6
DEGREES OF TESTABILITY
Theories may be more, or less, severely testable; that is to say, more, or less, easily falsifiable. The degree of their testability is of significance for the selection of theories.
In this chapter, I shall compare the various degrees of testability or falsifiability of theories through comparing the classes of their potential falsifiers. This investigation is quite independent of the question whether or not it is possible to distinguish in an absolute sense between falsifiable and non-falsifiable theories. Indeed one might say of the present chapter that it ‘relativizes’ the requirement of falsifiability by showing falsifiability to be a matter of degree.
31 A PROGRAMME AND AN ILLUSTRATION
A theory is falsifiable, as we saw in section 23, if there exists at least one non-empty class of homotypic basic statements which are forbidden by it; that is, if the class of its potential falsifiers is not empty. If, as in section 23, we represent the class of all possible basic statements by a circular area, and the possible events by the radii of the circle, then we can say: At least one radius—or perhaps better, one narrow sector whose width may represent the fact that the event is to be ‘observable’—must be incompatible with the theory and ruled out by it. One might then represent the potential falsifiers of various theories by sectors of various widths. And according to the greater and lesser width of the sectors ruled out by them, theories might then be said to have more, or fewer, potential falsifiers. (The question whether this ‘more’ or ‘fewer’ could be made at all precise will be left open for the moment.) It might then be said, further, that if the class of potential falsifiers of one theory is ‘larger’ than that of another, there will be more opportunities for the first theory to be refuted by experience; thus compared with the second theory, the first theory may be said to be ‘falsifiable in a higher degree’. This also means that the first theory says more about the world of experience than the second theory, for it rules out a larger class of basic statements. Although the class of permitted statements will thereby become smaller, this does not affect our argument; for we have seen that the theory does not assert anything about this class. Thus it can be said that the amount of empirical information conveyed by a theory, or its empirical content, increases with its degree of falsifiability.
Let us now imagine that we are given a theory, and that the sector representing the basic statements which it forbids becomes wider and wider. Ultimately the basic statements not forbidden by the theory will be represented by a narrow remaining sector. (If the theory is to be consistent, then some such sector must remain.) A theory like this would obviously be very easy to falsify, since it allows the empirical world only a narrow range of possibilities; for it rules out almost all conceivable, i.e. logically possible, events. It asserts so much about the world of experience, its empirical content is so great, that there is, as it were, little chance for it to escape falsification.
Now theoretical science aims, precisely, at obtaining theories which are easily falsifiable in this sense. It aims at restricting the range of permitted events to a minimum; and, if this can be done at all, to such a degree that any further restriction would lead to an actual empirical falsification of the theory. If we could be successful in obtaining a theory such as this, then this theory would describe ‘our particular world’ as precisely as a theory can; for it would single out the world of ‘our experience’ from the class of all logically possible worlds of experience with the greatest precision attainable by theoretical science. All the events or classes of occurrences which we actually encounter and observe, and only these, would be characterized as ‘permitted’.*1
32 HOW ARE CLASSES OF POTENTIAL FALSIFIERS TO BE COMPARED?
The classes of potential falsifiers are infinite classes. The intuitive ‘more’ and ‘fewer’ which can be applied without special safeguards to finite classes cannot similarly be applied to infinite classes.
We cannot easily get round this difficulty; not even if, instead of the forbidden basic statements or occurrences, we consider, for the purpose of comparison, classes of forbidden events, in order to ascertain which of them contains ‘more’ forbidden events. For the number of events forbidden by an empirical theory is also infinite, as may be seen from the fact that the conjunction of a forbidden event with any other event (whether forbidden or not) is again a forbidden event.
I shall consider three ways of giving a precise meaning, even 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 neighbourhood relations. The concept of dimension which allows us to compare 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 composition of basic statements may be linked with the concept of dimension. 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 composition; 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 β be also elements of a class β, so that β is a subclass of β (in symbols: β ⊂ β). Then either all elements of β are in their turn also elements of β—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 β. In the latter case the elements of β which do not belong to β form ‘the difference class’ or the complement of β with respect to β, and β 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 therefore 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 compared with the help of the subclass relation: they are non-comparable with respect to this relation.
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 statements 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, identical. (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 statement (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 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 falsifiability 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 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 be made more ‘dense’ by introducing new intermediate classes. All these sequences begin in this particular case with 1 and end with the empty class, since the latter is included in every class. (The empty class cannot be depicted in our diagram on the left, just because it is a subclass of every class and would therefore have to appear, so to speak, everywhere.) If we choose to identify class 1 with the class of all possible basic statements, then 1 becomes the contradiction (c); and 0 (corresponding to the empty class) may then denote the tautology (t). It is possible to pass from 1 to the empty class, or from (c) to (t) by various paths; some of these, as can be seen from the right hand diagram, may cross one another. We may therefore say that the structure of the relation is that of a lattice (a ‘lattice of sequences’ ordered by the arrow, or the subclass relation). There are nodal points (e.g. statements 4 and 5) in which the lattice is partially connected. The relation is totally connected only in the universal class and in the empty class, corresponding to the contradiction c and tautology t.
Is it possible to arrange the degrees of falsifiability of various statements on one scale, i.e. to correlate, with the various statements, numbers which order them according to their falsifiability? Clearly, we cannot possibly order all statements in this way;*1 for if we did, we should be arbitrarily making the non-comparable statements comparable. There is, however, nothing to prevent us from picking out one of the sequences from the lattice, and indicating the order of its statements by numbers. In so doing we should have to proceed in such a way that a statement which lies nearer to the contradiction c is always given a higher number than one which lies nearer to the tautology t. Since we have already assigned the numbers 0 and 1 to tautology and contradiction respectively, we should have to assign proper fractions to the empirical statements of the selected sequence.
I do not really intend, however, to single out one of the sequences. Also, the assignment of numbers to the statements of the sequence would be entirely arbitrary. Nevertheless, the fact that it is possible to assign such fractions is of great interest, especially because of the light it throws upon the connection between degree of falsifiability and the idea of probability. Whenever we can compare the degrees of falsifiability of two statements, we can say that the one which is the less falsifiable is also the more probable, by virtue of its logical form. This probability I call*2 ‘logical probability’;1 it must not be confused with that numerical probability which is employed in the theory of games of chance, and in statistics. The logical probability of a statement is complementary to its degree of falsifiability: it increases with decreasing degree of falsifiability. The logical probability 1 corresponds to the degree 0 of falsifiability, and vice versa. The better testable statement, i.e. the one with the higher degree of falsifiability, is the one which is logically less probable; and the statement which is less well testable is the one which is logically more probable.
As will be shown in section 72, numerical probability can be linked with logical probability, and thus with degree of falsifiability. It is possible to interpret numerical probability as applying to a subsequence (picked out from the logical probability relation) for which a system of measurement can be defined, on the basis of frequency estimates.
These observations on the comparison of degrees of falsifiability do not hold only for universal statements, or for systems of theories; they can be extended so as to apply to singular statements. Thus they hold, for example, for theories in conjunction with initial conditions. In this case the class of potential falsifiers must not be mistaken for a class of events—for a class of homotypic basic statements—since it is a class of occurrences. (This remark has some bearing on the connection between logical and numerical probability which will be analysed in section 72.)
35 EMPIRICAL CONTENT, ENTAILMENT, AND DEGREES OF FALSIFIABILITY
It was said in section 31 that what I call the empirical content of a statement increases with its degree of falsifiability: the more a statement forbids, the more it says about the world of experience. (Cf. section 6.) What I call ‘empirical content’ is closely related to, but not identical with, the concept ‘content’ as defined, for instance, by Carnap.1 For the latter I will use the term ‘logical content’, to distinguish it from that of empirical content.
I define the empirical content of a statement p as the class of its potential falsifiers (cf. section 31). The logical content is defined, with the help of the concept of derivability, as the class of all non-tautological statements which are derivable from the statement in question. (It may be called its ‘consequence class’.) So the logical content of p is at least equal to (i.e. greater than or equal to) that of a statement q, if q is derivable from p (or, in symbols, if ‘p → q’*1). If the derivability is mutual (in symbols, ‘p↔q’) then p and q are said to be of equal content.2 If q is derivable from p, but not p from q, then the consequence class of q must be a proper sub-set of the consequence class of p; and p then possesses the larger consequence class, and thereby the greater logical content (or logical force*2).
It is a consequence of my definition of empirical content that the comparison of the logical and of the empirical contents of two statements p and q leads to the same result if the statements compared contain no metaphysical elements. We shall therefore require the following: (a) two statements of equal logical content must also have equal empirical content; (b) a statement p whose logical content is greater than that of a statement q must also have greater empirical content, or at least equal empirical content; and finally, (c) if the empirical content of a statement p is greater than that of a statement q, then the logical content must be greater or else non-comparable. The qualification in (b) ‘or at least equal empirical content’ had to be added because p might be, for example, a conjunction of q with some purely existential statement, or with some other kind of metaphysical statement to which we must ascribe a certain logical content; for in this case the empirical content of p will not be greater than that of q. Corresponding considerations make it necessary to add to (c) the qualification ‘or else non-comparable’.*3
In comparing degrees of testability or of empirical content we shall therefore as a rule—i.e. in the case of purely empirical statements— arrive at the same results as in comparing logical content, or derivability-relations. Thus it will be possible to base the comparison of degrees of falsifiability to a large extent upon derivability relations. Both relations show the form of lattices which are totally connected in the self-contradiction and in the tautology (cf. section 34). This may be expressed by saying that a self-contradiction entails every statement and that a tautology is entailed by every statement. Moreover, empirical statements, as we have seen, can be characterized as those whose degree of falsifiability falls into the open interval which is bounded by the degrees of falsifiability of self-contradictions on the one side, and of tautologies on the other. Similarly, synthetic statements in general (including those which are non-empirical) are placed, by the entailment relation, in the open interval between self-contradiction and tautology.
To the positivist thesis that all non-empirical (metaphysical) statements are ‘meaningless’ there would thus correspond the thesis that my distinction between empirical and synthetic statements, or between empirical and logical content, is superfluous; for all synthetic statements would have to be empirical—all that are genuine, that is, and not mere pseudo-statements. But this way of using words, though feasible, seems to me more likely to confuse the issue than to clarify it.
Thus I regard the comparison of the empirical content of two statements as equivalent to the comparison of their degrees of falsifiability. This makes our methodological rule that those theories should be given preference which can be most severely tested (cf. the anticonventionalist rules in section 20) equivalent to a rule favouring theories with the highest possible empirical content.
36 LEVELS OF UNIVERSALITY AND DEGREES OF PRECISION
There are other methodological demands which may be reduced to the demand for the highest possible empirical content. Two of these are outstanding: the demand for the highest attainable level (or degree) of universality, and the demand for the highest attainable degree of precision.
With this in mind we may examine the following conceivable natural laws:
p: All heavenly bodies which move in closed orbits move in circles: or more briefly: All orbits of heavenly bodies are circles.
q: All orbits of planets are circles.
r: All orbits of heavenly bodies are ellipses.
s: All orbits of planets are ellipses.
The deducibility relations holding between these four statements are shown by the arrows in our diagram. From p all the others follow; from q follows s, which also follows from r; so that s follows from all the others.
Moving from p to q the degree of universality decreases; and q says less than p because the orbits of planets form a proper subclass of the orbits of heavenly bodies. Consequently p is more easily falsified than q: if q is falsified, so is p, but not vice versa. Moving from p to r, the degree of precision (of the predicate) decreases: circles are a proper subclass of ellipses; and if r is falsified, so is p, but not vice versa. Corresponding remarks apply to the other moves: moving from p to s, the degree of both universality and precision decreases; from q to s precision decreases; and from r to s, universality. To a higher degree of universality or precision corresponds a greater (logical or) empirical content, and thus a higher degree of testability.
Both universal and singular statements can be written in the form of a ‘universal conditional statement’ (or a ‘general implication’ as it is often called). If we put our four laws in this form, then we can perhaps see more easily and accurately how the degrees of universality and the degrees of precision of two statements may be compared.
A universal conditional statement (cf. note 6 to section 14) may be written in the form: ‘(x) (φx → fx)’ or in words: ‘All values of x which satisfy the statement function φx also satisfy the statement function fx.’ The statement s from our diagram yields the following example: ‘(x) (x is an orbit of a planet→ x is an ellipse)’ which means: ‘Whatever x may be, if x is an orbit of a planet then x is an ellipse.’ Let p and q be two statements written in this ‘normal’ form; then we can say that p is of greater universality than q if the antecedent statement function of p (which may be denoted by ‘φpx’) is tautologically implied by (or logically deducible from), but not equivalent to, the corresponding statement function of q (which may be denoted by ‘φqx’); or in other words, if ‘(x) (φqx → φpx)’ is tautological (or logically true). Similarly we shall say that p has greater precision than q if ‘(x) (fpx → fqx)’ is tautological, that is if the predicate (or the consequent statement function) of p is narrower than that of q, which means that the predicate of p entails that of q.*1
This definition may be extended to statement functions with more than one variable. Elementary logical transformations lead from it to the derivability relations which we have asserted, and which may be expressed by the following rule:1 If of two statements both their universality and their precision are comparable, then the less universal or less precise is derivable from the more universal or more precise; unless, of course, the one is more universal and the other more precise (as in the case of q and r in my diagram).2
We could now say that our methodological decision—sometimes metaphysically interpreted as the principle of causality—is to leave nothing unexplained, i.e. always to try to deduce statements from others of higher universality. This decision is derived from the demand for the highest attainable degree of universality and precision, and it can be reduced to the demand, or rule, that preference should be given to those theories which can be most severely tested.*2
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 complementary). 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 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 proximity; they cannot coincide, that is, coalesce into one point. However trite this remark might be in another context, it is important for the question 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 exceeding 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, 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 background 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 successful in guiding our choice between theories. Thus we may now say that Pauli’s exclusion principle, mentioned by way of example in section 20, indeed turns out to be highly satisfactory as an auxiliary hypothesis. For it greatly increases the degree of precision and, with it, the degree of testability, of the older quantum theory (like the corresponding statement of the new quantum theory which asserts that antisymmetrical states are realized by electrons, and symmetrical ones by uncharged, and by certain multiply charged, particles).
For many purposes, however, comparison by means of the subclass relation does not suffice. Thus Frank, for example, has pointed out that statements of a high level of universality—such as the principle of the conservation of energy in Planck’s formulation—are apt to become tautological, and to lose their empirical content, unless the initial conditions can be determined ‘... by few measurements,... i.e. by means of a small number of magnitudes characteristic of the state of the system’.1 The question of the number of parameters which have to be ascertained, and to be substituted in the formulae, cannot be elucidated with the help of the sub-class relation, in spite of the fact that it is evidently closely connected with the problem of testability and falsifiability, and their degrees. The fewer the magnitudes which are needed for determining the initial conditions, the less composite*1 will be the basic statements which suffice for the falsification of the theory; for a falsifying basic statement consists of the conjunction of the initial conditions with the negation of the derived prediction (cf. section 28). Thus it may be possible to compare theories as to their degree of testability by ascertaining the minimum degree of composition which a basic statement must have if it is to be able to contradict the theory; provided always that we can find a way to compare basic statements in order to ascertain whether they are more (or less) composite, i.e. compounds of a greater (or a smaller) number of basic statements of a simpler kind. All basic statements, whatever their content, whose degree of composition does not reach the requisite minimum, would be permitted by the theory simply because of their low degree of composition.
But any such programme is faced with difficulties. For generally it is not easy to tell, merely by inspecting it, whether a statement is composite, i.e. equivalent to a conjunction of simpler statements. In all statements there occur universal names, and by analysing these one can often break down the statement into conjunctive components. (For example, the statement ‘There is a glass of water at the place k’ might perhaps be analysed, and broken down into the two statements ‘There is a glass containing a fluid at the place k’ and ‘There is water at the place k’.) There is no hope of finding any natural end to the dissection of statements by this method, especially since we can always introduce new universals defined for the purpose of making a further dissection possible.
With a view to rendering comparable the degrees of composition of all basic statements, it might be suggested that we should choose a certain class of statements as the elementary or atomic ones,2 from which all other statements could then be obtained by conjunction and other logical operations. If successful, we should have defined in this way an ‘absolute zero’ of composition, and the composition of any statement could then be expressed, as it were, in absolute degrees of composition.*2 But for the reason given above, such a procedure would have to be regarded as highly unsuitable; for it would impose serious restrictions upon the free use of scientific language.*3
Yet it is still possible to compare the degrees of composition of basic statements, and thereby also those of other statements. This can be done by selecting arbitrarily a class of relatively atomic statements, which we take as a basis for comparison. Such a class of relatively atomic statements can be defined by means of a generating schema or matrix (for example, ‘There is a measuring apparatus for... at the place..., the pointer of which lies between the gradation marks... and...’). We can then define as relatively atomic, and thus as equi-composite, the class of all statements obtained from this kind of matrix (or statement function) by the substitution of definite values. The class of these statements, together with all the conjunctions which can be formed from them may be called a ‘field’. A conjunction of n different relatively atomic statements of a field may be called an ‘n-tuple of the field’; and we can say that the degree of its composition is equal to the number n.
If there exists, for a theory t, a field of singular (but not necessarily basic) statements such that, for some number d, the theory t cannot be falsified by any d-tuple of the field, although it can be falsified by certain d + 1-tuples, then we call d the characteristic number of the theory with respect to that field. All statements of the field whose degree of composition is less than d, or equal to d, are then compatible with the theory, and permitted by it, irrespective of their content.
Now it is possible to base the comparison of the degree of testability of theories upon this characteristic number d. But in order to avoid inconsistencies which might arise through the use of different fields, it is necessary to use a somewhat narrower concept than that of a field, namely that of a field of application. If a theory t is given, we say that a field is a field of application of the theory t if there exists a characteristic number d of the theory t with respect to this field, and if, in addition, it satisfies certain further conditions (which are explained in appendix i).
The characteristic number d of a theory t, with respect to a field of application, I call the dimension of t with respect to this field of application. The expression ‘dimension’ suggests itself because we can think of all possible n-tuples of the field as spatially arranged (in a configuration space of infinite dimensions). If, for example, d = 3, then those statements which are admissible because their composition is too low form a three-dimensional sub-space of this configuration. Transition from d = 3 to d = 2 corresponds to the transition from a solid to a surface. The smaller the dimension d, the more severely restricted is the class of those permitted statements which, regardless of their content, cannot contradict the theory owing to their low degree of composition; and the higher will be the degree of falsifiability of the theory.
The concept of the field of application has not been restricted to basic statements, but singular statements of all kinds have been allowed to be statements belonging to a field of application. But by comparing their dimensions with the help of the field, we can estimate the degree of composition of the basic statements. (We assume that to highly composite singular statements there correspond highly composite basic statements.) It thus can be assumed that to a theory of higher dimension, there corresponds a class of basic statements of higher dimension, such that all statements of this class are permitted by the theory, irrespective of what they assert.
This answers the question of how the two methods of comparing degrees of testability are related—the one by means of the dimension of a theory, and the other by means of the subclass relation. There will be cases in which neither, or only one, of the two methods is applicable. In such cases there is of course no room for conflict between the two methods. But if in a particular case both methods are applicable, then it may conceivably happen that two theories of equal dimensions may yet have different degrees of falsifiability if assessed by the method based upon the subclass relation. In such cases, the verdict of the latter method should be accepted, since it would prove to be the more sensitive method. In all other cases in which both methods are applicable, they must lead to the same result; for it can be shown, with the help of a simple theorem of the theory of dimension, that the dimension of a class must be greater than, or equal to, that of its subclasses.3
39 THE DIMENSION OF A SET OF CURVES
Sometimes we can identify what I have called the ‘field of application’ of a theory quite simply with the field of its graphic representation, i.e. the area of a graph-paper on which we represent the theory by graphs: each point of this field of graphic representation can be taken to correspond to one relatively atomic statement. The dimension of the theory with respect to this field (defined in appendix 1) is then identical with the dimension of the set of curves corresponding to the theory. I shall discuss these relations with the help of the two statements q and s of section 36. (Our comparison of dimensions applies to statements with different predicates.) The hypothesis q—that all planetary orbits are circles—is threedimensional: for its falsification at least four singular statements of the field are necessary, corresponding to four points of its graphic representation. The hypothesis s, that all planetary orbits are ellipses, is fivedimensional, since for its falsification at least six singular statements are necessary, corresponding to six points of the graph. We saw in section 36 that q is more easily falsifiable than s: since all circles are ellipses, it was possible to base the comparison on the subclass relation. But the use of dimensions enables us to compare theories which previously we were unable to compare. For example, we can now compare a circlehypothesis with a parabola-hypothesis (which is four dimensional). Each of the words ‘circle’, ‘ellipse’, ‘parabola’ denotes a class or set of curves; and each of these sets has the dimension d if d points are necessary and sufficient to single out, or characterize, one particular curve of the set. In algebraic representation, the dimension of the set of curves depends upon the number of parameters whose values we may freely choose. We can therefore say that the number of freely determinable parameters of a set of curves by which a theory is represented is characteristic for the degree of falsifiability (or testability) of that theory.
In connection with the statements q and s in my example I should like to make some methodological comments on Kepler’s discovery of his laws.*1
I do not wish to suggest that the belief in perfection—the heuristic principle that guided Kepler to his discovery—was inspired, consciously or unconsciously, by methodological considerations regarding degrees of falsifiability. But I do believe that Kepler owed his success in part to the fact that the circle-hypothesis with which he started was relatively easy to falsify. Had Kepler started with a hypothesis which owing to its logical form was not so easily testable as the circle hypothesis, he might well have got no result at all, considering the difficulties of calculations whose very basis was ‘in the air’—adrift in the skies, as it were, and moving in a way unknown. The unequivocal negative result which Kepler reached by the falsification of his circle hypothesis was in fact his first real success. His method had been vindicated sufficiently for him to proceed further; especially since even this first attempt had already yielded certain approximations.
No doubt, Kepler’s laws might have been found in another way. But I think it was no mere accident that this was the way which led to success. It corresponds to the method of elimination which is applicable only if the theory is sufficiently easy to falsify—sufficiently precise to be capable of clashing with observational experience.
40 TWO WAYS OF REDUCING THE NUMBER OF DIMENSIONS OF A SET OF CURVES
Quite different sets of curves may have the same dimension. The set of all circles, for example, is three-dimensional; but the set of all circles passing through a given point is a two-dimensional set (like the set of straight lines). If we demand that the circles should all pass through two given points, then we get a one-dimensional set, and so on. Each additional demand that all curves of a set should pass through one more given point reduces the dimensions of the set by one.
The number of dimensions can also be reduced by methods other than that of increasing the number of given points. For example the set of ellipses with given ratio of the axes is four-dimensional (as is that of parabolas), and so is the set of ellipses with given numerical eccentricity. The transition from the ellipse to the circle, of course, is equivalent to specifying an eccentricity (the eccentricity 0) or a particular ratio of the axes (unity).
| zero dimensional classes1 |
one dimensional classes |
two dimensional classes |
three dimensional classes |
four dimensional classes |
| - | - | straight line | circle | parabola |
| - | straight line through one given point |
circle through one given point |
parabola through one given point |
conic through one given point |
| straight line through two given points |
circle through two given points |
parabola through two given points |
conic through two given points |
- |
| circle through three given points |
parabola through three given points |
conic through three given points |
- | - |
As we are interested in assessing degrees of falsifiability of theories we will now ask whether the various methods of reducing the number of dimensions are equivalent for our purposes, or whether we should examine more closely their relative merits. Now the stipulation that a curve should pass through a certain singular point (or small region) will often be linked up with, or correspond to, the acceptance of a certain singular statement, i.e. of an initial condition. On the other hand, the transition from, say, an ellipse-hypothesis to a circle-hypothesis, will obviously correspond to a reduction of the dimension of the theory itself. But how are these two methods of reducing the dimensions to be kept apart? We may give the name ‘material reduction’ to that method of reducing dimensions which does not operate with stipulations as to the ‘form’ or ‘shape’ of the curve; that is, to reductions through specifying one or more points, for example, or by some equivalent specification. The other method, in which the form or shape of the curve becomes more narrowly specified as, for example, when we pass from ellipse to circle, or from circle to straight line, etc., I will call the method of ‘formal reduction’ of the number of dimensions.
It is not very easy, however, to get this distinction sharp. This may be seen as follows. Reducing the dimensions of a theory means, in algebraic terms, replacing a parameter by a constant. Now it is not quite clear how we can distinguish between different methods of replacing a parameter by a constant. The formal reduction, by passing from the general equation of an ellipse to the equation of a circle, can be described as equating one parameter to zero, and a second parameter to one. But if another parameter (the absolute term) is equated to zero, then this would mean a material reduction, namely the specification of a point of the ellipse. I think, however, that it is possible to make the distinction clear, if we see its connection with the problem of universal names. For material reduction introduces an individual name, formal reduction a universal name, into the definition of the relevant set of curves.
Let us imagine that we are given a certain individual plane, perhaps by ‘ostensive definition’. The set of all ellipses in this plane can be defined by means of the general equation of the ellipse; the set of circles, by the general equation of the circle. These definitions are independent of where, in the plane, we draw the (Cartesian) co-ordinates to which they relate; consequently they are independent of the choice of the origin, and the orientation, of the co-ordinates. A specific system of coordinates can be determined only by individual names; say, by ostensively specifying its origin and orientation. Since the definition of the set of ellipses (or circles) is the same for all Cartesian co-ordinates, it is independent of the specification of these individual names: it is invariant with respect to all co-ordinate transformations of the Euclidean group (displacements and similarity transformations).
If, on the other hand, one wishes to define a set of ellipses (or circles) which have a specific, an individual point of the plane in common, then we must operate with an equation which is not invariant with respect to the transformations of the Euclidean group, but relates to a singular, i.e. an individually or ostensively specified, co-ordinate system. Thus it is connected with individual names.2
The transformations can be arranged in a hierarchy. A definition which is invariant with respect to a more general group of transformations is also invariant with respect to more special ones. For each definition of a set of curves, there is one—the most general— transformation group which is characteristic of it. Now we can say: The definition D1 of a set of curves is called ‘equally general’ to (or more general than) a definition D2 of a set of curves if it is invariant with respect to the same transformation group as is D2 (or a more general one). A reduction of the dimension of a set of curves may now be called formal if the reduction does not diminish the generality of the definition; otherwise it may be called material.
If we compare the degree of falsifiability of two theories by considering their dimensions, we shall clearly have to take into account their generality, i.e. their invariance with respect to co-ordinate transformations, along with their dimensions.
The procedure will, of course, have to be different according to whether the theory, like Kepler’s theory, in fact makes geometrical statements about the world or whether it is ‘geometrical’ only in that it may be represented by a graph—such as, for example, the graph which represents the dependence of pressure upon temperature. It would be inappropriate to require of this latter kind of theory, or of the corresponding set of curves, that its definition should be invariant with respect to, say, rotations of the co-ordinate system; for in these cases, the different co-ordinates may represent entirely different things (the one pressure and the other temperature).
This concludes my exposition of the methods whereby degrees of falsifiability are to be compared. I believe that these methods can help us to elucidate epistemological questions, such as the problem of simplicity which will be our next concern. But there are other problems which are placed in a new light by our examination of degrees of falsifiability, as we shall see; especially the problem of the so-called ‘probability of hypotheses’ or of corroboration.
Addendum, 1972
One of the more important ideas of this book is that of the (empirical, or informative) content of a theory. (‘Not for nothing do we call the laws of nature “laws”: the more they prohibit, the more they say.’ Cp. pp. 18–19 above, and pp. 95 f.)
Two points were stressed by me in the preceding chapter: (1) The content or the testability (or the simplicity: see ch. vii) of a theory may have degrees, which may thus be said to relativize the idea of falsifiability (whose logical basis remains the modus tollens). (2) The aim of science— the growth of knowledge—can be identified with the growth of the content of our theories. (See my paper ‘The Aim of Science’, in Ratio I, 1957, pp. 24–35 and (revised) in Contemporary Philosophy, ed. R. Klibansky, 1969, pp. 129–142; now also Chapter 5 of my book Objective Knowledge: An Evolutionary Approach, which is forthcoming at the Clarendon Press.)
the Clarendon Press.)
More recently I have developed these ideas further; see especially ch. 10 of my Conjectures and Refutations, 1963 and later editions (with the new Addenda). Two of the new points are: (3) A further relativization of the idea of content or testability with respect to the problem, or set of problems, under discussion. (Already in 1934 I relativized these ideas with respect to a field of application; see my old Appendix i.) (4) The introduction of the idea of the truth content of a theory and of its approximation or nearness to truth (‘verisimilitude’).
*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 denumerable (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).
*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 ‘complex’, 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 translated 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.
*1 See section 38, and the appendices i, *vii, and *viii.
*2 See however now appendix *vii.
*1 I still believe that the attempt to make all statements comparable by introducing a metric must contain an arbitrary, extra-logical element. This is quite obvious in the case of statements such as ‘All adult men are more than two feet high’ (or ‘All adult men are less than nine feet high’); that is to say, statements with predicates stating a measurable property. For it can be shown that the metric of content or falsifiability would have to be a function of the metric of the predicate; and the latter must always contain an arbitrary,
or at any rate an extra-logical element. Of course, we may construct artificial languages for which we lay down a metric. But the resulting measure will not be purely logical, however ‘obvious’ the measure may appear as long as only discrete, qualitative yes-or-no predicates (as opposed to quantitative, measurable ones) are admitted. See also appendix *ix, the Second and Third Notes.
*2 I now (since 1938; cf. appendix *ii) use the term ‘absolute logical probability’ rather than ‘logical probability’ in order to distinguish it from ‘relative logical probability’ (or ‘conditional logical probability’). See also appendices *iv, and *vii to *ix.
1 To this idea of logical probability (inverted testability) corresponds Bolzano’s idea of validity, especially when he applies it to the comparison of statements. For example, he describes the major propositions in a derivability relation as the statements of lesser validity, the consequents as those of greater validity (Wissenschaftslehre, 1837, Vol. II, 
157, No. 1). The relation of his concept of validity to that of probability is explained by Bolzano in op. cit. 147. Cf. also Keynes, A Treatise on Probability, 1921, p. 224. The examples there given show that my comparison of logical probabilities is identical with Keynes’s ‘comparison of the probability which we ascribe a priori to a generalization’. See also notes 1 to section 36 and 1 to section 83.
1 Carnap, Erkenntnis 2, 1932, p. 458.
*1 ‘p → q’ means, according to this explanation, that the conditional statement with the antecedent p and the consequent q is tautological, or logically true. (At the time of writing the text, I was not clear on this point; nor did I understand the significance of the fact that an assertion about deducibility was a meta-linguistic one. See also note *1 to section 18, above.) Thus ‘p → q’ may be read here: ‘p entails q’.
2 Carnap, op. cit., says: ‘The metalogical term “equal in content” is defined as “mutually derivable”.’ Carnap’s Logische Syntax der Sprache, 1934, and his Die Aufgabe der Wissenschaftslogik, 1934, were published too late to be considered here.
*2 If the logical content of p exceeds that of q, then we say also that p is logically stronger than q, or that its logical force exceeds that of q.
*3 See, again, appendix *vii.
*1 It will be seen that in the present section (in contrast to sections 18 and 35), the arrow is used to express a conditional rather than the entailment relation; cf. also note *1 to section 18.
1 We can write: [( øqx → øpx).(fpx→ fqx)] ¨ [( øpx→fpx) ¨ ( øqx) → fqx)] or for short: [( øq → øp).(fp→fq] ¨ (p → q). *The 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.
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.
*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.)
*3 These considerations are closely connected with, and supported by, some of the results discussed under points 8 ff. of my ‘Third Note’, reprinted in appendix *ix. See also section *15 of the Postscript for the significance of measurement for the ‘depth’ of theories.
1 Cf. Frank, Das Kausalgesetz und seine Grenzen, 1931, e.g. p. 24.
*1 For the term ‘composite’, see note *1 to section 32.
2 ‘Elementary propositions’ in Wittgenstein, Tractatus Logico-Philosophicus, Proposition 5: ‘Propositions are truth-functions of elementary propositions’. ‘Atomic propositions’ (as opposed to the composite ‘molecular propositions’) in Whitehead and Russell’s Principia Mathematica Vol. I. Introduction to 2nd edition, 1925, pp. xv f. C. K. Ogden translated Wittgenstein’s term ‘Elementarsatz’ as ‘elementary proposition’, (cf. Tractatus 4.21), while Bertrand Russell in his Preface to the Tractatus, 1922, p. 13, translated it as ‘atomic proposition’. The latter term has become more popular.
*2 Absolute degrees of composition would determine, of course, absolute degrees of content, and thus of absolute logical improbability. The programme here indicated of introducing improbability, and thus probability, by singling out a certain class of absolutely atomic statements (earlier sketched, for example, by Wittgenstein) has more recently been elaborated by Carnap in his Logical Foundations of Probability, 1950, in order to construct a theory of induction. See also the remarks on model languages in my Preface to the English Edition, 1958, above, where I allude to the fact that the third model language (Carnap’s language system) does not admit measurable properties. (Nor does it in its present form allow the introduction of a temporal or spatial order.)
*3 The words ‘scientific language’ were here used quite naïvely, and should not be interpreted in the technical sense of what is today called a ‘language system’. On the contrary, my main point was that we should remember the fact that scientists cannot use a ‘language system’ since they have constantly to change their language, with every new step they take. ‘Matter’, or ‘atom’, after Rutherford, and ‘matter’, or ‘energy’, after Einstein, meant something different from what they meant before: the meaning of these concepts is a function of the—constantly changing—theory.
3 Cf. Menger, Dimensionstheorie, 1928, p. 81. *The conditions under which this theorem holds can be assumed to be always satisfied by the ‘spaces’ with which we are concerned here.
*1 The views here developed were accepted, with acknowledgments, by W. C. Kneale, Probability and Induction, 1949, p. 230, and J. G. Kemeny, ‘The Use of Simplicity in Induction’, Philos. Review 57, 1953; see his footnote on p. 404.
1 We could also, of course, begin with the empty (over-determined) minus-onedimensional class.
2 On the relations between transformation groups and ‘individualization’ cf. Weyl, Philosophie der Mathematik u. Naturwissenschaft, 1927, p. 59, English edition pp. 73 f., where reference is made to Klein’s Erlanger Programm.