Quite similar is the case with the other member of this relation. It often happens that when a, or a´´, a´´´ is found, somewhat different things appear, which do not fit the concept as first constructed. Then we must multiply the experiences as much as possible in order to determine what constant elements are found in the concept B, and to form from these constant elements the corresponding concept B´. The improved proposition will then read: if A´ is, then B´ is also.
This entire process may be called the purification of the causal relation. By this term we express the general fact that in first forming such a regular connection, the proper concepts are very seldom brought into relation with one another at once. The cause of it is that at first we make use of existing concepts which had been formed for quite a different purpose. It must therefore be regarded as a special piece of good fortune if these old concepts should at once prove suited to the new purpose. Furthermore, the existing concepts are as a rule so vaguely characterized by their names, which we must employ to express the new relation, that for this reason also it is often necessary to determine empirically in what way the concept is to be definitely established.
The various sciences are constantly occupied with this work of the mutual adaptation of the concepts that enter into a causal relation. By way of example, we may take the "self-understood" proposition which we use when we call out to a careless child when it sticks its finger into the flame of a candle, "Fire burns!" We discover that there are self-luminous bodies which produce no increase of temperature, and therefore no sensation of pain. We discover that there are processes of combustion that develop no light, but heat enough to burn one's fingers. And, finally, the scientific investigation of this proposition arrives at the general expression that, as a rule, chemical processes are accompanied by the development of heat, but that, conversely, such processes may also be accompanied by the absorption of heat. In this way that casual sentence which we call out to the child develops into the extensive science of thermo-chemistry when it is subjected to the continuous purification of the causal relation, which is the general task of science.
It remains to be added that in this process of adapting concepts it is necessary also sometimes to follow the opposite course. This is the case when exceptions are noticed in a relation as expressed for the time being; when, therefore, the proposition if A is present, then B is present also, is in a great many instances valid, but occasionally fails. This is an indication that in the concept A an element is still lacking. This element, however, is present in the instances that tally, but absent in the negative cases, and its absence is not noticed because it is not contained in A. Then it is necessary to seek this part, and after it has been found, to embody it in the concept A, which thus passes into the new concept A´.
This case is the obverse of the former one. Here the more suitable concept proves to be less general than the concept accepted temporarily, while in the first case the improved concept is more general. Hence we formulate the rule: exceptions to the temporary rule require a limitation, while an unforeseen freedom requires an extension, of the accepted concept.
12. Induction.
The form of conclusion previously discussed, because it has been so, I expect it will continue to be so in the future, is the form through which each science has arisen and has won its real content, that is, its value for the judgment of the future. It is called inference by induction, and the sciences in which it is preponderatingly applied are called inductive sciences. They are also called experiential or empirical sciences. At the basis of this nomenclature is the notion that there are other sciences, the deductive or rational sciences, in which a reverse logical procedure is applied, whereby from general principles admitted to be valid in advance, according to an absolutely sure logical process, conclusions of like absolute validity are drawn. At the present time people are beginning to recognize the fact that the deductive sciences must give up these claims one by one, and that they already have given them up to a certain extent; partly because on closer study they prove to be inductive sciences, and partly because they must forego the title and rank of a science altogether. The latter alternative applies especially to those provinces of knowledge which have not been used in prophesying the future or cannot be so used.
To return to the inductive method—it is to be noted that Aristotle, who was the first to describe it, proposed two kinds of induction, the complete and the incomplete. The first has this form: since all things of a certain kind are so, each individual thing is so. While the incomplete induction merely says: since many things of a certain kind are so, presumably all things of this kind are so. One instantly perceives that the two conclusions are essentially different. The first lays claim to afford an absolutely certain result. But it rests upon the assumption that all the things of the kind in question are known and have been tested as to their behavior. This hypothesis is generally impossible of fulfilment, since we can never prove that there are not more things of the same kind other than those known to us or tested by us. Moreover, the conclusion is superfluous, as it merely repeats knowledge that we have already directly acquired, since we have tested all the things of the one kind, hence the special thing to which the predication refers.
On the other hand, the incomplete induction affirms something that has not yet been tested, and therefore involves as a condition an extension of our knowledge, sometimes an extremely important extension. To be sure, it must give up the claim to unqualified or absolute validity, but, to compensate, it acquires the irreplaceable advantage of lending itself to practical application. Indeed, in accordance with the scientific practice justified by experience, described on [p. 29], the scientific inductive conclusion assumes the form: because it has once been found to be so, it will always be so. From this appears the significance of this method for the enlargement of science, which, without it, would have had to proceed at an incomparably slower pace.