recently C. D. Broad1 and J. M. Keynes 2 have given rigorous demonstrations of the necessity of stating inductive conclusions in these terms. Because of its bearing upon the central problem of statistical induction the argument may be briefly summarized.

Keynes divides all generalizations based upon empirical evidence into two classes, those involving universal induction, and those in the form of inductive correlations. The former affirm invariable relations, though any degree of probability may attach to the conclusions. The latter process leads up to generalizations of the type described in earlier sections of this paper, to statistical laws, laws of "probable connection. For the present we are concerned with uni

versal induction.

Such an induction, it has been established, involves a formal fallacy (the fallacy of drawing a universal conclusion from a particular premise) unless the conclusion be couched in terms of probability. No universal generalization based upon a study of instances, no matter how numerous, can be known to be a certain truth. So much needs no demonstration. But more than this acknowledgment is necessary to render an induction valid. If such a generalization is to be true with any finite degree of probability, some assumption must be made about the nature of the universe from which the facts were drawn and to which the conclusion applies. The argument from particular facts to a general law must proceed from some premise about nature, in addition to that premise which takes account of the instances studied.

The nature of this "missing premise" about nature may be made. clear by assuming the universe from which the instances were taken to be characterized by utter chaos, with an infinite multiplicity of causes and an infinite variety of phenomena, possessing nothing of uniformity, stability or order. General laws might be assumed, after a study of cases drawn from such a universe, but no finite probability would attach to these generalizations. In order that there should be a reasonable degree of probability in favor of the truth of an inductive conclusion it is necessary that we make some assumption about the orderliness of nature and the degree of variety to be expected in the universe to which the conclusion applies.

This missing premise upon which all inductions rest has been

1 "On the Relation Between Induction and Probability" Mind, N. S., xxvii, 1918, and xxix, 1920.

2 A Treatise on Probability, 1921.

3 Loc. cit., 220,

vaguely described as the uniformity of nature. In different forms it enters into the reasoning of students in many fields, for some such assumption is a necessary bed-rock of all scientific thinking. Josiah Royce, following Charles Peirce, finds it in "the more or less systematic tendency towards a mutual assimilation of the fortunes, the characters, or the mutual relations of the members of (an) ... aggregate," and in this urge towards mutual assimilation he finds not only "nature's principal tendency" but, as well, the expression of an "unconscious teleology" in nature.1 Karl Pearson sees it in "the routine of experience" traced back to a "routine of perception" which is an essential condition of knowledge.2 Keynes finds the justification for the inductive process in "the limitation to the amount of independent variety" in nature. Defining a finite system, as distinguished from a collection of heterogeneous facts, as one in which "the number of premises, or the amount of independent variety (is) . . . less than the number of its members," he contends that probable knowledge of facts or propositions belonging to a finite system can be obtained by means of an inductive argument. We require a priori a finite probability in favor of an inductive conclusion, according to Keynes, the methods of empirical proof being used merely to strengthen such a probability. This initial probability is based on the assumption of a limited amount of independent variety in nature.3

C. D. Broad finds the missing premise about nature in a theory of "natural kinds," based on the assumption "that nature consists of a comparatively few kinds of permanent substances, that their changes are all subject to laws, and that the variety of nature. is due to varying combinations of the few elementary substances." In other words, "experience has suggested a simple ground plan

1 Royce, loc. cit., 565.

2 The Grammar of Sciencc, chaps. III and IV.

3 Keynes, loc. cit., 251-3.

One point which Keynes emphasizes is worthy of note at this stage of the argument. A universal induction, leading to a generalization to which there are no exceptions, is justified only on the assumption "that a finite degree of probability always exists that there is not, in any given case, a plurality of causes." In "dealing with inductive correlation, where we do not claim universality for our conclusions, it would be sufficient for us to assume that the number of distinct generators, to which a given property can be due, is always finite." The fact that the former assumption is hardly plausible "strongly suggests," writes Keynes, "that our conclusions should be in the form of inductive correlations, rather than of universal generalizations." This furnishes an interesting logical confirmation of the statistical view of law, as presented

of the material world to us, and it is reasonable to suppose that the plan extends beyond what we have actually experienced." When we feel that we have an understanding of the ground plan, the fundamental structure of phenomena in a given field, considerable weight is attached to inductions; otherwise a very slight degree of probability attaches to them. Until such a ground plan is discovered in a given field, says Broad, investigation is almost at a prescientific level.1

Granting the validity of some such assumption about nature, universal inductions may possess some finite degree of probability. They will never attain to certainty, nor may the degree of probability be expressed numerically."The logical position is (a) that those inductions which we regard as highly probable are so relatively to the belief that we really have got hold of the general ground plan of nature in the region of phenomena under investigation; (b) the evidence for this is never of the nature of a 'knockdown' proof and no numerical probability can be assigned to it."2 If we could assume a definite number of independent influences at work this probability might be measured, but such an assumption cannot usually be made.

With these arguments in mind, the particular problem of statistical induction may be considered. Induction of this type consists, simply, in the application of certain statistical results to objects or events not included in the sample studied, these results corresponding to the generalizations which are the usual objects of the inductive process. Thus an average of earnings of industrial employees is determined by the study of a sample group, and is accepted as applying to the entire group from which the sample was drawn. An index number of wholesale prices, computed from quotations on a limited number of commodities, is taken as representative of all wholesale prices. Or a statistically derived law, such as the equation describing the relationship between alfalfa yield and inches of water used in irrigation, is accepted as of general validity.

This process of generalizing from statistical results is a common

A sample is studied, the "probable error" of the result determined by methods the logic of which is seldom investigated, and the findings are presented as of general significance, with few, if any, qualifications. This procedure is by no means of unassailable soundness, and cannot be followed without due regard to possible

1 Loc. cit., xxix, 42-5,

2 Ibid., 44,

pitfalls. Yet this sort of induction plays, and must play, such an important part in economic research that the economist must give attention to the strength or weakness of the logic behind it.

Quantitative inference of this type differs in no wise from the ordinary process of induction, except in that one of the premises is in quantitative form, and that the conclusion asserts only a probable connection or extends an average value, which may or may not hold in any given case. Both evidence and conclusion deal with only probable and approximate relationships or average values, and in this respect accord more closely with actual experience than do the premises and conclusions of universal inductions.

The problem at issue in the discussion of the validity of this process relates to the reliability of the results, to the stability, when applied beyond the sample, of the averages, ratios or equations computed. The whole practical problem of statistics centers about the stability of such results, and the limits to such stability when the results are generalized in this way.

The factors in the case are practically the same as in the case of universal induction. Since we are generalizing from a limited sample, some degree of probability must attach to the conclusion. To justify such a conclusion, moreover, an assumption about nature, similar to the assumption of uniformity suggested above, must be made. There will be an a priori element in this assumption. Finally, since the attempt is usually made to measure the precise degree of probability attaching to such a quantitative conclusion, a principle of probability enters more specifically into such arguments.

The assumption about nature which constitutes an implied premise in all statistical generalizations, and which corresponds to the conception of finite variety, is Quetelet's "stability of large numbers" and Clerk Maxwell's "new kind of regularity, the regularity of averages." The multiplicity of causes and the existence of variation in nature, which make statistical rather than mechanical methods necessary, are accompanied by a limitation of variety and a stability attaching to numbers in mass which bring some order out of the complexities of nature, sufficient order to render scientific method applicable and knowledge possible. That there is such stability is always assumed in generalizing from statistical results, and the assumption introduces an a priori element into such inductions.

The degree of probability attaching, in any given case, to this assumption of stability is incapable of precise evaluation, yet it is essential to the process of statistical induction that this stability be tested and, if possible, measured. If the results are to be applicable

beyond the sample, some idea of the limits within which the statistical measure is likely to fluctuate is a practical necessity. If 5 per cent of the men of a certain age, in a given sample, die within a period of twelve months, what are the probable limits within which this ratio will fluctuate when applied to the larger group from which the sample was taken? If 20 per cent of a sample group of wage-earners are unemployed more than 100 days a year, what are the probable limits to the fluctuations of this ratio when applied beyond the sample? The answers to these questions involve the theory of inverse or empirical probabilities. As to the soundness of this theory a question has been raised which Pearson has called the fundamental problem of practical statistics. For the very foundation of statistical induction, insofar as an attempt is made to measure the stability of the conclusions, rests upon the validity of determining probabilities empirically.

When we have a priori knowledge of the probability of a given event we are able to assign in advance the probable outcome of a trial or of a number of trials. Thus when a coin or die is tossed all the possible outcomes and their respective probabilities are known in advance. Our knowledge of probabilities is empirical, on the other hand, when based upon the results of a number of trials or upon statistical frequencies. It is a probability "into the determination of which actual experience has entered as a dominant factor." The term "inverse probabilities" has also been applied to the latter type, as the probability ratios are derived by reasoning backward from the results to the probabilities. In practically all cases in which the statistical method is applied to the data of economics our knowledge of probabilities is of this empirical character. Whenever a generalization is based upon a statistical study, whether it relate to mortality rates, income distribution, the correlation between variables, or any similar material, it applies to the larger field not with certainty but with a degree of probability, and insofar as this degree of probability is known it is known only empirically.

The issues at stake in the controversy as to the validity of thus computing probabilities from the results of experience may be briefly indicated, without going into the details of the argument. Formulae associated primarily with the names of Bayes and Laplace have been developed for computing from the results obtained from a limited sample the probability of securing similar results in a study of the larger group from which the sample was drawn. More exactly, these methods are employed for determining the probability that the subsequent results will fall within certain stated limits. Such