. This is one class, and its relation to
is not shared by any other class. We may call the class of numbers that are less than
the "proper ancestry" of
, in the sense in which we spoke of ancestry and posterity in connection with mathematical induction. Then "proper ancestry" is a one-many relation (one-many will always be used so as to include one-one), since each number determines a single class of numbers as constituting its proper ancestry. Thus the relation less than can be replaced by being a member of the proper ancestry of. In this way a one-many relation in which the one is a class, together with membership of this class, can always formally replace a relation which is not one-many. Peano, who for some reason always instinctively conceives of a relation as one-many, deals in this way with those that are naturally not so. Reduction to one-many relations by this method, however, though possible as a matter of form, does not represent a technical simplification, and there is every reason to think that it does not represent a philosophical analysis, if only because classes must be regarded as "logical fictions." We shall therefore continue to regard one-many relations as a special kind of relations.
One-many relations are involved in all phrases of the form "the so-and-so of such-and-such." "The King of England," "the wife of Socrates," "the father of John Stuart Mill," and so on, all describe some person by means of a one-many relation to a given term. A person cannot have more than one father, therefore "the father of John Stuart Mill" described some one person, even if we did not know whom. There is much to say on the subject of descriptions, but for the present it is relations that we are concerned with, and descriptions are only relevant as exemplifying the uses of one-many relations. It should be observed that all mathematical functions result from one-many relations: the logarithm of
, the cosine of