K. Popper - Logic scientific discovery
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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