The Principles of Mathematics (1903)

§ 29

Other assumptions required are that the negation of a relation is a relation, and that the logical product of a class of relations (i.e. the assertion of all of them simultaneously) is a relation. Also the relative product of two relations must be a relation. The relative product of two relations R, S is the relation which holds between x and z whenever there is a term y towhich x has the relation R and which has to z the relation S. Thus the relation of a maternal grandfather to his grandson is the relative produt of father and mother; that of a paternal grandmother to her grandson is the relative product of mother and father; that of a grandparent to grandchild is the relative product of parent and parent. The relative product, as these instances show, is not in general commutative, and does not in general obey the law of tautology. The relative product is a notion of very great importance. Since it does not obey the law of tautology, it leads to powers of relations: the square of the relation to parent and child is the relation of grandparent and grandchild, and so on. Peirce and Schröder consider also what they call the relative sum of two relations R and S, which holds between x and z when, if y be any other term whatever, either x has to y the relation R or y has to z the relation S. This is a complicated notion, which I found no occasion to employ, and which is only introduced in order to preserve the duality of addition and multiplication. This duality has a certain technical charm when the subject is considered as an independent branch of mathematics; but when it is considered solely in relation to the principles of mathematics, the duality in question appears devoid of all philosophical importance.(§ 29 ¶ 1)