculability which makes one disposed to believe—after many unsuccessful attempts— that all known rational methods of prediction must fail in their case. We have, as it were, the feeling that not a scientist but only a prophet could predict them. And yet, it is just this incalculability that makes us conclude that the calculus of probability can be applied to these events. This somewhat paradoxical conclusion from incalculability to cal- culability (i.e. to the applicability of a certain calculus) ceases, it is true, to be paradoxical if we accept the subjective theory. But this way of avoiding the paradox is extremely unsatisfactory. For it entails the view that the probability calculus is not a method of calculating predictions, in contradistinction to all the other methods of empirical science. It is, according to the subjective theory, merely a method for carrying out logical transformations of what we already know; or rather what we do not know; for it is just when we lack knowledge that we carry out these transformations.1 This conception dissolves the paradox indeed, but it does not explain how a statement of ignorance, interpreted as a frequency statement, can be empirically tested and corroborated. Yet this is precisely our problem. How can we explain the fact that from incalculability—that is, from ignorance— we may draw conclusions which we can interpret as statements about empirical frequencies, and which we then ﬁnd brilliantly corroborated in practice? Even the frequency theory has not up to now been able to give a satisfactory solution of this problem—the fundamental problem of the theory of chance, as I shall call it. It will be shown in section 67 that this problem is connected with the ‘axiom of convergence’ which is an integral part 1 Waismann, Erkenntnis 1, 1930, p. 238, says: ‘There is no other reason for introducing the concept of probability than the incompleteness of our knowledge.’ A similar view is held by C. Stumpf (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Klasse, 1892, p. 41). *I believe that this widely held view is responsible for the worst confusions. This will be shown in detail in my Postscript, chapters *ii and *v. some structural components of a theory of experience138 of the theory in its present form. But it is possible to ﬁnd a satisfactory solution within the framework of the frequency theory, after this axiom has been eliminated. It will be found by analysing the assump- tions which allow us to argue from the irregular succession of single occurrences to the regularity or stability of their frequencies. 50 THE FREQUENCY THEORY OF VON MISES A frequency theory which provides a foundation for all the principal theorems of the calculus of probability was ﬁrst proposed by Richard von Mises.1 His fundamental ideas are as follows. The calculus