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

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in
the case of infinite classes, to the intuitive ‘more’ or ‘fewer,’ in order
to find out whether any of them may be used for the purpose of
comparing classes of forbidden events.

(1) The concept of the cardinality (or power) of a class. This concept
cannot help us to solve our problem, since it can easily be shown that
the classes of potential falsifiers have the same cardinal number for all
theories.1

(2) The concept of dimension. The vague intuitive idea that a cube in
some way contains more points than, say, a straight line can be clearly
formulated in logically unexceptionable terms by the set-theoretical
concept of dimension. This distinguishes classes or sets of points
according to the wealth of the ‘neighbourhood relations’ between their
elements: sets of higher dimension have more abundant neighbour-
hood relations. The concept of dimension which allows us to compare

*1 For further remarks concerning the aims of science, see appendix *x, and section *15 of
the Postscript, and my paper ‘The Aim of Science’, Ratio 1, 1957, pp. 24–35.
1 Tarski has proved that under certain assumptions every class of statements is denumer-
able (cf. Monatshefte f. Mathem. u. Physik 40, 1933, p. 100, note 10). *The concept of measure
is inapplicable for similar reasons (i.e. because the set of all statements of a language is
denumerable).

degrees of testability 97

classes of ‘higher’ and ‘lower’ dimension, will be used here to tackle
the problem of comparing degrees of testability. This is possible
because basic statements, combined by conjunction with other basic
statements, again yield basic statements which, however, are ‘more
highly composite’ than their components; and this degree of com-
position of basic statements may be linked with the concept of dimen-
sion. However, it is not the compositions of the forbidden events but
that of the permitted ones which will have to be used. The reason is
that the events forbidden by a theory can be of any degree of com-
position; on the other hand, some of the permitted statements are
permitted merely because of their form or, more precisely, because
their degree of composition is too low to enable them to contradict the
theory in question; and this fact can be used for comparing
dimensions.*1

(3) The subclass relation. Let all elements of a class α be also elements
of a class β, so that α is a subclass of β (in symbols: α ⊂ β). Then either
all elements of β are in their turn also elements of α—in which case the
two classes are said to have the same extension, or to be identical—or
there are elements of β which do not belong to α. In the latter case the
elements of β which do not belong to α form ‘the difference class’ or
the complement of α with respect to β, and α is a proper subclass of β. The
subclass relation corresponds very well to the intuitive ‘more’ and
‘fewer’, but it suffers from the disadvantage that this relation can only
be used to compare the two classes if one includes the other. If there-
fore two classes of potential falsifiers intersect, without one being
included in the other, or if they have no common elements, then the
degree of falsifiability of the corresponding theories cannot be com-
pared with the help of the subclass relation: they are non-comparable
with respect to this relation.

*1 The German term ‘komplex’ has been translated here and in similar passages by ‘composite’
rather than by ‘complex’. The reason is that it does not denote, as does the English ‘com-
plex’, the opposite of ‘simple’. The opposite of ‘simple’ (‘einfach’) is denoted, rather, by
the German ‘kompliziert’. (Cf. the first paragraph of section 41 where ‘kompliziert’ is trans-
lated by ‘complex’.) In view of the fact that degree of simplicity is one of the major topics of
this book, it would have been misleading to speak here (and in section 38) of degree of
complexity. I therefore decided to use the term ‘degree of composition’ which seems to fit the
context very well.

some structural components of a theory of experience98

33 DEGREES OF FALSIFIABILITY COMPARED BY
MEANS OF THE SUBCLASS RELATION

The following definitions are introduced provisionally, to be
improved later in the course of our discussion of the dimensions of
theories.*1

(1) A statement x is said to be ‘falsifiable in a higher degree’ or
‘better testable’ than a statement y, or in symbols: Fsb(x) > Fsb(y), if and
only if the class of potential falsifiers of x includes the class of the
potential falsifiers of y as a proper subclass.

(2) If the classes of potential falsifiers of the two statements x and y
are identical, then they have the same degree of falsifiability, i.e.
Fsb(x) = Fsb(y).

(3) If neither of the classes of potential falsifiers of the two state-
ments includes the other as a proper subclass, then the two statements
have non-comparable degrees of falsifiability (Fsb(x) || Fsb(y)).

If (1) applies, there will always be a non-empty complement class.
In the case of universal statements, this complement class must be
infinite. It is not possible, therefore, for the two (strictly universal)
theories to differ in that one of them forbids a finite number of single
occurrences permitted by the other.

The classes of potential falsifiers of all tautological and metaphysical
statements are empty. In accordance with (2) they are, therefore, iden-
tical. (For empty classes are subclasses of all classes, and hence also of
empty classes, so that all empty classes are identical; which may be
expressed by saying that there exists only one empty class.) If we denote
an empirical statement by ‘e’, and a tautology or a metaphysical state-
ment (e.g. a purely existential statement) by ‘t’ or ‘m’ respectively, then
we may ascribe to tautological and metaphysical statements a zero
degree of falsifiability and we can write: Fsb(t) = Fsb(m) = 0, and
Fsb(e) > 0.

A self-contradictory statement (which we may denote by ‘c’) may be
said to have the class of all logically possible basic statements as its class
of potential falsifiers. This means that any statements whatsoever is
comparable with a self-contradictory statement as to its degree of

*1 See section 38, and the appendices i, *vii, and *viii.

degrees of testability 99

falsifiability. We have Fsb(c) > Fsb(e) > 0.*2 If we arbitrarily put
Fsb(c) = 1, i.e. arbitrarily assign the number 1 to the degree of falsifi-
ability of a self-contradictory statement, then we may even define an
empirical statement e by the condition 1 > Fsb(e) > 0. In accordance
with this formula, Fsb(e) always falls within the interval between 0 and
1, excluding these limits, i.e. within the ‘open interval’ bounded by
these numbers. By excluding contradiction and tautology (as well as
metaphysical statements) the formula expresses at the same time both
the requirement of consistency and that of falsifiability.

34 THE STRUCTURE OF THE SUBCLASS
RELATION. LOGICAL PROBABILITY

We have defined the comparison of the degree of falsifiability of two
statements with the help of the subclass relation; it therefore shares all
the structural properties of the latter. The question of comparability can
be elucidated with the help of a diagram (fig. 1), in which certain
subclass relations are depicted on the left, and the corresponding

Figure 1

*2 See however now appendix *vii.

some structural components of a theory of experience100

testability relations on the right. The Arabic numerals on the right
correspond to the Roman numerals on the left in such a way that a
given Roman numeral denotes the class of the potential falsifiers of that
statement which is denoted by the corresponding Arabic numeral. The
arrows in the diagram showing the degrees of testability run from
the better testable or better falsifiable statements to those which are
not so well testable. (They therefore correspond fairly precisely to
derivability-arrows; see section 35.)

It will be seen from the diagram that various sequences of subclasses
can be distinguished and traced, for example the sequence i–ii–iv or
i–iii–v, and that these could