The Principles of Mathematics (1903)

§ 99

Mr Bradley, in Appearance and Reality, Chapter III, has based an argument against the reality of relations upon the endless regress arising from the fact that a relation which relates two terms must be related to each of them. The endless regress is undeniable, if relational propositions are taken to be ultimate, but it is very doubtful whether it forms any logical difficulty. We have already had occasion (§ 55) to distinguish two kinds of regress, the one proceeding merely to perpetually new implied propositions, the other in the meaning of a proposition itself; of these two kinds, we agreed that the former, since the solution of the problem of infinity, has ceased to be objectionable, while the latter remains inadmissible. We have to inquire which kind of regress occurs in the present instance. It may be urged that it is part of the very meaning of a relational proposition that the relation involved should have to the terms the relation expressed in saying that it relates them, and that this is what makes the distinction, which we formerly (§ 54) left unexplained, between a relating relation and a relation in itself. It may be urged, however, against this view, that the assertion of a relation between the relation and the terms, though implied, is no part of the original proposition, and that a relating relation is distinguished from a relation in itself by the indefinable element of assertion which distinguishes a proposition from a concept. Against this it might be retorted that, in the concept difference of a and b, difference relates a and b just as much as in the proposition a and b differ; but to this it may be rejoined that we found the difference of a and b, except in so far as some specific point of difference may be in question, to be indistinguishable from bare difference. Thus it seems impossible to prove that the endless regress involved is of the objectionable kind. We may distinguish, I think, between a exceeds b and a is greater than b, though it would be absurd to deny that people usually mean the same thing by these two propositions. On the principle, from which I can see no escape, that every genuine word must have some meaning, the is and than must form part of a is greater than b, which thus contains more than two terms and a relation. The is seems to state that a has to greater the relation of referent, while the than states similarly that b has to greater the relation of relatum. But a exceeds b may be held to express solely the relation of a to b, without including any of the implications of further relations. Hence we shall have to conclude that a relational proposition aRb does not include in its meaning any relation of a or b to R, and that the endless regress, though undeniable, is logically quite harmless. With these remarks, we may leave the further theory of relations to later Parts of the present work.(§ 99 ¶ 1)