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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