66

COMPLETE ENUMERATION

3

85

planets. We can say then with perfect safety, that all the planets revolve about the sun. The universal statement is warranted because each of the instances which it covers has been observed. We are saying no more in the conclusion than we had already said in the several statements on which the conclusion is based. The universal is, in fact, simply a summary way of expressing what had already been said. It is merely a telescoping" of the other statements, as it were. This act of basing a general statement on a complete enumeration of the particular cases which it covers has been called Perfect Induction. It was so called because the conclusion is one which possesses complete certainty, whereas most inductive inferences are more or less uncertain. It might seem then that this was the solution of the problem raised above; you are sure of your universal if you have seen all the particulars which it covers. But how can we be sure that we have counted all the particulars? The field of observation may be so small and so easily explored that every existing case may be observed. But even if all existing cases have been observed, how can we be sure that others may not arise, and that they may not differ from those we have observed? We may have such knowledge about a class of objects as will enable us to say that if any other members of the class should come into existence they would be like those already known. We may know that the sum of the angles of every plane triangle which may ever exist will be equal to two right angles, not because we have counted cases, but because we know that this necessarily follows from the properties essen3 Leaving the asteroids out of consideration.

tial to all triangles. However numerous the class which has been completely observed, the knowledge that each of the observed members stands in certain relations does not by itself assure us that other conceivable members of the class would be like them in this respect. Complete enumeration is useful as an abbreviated way of stating certain kinds of information, but it throws no light on the methods of discovering unconditional connections.4

The judgments which result from the complete enumeration of cases have been called, by some writers, Enumerative Judgments and by others, Collective Judgments.

How Generalizations can be Verified-It appears then that enumeration of all the existing members of a class does not enable us to establish laws. Anything short of that might seem to leave us still farther from that goal. And it is of course true, as appeared on page 84, that an incomplete enumeration of instances furnishes no verification. Then if verification is possible at all it can not rest on mere enumeration, or counting of cases. Suppose that the observation of one

4 It may be well to note one case in which a statement in the universal form must be distinguished from a law. As an example, we may take, "Every three-sided figure is a triangle." This is not an inductive inference; it is not based upon the observation of individual instances at all. It is true in all cases, but it is true because we have previously said, "If any figure has three sides we will call it a triangle." In other words, it is true by definition. It is like an inductive generalization in applying to all possible cases, past, present and to come, real and imaginary, etc., but it is not based upon the observation of individual facts. Other judgments and operations, which must be distinguished from those which are present in induction properly so called, will be discussed in a later chapter; and still others may be found by referring to Mill's Logic, Book III, chapter ii.

VERIFICATION

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or more instances in which B has followed A has suggested to us the inference that A and B are causally related. Let us ask ourselves what consequences would follow upon the truth of this inference. In the first place we could conclude that if B were present in any case A must have been present also; and, again, if A were absent in any case, its supposed effect B must have been absent also; or if either A or B varied in amount or degree the other should show a corresponding variation. All these things should be true of phenomena which are causally related. Phenomena which failed to satisfy such conditions could not be unconditionally connected. Suppose we had inferred that absence of oxygen would cause death. If that is true, an animal immersed in nitrogen should die. If experiment showed that an animal could live under such conditions, our inference would, of course, be disproved; but suppose the animal did die, would the inference be proved? Not necessarily. Perhaps the nitrogen acted as a poison or perhaps the death of the animal was due to rough handling, etc. Our inductive inference would be completely verified only if we could show that death could not have been due to anything except the absence of oxygen. If we could be sure that all the circumstances which were present before the experiment remained precisely the same with the one exception that oxygen was present in the first case and absent in the second, then we should have shown a necessary connection between the absence of oxygen and the occurrence of death. Nothing else could have been the cause because all were present when death did not occur. If a second circumstance were present when the phenomenon occurred and

absent when it did not occur, it would dispute with the first the right to be called the cause and no final conclusion would be possible. When all other possibilities can be excluded, the one which remains is the cause. When no other inference is consistent with the facts, the one which is consistent must be accepted as true. We can say, then, that an inductive inference is completely verified when we have found facts which are consistent with its truth and inconsistent with any possible rival inference; or more briefly, when it fits the facts and no alternative inference does. We establish one inference by eliminating all others. We reason that the phenomenon under investigation has some cause; this other phenomenon, A, may be the cause; it fulfills the requirements and no other does; therefore, this one is and must be the cause in question. There are several ways of selecting or grouping instances so as to show that some one factor alone satisfies the requirements. These are known as the INDUCTIVE METHODS.

Observation and Analysis are Presupposed.-One thing should not be forgotten. It is that the application of such principles as these presupposes very careful observation; if we are to be certain that no other circumstance is present when a given phenomenon is present or absent when it is absent, we must have observed all the other circumstances. In ordinary observation we note only a few of the circumstances; if we are untrained observers it may be impossible for us to observe more than a few. It is quite impossible for a child to observe in a flower all that a trained botanist can observe there. Accurate observation presupposes analysis, i. e., breaking up the total complex phenome

TEST CONDITIONS

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non into its element. The beginner in any science is unable to handle the facts properly because he is unable to analyse them; he sees only their most obvious characteristics.

Postponing Inference till Test Conditions are Present-Before we begin the more detailed examination of the methods of verifying an inductive inference, there is one more statement to be made, namely this: instead of making a generalization and then searching for means of verifying it, we may refrain from drawing any inference until we have before us a group of facts which will make it possible to draw a correct inference. In other words, we may make our inference under test conditions. Suppose, for example, that we are trying to discover the cause of eclipses. Before making any theories on the subject we might observe a number of cases. If we found that whenever an eclipse occurred there was an opaque body between us and the source of light and that at other times everything was the same except that there was no body in that position, we should infer at once that the presence of the opaque body in that position was the cause of the eclipse. And so in any other case we might form no theory until we had facts which would make it possible to form a cor

rect one.

66 If a chemist discovers a new element, he will proceed to try a variety of experiments in order to determine the proportions in which it will combine with other elements as well as to discover the various properties of such combinations. Supposing such experiments to have been properly conducted, the inductions at which he arrives will be perfectly valid, though he