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# K. Popper - Logic scientific discovery

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What I attempted was merely this: Some great scientists and philosophers have made assertions about simplicity and its value for science. I suggested that some of these 1 Schlick, ibid., p. 148. simplicity 131 assertions can be better understood if we assume that when speaking about simplicity they sometimes had testability in mind. This eluci- dates even some of Poincaré’s examples, though it clashes with his views. Today I should stress two further points: (1) We can compare theor- ies with respect to testability only if at least some of the problems they are supposed to solve coincide. (2) Ad hoc hypotheses cannot be compared in this way. some structural components of a theory of experience132 8 PROBABILITY In this chapter I shall only deal with the probability of events and the problems it raises. They arise in connection with the theory of games of chance, and with the probabilistic laws of physics. I shall leave the problems of what may be called the probability of hypotheses—such ques- tions as whether a frequently tested hypothesis is more probable than one which has been little tested—to be discussed in sections 79 to 85 under the title of ‘Corroboration’. Ideas involving the theory of probability play a decisive part in mod- ern physics. Yet we still lack a satisfactory, consistent deﬁnition of probability; or, what amounts to much the same, we still lack a satisfac- tory axiomatic system for the calculus of probability. The relations between probability and experience are also still in need of clariﬁca- tion. In investigating this problem we shall discover what will at ﬁrst seem an almost insuperable objection to my methodological views. For although probability statements play such a vitally important rôle in empirical science, they turn out to be in principle impervious to strict falsiﬁcation. Yet this very stumbling block will become a touchstone upon which to test my theory, in order to ﬁnd out what it is worth. Thus we are confronted with two tasks. The ﬁrst is to provide new founda- tions for the calculus of probability. This I shall try to do by developing the theory of probability as a frequency theory, along the lines followed by Richard von Mises, but without the use of what he calls the ‘axiom of convergence’ (or ‘limit axiom’), and with a somewhat weakened ‘axiom of randomness’. The second task is to elucidate the relations between probability and experience. This means solving what I call the problem of decid- ability of probability statements. My hope is that these investigations will help to relieve the present unsatisfactory situation in which physicists make much use of prob- abilities without being able to say, consistently, what they mean by ‘probability’.*1 47 THE PROBLEM OF INTERPRETING PROBABILITY STATEMENTS I shall begin by distinguishing two kinds of probability statements: those which state a probability in terms of numbers—which I will call numerical probability statements—and those which do not. Thus the statement, ‘The probability of throwing eleven with two (true) dice is 1/18’, would be an example of a numerical probability statement. Non-numerical probability statements can be of various kinds. ‘It is very probable that we shall obtain a homogeneous mixture *1 Within the theory of probability, I have made since 1934 three kinds of changes. (1) The introduction of a formal (axiomatic) calculus of probabilities which can be interpreted in many ways—for example, in the sense of the logical and of the frequency interpretations discussed in this book, and also of the propensity interpretation discussed in my Postscript. (2) A simpliﬁcation of the frequency theory of probability through carrying out, more fully and more directly than in 1934, that programme for reconstructing the frequency theory which underlies the present chapter. (3) The replacement of the objective interpretation of probability in terms of fre- quency by another objective interpretation—the propensity interpretation—and the replace- ment of the calculus of frequencies by the neo-classical (or measure-theoretical) formalism. The ﬁrst two of these changes date back to 1938 and are indicated in the book itself (i.e. in this volume): the ﬁrst by some new appendices, *ii to *v, and the second—the one which aﬀects the argument of the present chapter—by a number of new footnotes to this chapter, and by the new appendix *vi. The main change is described here in footnote *1 to section 57. The third change (which I ﬁrst introduced, tentatively, in 1953) is explained and developed in the Postscript, where it is also applied to the problems of quantum theory. some structural components of a theory of experience134 by mixing water and alcohol’, illustrates one kind of statement which, suitably interpreted, might perhaps be transformed into a numerical probability statement. (For example, ‘The probability of obtaining . . . is very near to 1’.) A very diﬀerent kind of non-numerical probability statement would be, for instance, ‘The discovery of a physical eﬀect which contradicts the quantum theory is highly improbable’; a state- ment which, I believe, cannot be transformed into a numerical prob- ability statement, or put on a par with one, without distorting its meaning. I shall deal ﬁrst with numerical probability statements; non- numerical ones, which I think less important, will be considered afterwards. In connection with every numerical probability statement, the ques- tion arises: ‘How are we to interpret a statement of this kind and, in particular, the numerical assertion it makes?’ 48 SUBJECTIVE AND OBJECTIVE INTERPRETATIONS The classical (Laplacean) theory of probability deﬁnes the numerical value of a probability as the quotient obtained by dividing the number of favourable cases by the number of equally possible cases. We might disregard the logical objections which have been raised against this deﬁnition,1 such as that ‘equally possible’ is only another expression for ‘equally probable’. But even then we could hardly accept this deﬁn- ition as providing an unambiguously applicable interpretation. For there are latent in it several diﬀerent interpretations which I will classify as subjective and objective. A subjective interpretation of probability theory is suggested by the fre- quent use of expressions with a psychological ﬂavour, like ‘mathemat- ical expectation’ or, say, ‘normal law of error’, etc.; in its original form it is psychologistic. It treats the degree of probability as a measure of the feel- ings of certainty or uncertainty, of belief or doubt, which may be 1 Cf. for example von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, 1928, pp. 62 ﬀ.; 2nd edition, 1936, pp. 84 ﬀ.; English translation by J. Neyman, D. Sholl, and E. Rabinowitsch, Probability, Statistics and Truth, 1939, pp. 98 ﬀ. *Although the classical deﬁnition is often called ‘Laplacean’ (also in this book), it is at least as old as De Moivre’s Doctrine of Chances, 1718. For an early objection against the phrase ‘equally possible’, see C. S. Peirce, Collected Papers 2, 1932 (ﬁrst published 1878), p. 417, para. 2, 673. probability 135 aroused in us by certain assertions or conjectures. In connection with some non-numerical statements, the word ‘probable’ may be quite satisfactorily translated in this way; but an interpretation along these lines does not seem to me very satisfactory for numerical probability statements. A newer variant of the subjective interpretation,*1 however, deserves more serious consideration here. This interprets probability statements not psychologically but logically, as assertions about what may be called the ‘logical proximity’2 of statements. Statements, as we all know, can stand in various logical relations to one another, like derivability, incompatibility, or mutual independence; and the