It will be observed that if we take for the relation "parent," "ancestor" and "posterity" will have the usual meanings, except that a person will be included among his own ancestors and posterity. It is, of course, obvious at once that "ancestor" must be capable of definition in terms of "parent," but until Frege developed his generalised theory of induction, no one could have defined "ancestor" precisely in terms of "parent." A brief consideration of this point will serve to show the importance of the theory. A person confronted for the first time with the problem of defining "ancestor" in terms of "parent" would naturally say that is an ancestor of if, between and , there are a certain number of people, , , ..., of whom is a child of , each is a parent of the next, until the last, who is a parent of . But this definition is not adequate unless we add that the number of intermediate terms is to be finite. Take, for example, such a series as the following:— Here we have first a series of negative fractions with no end, and then a series of positive fractions with no beginning. Shall we say that, in this series, is an ancestor of ? It will be so according to the beginner's definition suggested above, but it will not be so according to any definition which will give the kind of idea that we wish to define. For this purpose, it is essential that the number of intermediaries should be finite. But, as we saw, "finite" is to be defined by means of mathematical induction, and it is simpler to define the ancestral relation generally at once than to define it first only for the case of the relation of to , and then extend it to other cases. Here, as constantly elsewhere, generality from the first, though it may [Pg 26] require more thought at the start, will be found in the long run to economise thought and increase logical power.