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# K. Popper - Logic scientific discovery

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elementary character of this formula, asserted in the text, becomes clear if we write: ‘[(a → b).(c → d)] → [(b → c) → (a → d)]’. We then put, in accordance with the text, ‘p’ for ‘b → c’ and ‘q’ for ‘a → d’, etc. 2 What I call higher universality in a statement corresponds roughly to what classical logic might call the greater ‘extension of the subject’; and what I call greater precision corresponds to the smaller extension, or the ‘restriction of the predicate’. The rule concerning the derivability relation, which we have just discussed, can be regarded as clarifying and combining the classical ‘dictum de omni et nullo’ and the ‘nota-notae’ principle, the ‘fundamental principle of mediate predication’. Cf. Bolzano, Wissenschaftslehre II, 1837, §263, Nos. 1 and 4; Külpe, Vorlesungen über Logik (edited by Selz, 1923), §34, 5, and 7. *2 See now also section *15 and chapter *iv of my Postscript, especially section *76, text to note 5. degrees of testability 107 37 LOGICAL RANGES. NOTES ON THE THEORY OF MEASUREMENT If a statement p is more easy to falsify than a statement q, because it is of a higher level of universality or precision, then the class of the basic statements permitted by p is a proper subclass of the class of the basic statements permitted by q. The subclass-relationship holding between classes of permitted statements is the opposite of that holding between classes of forbidden statements (potential falsiﬁers): the two relationships may be said to be inverse (or perhaps comple- mentary). The class of basic statements permitted by a statement may be called its ‘range’.1 The ‘range’ which a statement allows to reality is, as it were, the amount of ‘free play’ (or the degree of freedom) which it allows to reality. Range and empirical content (cf. section 35) are converse (or complementary) concepts. Accordingly, the ranges of two statements are related to each other in the same way as are their logical probabilities (cf. sections 34 and 72). I have introduced the concept of range because it helps us to handle certain questions connected with degree of precision in measurement. Assume that the consequences of two theories diﬀer so little in all ﬁelds of application that the very small diﬀerences between the calculated observable events cannot be detected, owing to the fact that the degree of precision attainable in our measurements is not suﬃciently high. It will then be impossible to decide by experiment between the two theories, without ﬁrst improving our technique of measurement.*1 This shows that the prevailing technique of measurement determines a certain range—a region within which discrepancies between the observations are permitted by the theory. Thus the rule that theories should have the highest attainable degree of testability (and thus allow only the narrowest range) entails the 1 The concept of range (Spielraum) was introduced by von Kries (1886); similar ideas are found in Bolzano. Waismann (Erkenntnis 1, 1930, pp. 228 ﬀ.) attempts to combine the theory of range with the frequency theory; cf. section 72. *Keynes gives (Treatise, p. 88) ‘ﬁeld’ as a translation of ‘Spielraum’, here translated as ‘range’; he also uses (p.224) ‘scope’ for what in my view amounts to precisely the same thing. *1 This is a point which, I believe, was wrongly interpreted by Duhem. See his Aim and Structure of Physical Theory, pp. 137 ﬀ. some structural components of a theory of experience108 demand that the degree of precision in measurement should be raised as much as possible. It is often said that all measurement consists in the determination of coincidences of points. But any such determination can only be correct within limits. There are no coincidences of points in a strict sense.*2 Two physical ‘points’—a mark, say, on the measuring-rod, and another on a body to be measured—can at best be brought into close proxim- ity; they cannot coincide, that is, coalesce into one point. However trite this remark might be in another context, it is important for the ques- tion of precision in measurement. For it reminds us that measurement should be described in the following terms. We ﬁnd that the point of the body to be measured lies between two gradations or marks on the measuring-rod or, say, that the pointer of our measuring apparatus lies between two gradations on the scale. We can then either regard these gradations or marks as our two optimal limits of error, or proceed to estimate the position of (say) the pointer within the interval of the gradations, and so obtain a more accurate result. One may describe this latter case by saying that we take the pointer to lie between two imaginary gradation marks. Thus an interval, a range, always remains. It is the custom of physicists to estimate this interval for every measurement. (Thus following Millikan they give, for example, the elementary charge of the electron, measured in electrostatic units, as e = 4.774.10 − 10, adding that the range of imprecision is ± 0.005.10 − 10.) But this raises a problem. What can be the purpose of replacing, as it were, one mark on a scale by two—to wit, the two bounds of the interval—when for each of these two bounds there must again arise the same question: what are the limits of accuracy for the bounds of the interval? Giving the bounds of the interval is clearly useless unless these two bounds in turn can be ﬁxed with a degree of precision greatly exceed- ing what we can hope to attain for the original measurement; ﬁxed, that is, within their own intervals of imprecision which should thus be smaller, by several orders of magnitude, than the interval they determine for the value of the original measurement. In other words, *2 Note that I am speaking here of measuring, not of counting. (The diﬀerence between these two is closely related to that between real numbers and rational numbers.) degrees of testability 109 the bounds of the interval are not sharp bounds but are really very small intervals, the bounds of which are in their turn still much smaller intervals, and so on. In this way we arrive at the idea of what may be called the ‘unsharp bounds’ or ‘condensation bounds’ of the interval. These considerations do not presuppose the mathematical theory of errors, nor the theory of probability. It is rather the other way round; by analysing the idea of a measuring interval they furnish a back- ground without which the statistical theory of errors makes very little sense. If we measure a magnitude many times, we obtain values which are distributed with diﬀerent densities over an interval—the interval of precision depending upon the prevailing measuring technique. Only if we know what we are seeking—namely the condensation bounds of this interval—can we apply to these values the theory of errors, and determine the bounds of the interval.*3 Now all this sheds some light, I think, on the superiority of methods that employ measurements over purely qualitative methods. It is true that even in the case of qualitative estimates, such as an estimate of the pitch of a musical sound, it may sometimes be possible to give an interval of accuracy for the estimates; but in the absence of measurements, any such interval can be only very vague, since in such cases the concept of condensation bounds cannot be applied. This concept is applicable only where we can speak of orders of magnitude, and therefore only where methods of measurement are deﬁned. I shall make further use of the concept of condensation bounds of intervals of precision in section 68, in connection with the theory of probability. 38 DEGREES OF TESTABILITY COMPARED BY REFERENCE TO DIMENSIONS Till now we have discussed the comparison of theories with respect to their degrees of testability only in so far as they can be compared with the help of the subclass-relation. In some cases this method is quite *3 These considerations are closely connected with, and supported by, some of the results discussed under points 8 ﬀ. of my ‘Third Note’, reprinted