The real bearing of this law can, it appears to me, best be derived from a study of its mathematical expression. This is, according to the notation of Professor Boole, _x_^{2}=_x_. As such, it presents a fundamental equation of thought, and it is because it is of the second degree that we classify in pairs or opposites. This equation can only be satisfied by assigning to _x_ the value of 1 or 0. The “universal type of form” is therefore _x_(1-_x_)=0. This algebraic notation shows that there is, not two, but only one thought in the antithesis; that it is made up of a thought and its expressed limit; and, therefore, that the so-called “law of contradictories” does not concern contradictories at all, in pure logic. This result was seen, though not clearly, by Dr. Thompson, who indicated the proper relation of the members of the formula as a positive and a privative. He, however, retained Hamilton’s doctrine that “privative conceptions enter into and assist the higher processes of the reason in all that it can know of the absolute and infinite;” that we must, “from the seen realize an unseen world, not by extending to the latter the properties of the former, but by assigning to it attributes entirely opposite.”[31-1] The error that vitiates all such reasoning is the assumption that the privative is an independent thought, that a thought and its limitation are two thoughts; whereas they are but the two aspects of the one thought, like two sides to the one disc, and the absurdity of speaking of them as separate thoughts is as great as to speak of a curve seen from its concavity as a different thing from the same curve regarded from its convexity. The privative can help us nowhere and to nothing; the positive only can assist our reasoning. This elevation of the privative into a contrary, or a contradictory, has been the bane of metaphysical reasoning. From it has arisen the doctrine of the synthesis of an affirmative and a negative into a higher conception, reconciling them both. This is the maxim of the Hegelian logic, which starts from the synthesis of Being and Not-being into the Becoming, a very ancient doctrine, long since offered as an explanation of certain phenomena, which I shall now touch upon.