In Part IV we considered what philosophy has to say about the universe. The function of philosophy, according to the view advocated in this volume, is somewhat different from that which has been assigned to it by a large and influential school. Take, _e.g._ Kant’s antinomies. He argues (1) that space must be infinite, (2) that space cannot be infinite; and he deduces that space is subjective. The non-Euclideans refuted the argument that it must be infinite, and Georg Cantor refuted the argument that it cannot be. Formerly, _a priori_ logic was used to prove that various hypotheses which looked possible were impossible, leaving only one possibility, which philosophy therefore pronounced true. Now _a priori_ logic is used to prove the exact contrary, namely, that hypotheses which looked impossible are possible. Whereas logic was formerly counsel for the prosecution, it is now counsel for the defence. The result is that many more hypotheses are at large than was formerly the case. Formerly, to revert to the instance of space, it appeared that experience left only one kind of space to logic, and logic showed this one kind to be impossible. Now, logic presents many kinds of space as possible apart from experience, and experience only partially decides between them. Thus, while our knowledge of what is has become less than it was formerly supposed to be, our knowledge of what may be is enormously increased. Instead of being shut in within narrow walls, of which every nook and cranny could be explored, we find ourselves in an open world of free possibilities, where much remains unknown because there is so much to know. The attempt to prescribe to the universe by means of _a priori_ principles has broken down; logic, instead of being, as formerly, a bar to possibilities, has become the great liberator of the imagination, presenting innumerable alternatives which are closed to unreflective common sense, and leaving to experience the task of deciding, where decision is possible, between the many worlds which logic offers for our choice.