K. Popper - Logic scientific discovery
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K. Popper - Logic scientific discovery

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(γ) = αF″ (γ)

1 Von Mises’s term is ‘choice’ (‘Auswahl’).
2 The answer to this question is given by the general division theorem (cf. appendix ii).

probability 145

In this case we say (following Hausdorff3) that the properties β and γ
are ‘mutually independent, within the reference-class α’. The relation of
independence is a three-termed relation and is symmetrical in the
properties β and γ.4 If two properties β and γ are (mutually)
independent within a reference-class α we can also say that the prop-
erty γ is, within α, insensitive to the selection of β-elements; or perhaps
that the reference-class α is, with respect to this property γ, insensitive
to a selection according to the property β.

The mutual independence, or insensitiveness, of β and γ within α
could also—from the point of view of the subjective theory—be inter-
preted as follows: If we are informed that a particular element of the
class α has the property β, then this information is irrelevant if β and γ
are mutually independent within α; irrelevant namely, to the question
whether this element also has the property γ, or not.*1 If, on the other
hand, we know that γ occurs more often (or less often) in the subclass
α.β (which has been selected from α according to β), then the infor-
mation that an element has the property β is relevant to the question
whether this element also has the property γ or not.5

3 Hausdorff, Berichte über die Verhandlungen der sächsischen Ges. d. Wissenschaften, Leipzig, mathem.-
physik. Klasse 53, 1901, p. 158.
4 It is even triply symmetrical, i.e. for α, β and γ, if we assume β and γ also to be finite. For
the proof of the symmetry assertion cf. appendix ii, (1s) and (1s). *The condition of
finitude for triple symmetry asserted in this note is insufficient. I may have intended to
express the condition that β and γ are bounded by the finite reference class α, or, most
likely, that α should be our finite universe of discourse. (These are sufficient conditions.)
The insufficiency of the condition, as formulated in my note, is shown by the following
counter-example. Take a universe of 5 buttons; 4 are round (α); 2 are round and black
(αβ); 2 are round and large (αγ); 1 is round, black, and large (αβγ); and 1 is square,
black, and large (�βγ). Then we do not have triple symmetry since αF″ (γ) ≠ βF″ (γ).
*1 Thus any information about the possession of properties is relevant, or irrelevant, if
and only if the properties in question are, respectively, dependent or independent. Rele-
vance can thus be defined in terms of dependence, but the reverse is not the case. (Cf. the
next footnote, and note *1 to section 55.)
5 Keynes objected to the frequency theory because he believed that it was impossible to
define relevance in its terms; cf. op. cit., pp. 103 ff. *In fact, the subjective theory cannot define
(objective) independence, which is a serious objection as 1 show in my Postscript, chapter *ii,
especially sections *40 to *43.

some structural components of a theory of experience146

54 FINITE SEQUENCES. ORDINAL SELECTION AND
NEIGHBOURHOOD SELECTION

Let us suppose that the elements of a finite reference-class α are numbered
(for instance that a number is written on each button in the box), and
that they are arranged in a sequence, in accordance with these ordinal
numbers. In such a sequence we can distinguish two kinds of selection
which have special importance, namely selection according to the
ordinal number of an element, or briefly, ordinal selection, and
selection according to its neighbourhood.

Ordinal selection consists in making a selection, from the sequence α, in
accordance with a property β which depends upon the ordinal number
of the element (whose selection is to be decided on). For example β
may be the property even, so that we select from α all those elements
whose ordinal number is even. The elements thus selected form a
selected sub-sequence. Should a property γ be independent of an ordinal
selection according to β, then we can also say that the ordinal selection is
independent with respect to γ; or we can say that the sequence α is,
with respect to γ, insensitive to a selection of β-elements.

Neighbourhood selection is made possible by the fact that, in ordering the
elements in a numbered sequence, certain neighbourhood relations are
created. This allows us, for example, to select all those members whose
immediate predecessor has the property γ; or, say, those whose first
and second predecessors, or whose second successor, have the property
γ; and so on.

Thus if we have a sequence of events—say tosses of a coin—we have
to distinguish two kinds of properties: its primary properties such as
‘heads’ or ‘tails’, which belong to each element independently of its
position in the sequence; and its secondary properties such as ‘even’ or
‘successor of tails’, etc., which an element acquires by virtue of its
position in the sequence.

A sequence with two primary properties has been called ‘alterna-
tive’. As von Mises has shown, it is possible to develop (if we are
careful) the essentials of the theory of probability as a theory of alterna-
tives, without sacrificing generality. Denoting the two primary proper-
ties of an alternative by the figures ‘1’ and ‘0’, every alternative can be
represented as a sequence of ones and zeros.

probability 147

Now the structure of an alternative can be regular, or it can be more or
less irregular. In what follows we will study this regularity or irregularity
of certain finite alternatives more closely.*1

55 N-FREEDOM IN FINITE SEQUENCES

Let us take a finite alternative α, for example one consisting of a
thousand ones and zeros regularly arranged as follows:

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 . . .(α)

In this alternative we have equal distribution, i.e. the relative frequen-
cies of the ones and the zeros are equal. If we denote the relative
frequency of the property 1 by ‘F″ (1)’ and that of 0 by ‘F″ (0)’, we can
write:

αF″ (1) = αF″ (0) =
1
2(1)

We now select from α all terms with the neighbourhood-property of
immediately succeeding a one (within the sequence α). If we denote this
property by ‘β’, we may call the selected sub-sequence ‘α.β’. It will
have the structure:

1 0 1 0 1 0 1 0 1 0 . . .(α.β)

This sequence is again an alternative with equal distribution. Moreover,
neither the relative frequency of the ones nor that of the zeros has
changed; i.e. we have

α.βF″ (1) = αF″ (1); α.βF″ (0) = αF″ (0).(2)

In the terminology introduced in section 53, we can say that the pri-
mary properties of the alternative α are insensitive to selection according
to the property β; or, more briefly, that α is insensitive to selection
according to β.

*1 I suggest that sections 55 to 64, or perhaps only 56 to 64, be skipped at first reading.
It may even be advisable to turn from here, or from the end of section 55, direct to
chapter 10.

some structural components of a theory of experience148

Since every element of α has either the property β (that of being the
successor of a one) or that of being the successor of a zero, we can
denote the latter property by ‘β

-
’. If we now select the members having

the property β
-
 we obtain the alternative:

0 1 0 1 0 1 0 1 0 . . .(α.β
-
)

This sequence shows a very slight deviation from equal distribution in
so far as it begins and ends with zero (since α itself ends with ‘0, 0’ on
account of its equal distribution). If α contains 2000 elements, then
α.β

-
 will contain 500 zeros, and only 499 ones. Such deviations from

equal distribution (or from other distributions) arise only on account
of the first or last elements: they can be made as small as we please by
making the sequence sufficiently long. For this reason they will be
neglected in what follows; especially since our investigations are to be
extended to infinite sequences, where these deviations vanish. Accord-
ingly, we shall say that the alternative α.β

-
 has equal distribution, and

that the alternative α is insensitive to the selection of elements having the
property β

-
. As