FIG. 1. AN ILLUSTRATION OF A PHYSICAL LAW.

Graph of an equation describing the relationship between terminal voltage and current in tungsten lamp filament.

(The points plotted represent observations of corresponding values of terminal voltage and current in a 118-volt tungsten lamp. The equation to the curve fitted to these points is y=0.01624x0.6)

(From Steinmetz)

perfect dependence. This is the type of relationship which the laws of the most accurate of the physical sciences describe, a relation which in many cases accords so closely with the mechanistic view that the application of the mechanical method involves no appreciable error.

The points plotted in Figure 2 represent the corresponding values of alfalfa yield and depth of irrigation in forty-four different experiments. The problem is identical with that faced in the preceding case, the objects being the expression of the functional relationship in the form of an equation, and the measurement of the degree of deviation from the average relationship so described. The curve fitted to the plotted points represents the functional relationship. The equation is given below the figure. The second measure required, that of degree of correlation, has in this case a value of .80, which indicates

1 The data were obtained from "The Economical Irrigation of Alfalfa in the Sacramento Valley"; Bull. No. 280 of the University of California Agricultural Experiment Station, May, 1917.

Graph of an equation describing the relationship between alfalfa yield and depth of irrigation water.

(The points plotted represent observations of alfalfa yield in tons per acre and depth of irrigation water in inches. The average in each class is distinguished from the other points. The equation to the curve fitted to these points is y=3.55+.252x -.002816x2)

a close connection, but not one of complete dependence, between alfalfa yield and depth of irrigation.

This latter is the type of relationship which economic laws approximate, and this is the general method by which such laws may be given precision and utility. An additional measure of the dispersion about the curve may be computed, making even more precise the description of relationship which this method affords. Here is no vague generalization, but an exact statement of relationship. This is the type of information which we may secure in a sphere in which statistical knowledge, in terms of averages, approximation and probabilities, is alone possible.

But this knowledge is restricted in scope. In describing a single group, averages are fully representative only when they relate to the cases included in the given sample. So, in describing relationships, our "laws" hold with validity, within the limits determined in the given situation, only when confined to the cases included in the sample studied. With respect to that group we may apply the results with absolute confidence. This is statistical description; results are applied only to objects actually measured or to events actually

observed.

The economist cannot be content with this. He seeks generalizations which will apply to a wider group, to events not observed, to cases not included in his sample. He seeks, that is, to employ the ordinary methods of induction, basing the logical process upon materials of a particular kind-statistical data. This process of statistical induction gives rise to problems quite distinct from those encountered when mere description is attempted. Since most economic studies which are. quantitative in character involve such a process of generalization, some of the assumptions and limitations of statistical induction may be set forth.

4. THE NATURE OF STATISTICAL INDUCTION

The time-worn controversy over the relative merits of induction and deduction in scientific research has perhaps not been settled, but it seems to have been shelved, so far as current discussion is concerned. The problem remains, but the controversy does not burn so brightly as it did a generation or two ago. Having reached the practical conclusion that the two methods must go hand in hand, the scientist has ignored such smoulderings of the controversy as have survived. There undoubtedly has been a greater emphasis on the quantitative aspect of research and a greater readiness to submit theories to the arbitrament of facts than at certain periods in the past, but this has not meant an out and out acceptance of the Baconian all-case method. In the field of economics today, while pure economists and empiricists may follow their separate bents, there is certainly no such sharp difference of opinion as to method as that which separated Hume from Bacon, or Bagehot from Cohn. Abstraction and empiricism have reached something approximating a working agreement in economics, if they have not in philosophy.

But while we may recognize that pure deduction and pure induction are alike unworkable as scientific methods-that, on the one hand, there is an inductive residue in even the most abstract piece of scientific reasoning, and, on the other, that the foundation of a universal generalization upon the mere multiplication of instances is logically quite invalid-there still remains a question as to the relative importance, in any given field, of deduction from general principles as opposed to the inductive methods of enumeration and analogy. The following conclusion is suggested.

The mechanical concepts of direct causal relationship and of invariant law do not accord strictly with the facts in any scientific field. Infinite knowledge and perfect certainty being impossible, some elements of ignorance and doubt must always be present in

scientific reasoning and prediction. But in certain of the physical sciences these elements are reduced to a minimum, and the concept of mechanism comes close to actual realization. In such fields, where the notions of perfect causality, absolute sameness and certainty most closely approximate the truth, the amount of inductive evidence needed to support a train of deductive reasoning will be small. Such inductive evidence will always be required, but the amount of such evidence necessary as a foundation or a verification of an argument will vary directly with the number of causes in operation and the degree of variation in nature.

It follows from this that as scientific method invades fields in which large errors are involved in applying the mechanistic concept, in which a multiplicity of causes operate and in which there is a high degree of variation in the data, a much larger body of inductive evidence is needed.1 The premises will be tinctured with a larger element of doubt, the initial degree of probability will be less, and therefore more facts and more varied facts will be needed to establish a reasonable degree of probability for the conclusions. Induction must play a commanding rôle in investigations in those fields in which the elements of approximation and probability bulk large. Where we must phrase our laws in terms of "tendencies," where "other things must be equal," where laws must be qualified because the effects of single causes cannot be isolated, and where the operation of many factors in a given case must be recognized-in other words, where the statistical view of nature applies in its fulness-induction can never be merely a halting companion to deduction but must play a dominant part in scientific investigations.

Economics constitutes such a field. A multiplicity of causes operate, variation is pronounced, some degree of uncertainty enters every argument, and an element of probability attaches to every conclusion. By purely a priori reasoning, moreover, a measure can rarely, if ever, be given to this element of probability. Such accuracy as may be given to the conclusions of economic reasoning must come from the test of facts. In such a field, where probability and uncertainty are ever present, the trains of deduction must rest upon solid inductive foundations. The tendency of some writers of the past to

1 The relation between the inductive process and the multiplicity of causes has been noted by Mill, as well as by later logicians. "The plurality of causes," Mill writes, "is the only reason why mere number of instances is of any importance in inductive inquiry." System of Logic, Book IV, Chap. X, 2. Quoted by Keynes, loc. cit., 269.

rear their structures of theory upon less solid foundations called from Cliffe Leslie the just criticism that economists, dwelling in the "region of assumption, conjecture and provisional generalization" too often "jumped to the laws without heed to the phenomena.'

If induction is to play such an important rôle in economic research, economists should employ this tool with a full knowledge of its possibilities and its limitations. The logical basis and the limitations of non-quantitative induction have perhaps been more fully set forth and are understood more generally than is true in the case of quantitative induction employing the methods of statistics. With the increased use of measurement in economics most inductive processes in this field are based upon statistical results. These results, in the form of averages, frequency ratios, coefficients of correlation or empirical equations, are usually taken to apply to objects or events not included in the sample on which they are based, and this extension corresponds to the usual inductive process of generalization from particulars. The logical justification for such an extension and the conditions upon which a generalization is valid have been discussed by logicians and students of probability, but these questions are generally ignored in actual statistical study.

As regards economics, it is perhaps not too much to say that we are here considering the central problem of economic research. If there be validity to the arguments set forth above as to the essentially statistical character of the data of economics and the laws of economics, and if inductive methods must play a large part in the derivation and verification of the laws of economics, no problem is more important than that of the logical soundness of induction based upon statistical results. Are we justified in generalizing from limited data? Does the fact that the data and the results happen to be in quantitative form make the process any more accurate than it would otherwise be? What validity have empirical laws? Admitting the existence of elements of approximation and probability in all such generalizations, can a method be developed for measuring the precise degree of probability in a given case? Can the logic of probabilities eliminate or lessen the uncertainty which attaches to all events in the world about us-in none, perhaps, to a greater degree than in the economic realm? These are questions of profound importance. Perhaps they cannot all be answered, but at least they cannot be ignored.

The conclusions of all inductive arguments must be expressed in terms of probability. They can never attain to a condition of certainty. This fact was recognized by Laplace and Jevons; more