CHAPTER VII. INDUCTION.

The process of inference by which all such truths are obtained may readily be exhibited in a precise symbolic form. We must have one premise specifying in a disjunctive form all the possible individuals which belong to a class; we resolve the class, in short, into its constituents. We then need a number of propositions, each of which affirms that one of the individuals possesses a certain property. Thus the premises must be of the forms

Now, if we substitute for each alternative of the first premise its description as found among the succeeding premises, we obtain

or

But for the aggregate of alternatives we may now substitute their equivalent as given in the first premise, namely A, so that we get the required result:

We should have reached the same result if the first premise had been of the form

We can always prove a proposition, if we find it more convenient, by proving its equivalent. To assert that all not-B’s are not-A’s, is exactly the same as to assert that all A’s are B’s. Accordingly we may ascertain that A = AB by first ascertaining that b = ab. If we observe, for instance, that all substances which are not solids are also not capable of double refraction, it follows necessarily that all double refracting substances are solids. We may convince ourselves that all electric substances are nonconductors of electricity, by reflecting that all good conductors do not, and in fact cannot, retain electric excitation. When we come to questions of probability it will be found desirable to prove, as far as possible, both the original proposition and its equivalent, as there is then an increased area of observation.