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between two contradictory extremes,” Sir William Hamilton defended with his wide learning those theories of the Conditioned and the Unconditioned, the Knowable and Unknowable, which banish religion from the realm of reason and knowledge to that of faith, and cleave an impassable chasm between the human and the divine intelligence. From this unfavorable ground his orthodox followers, Mansel and Mozley, defended with ability but poor success their Christianity against Herbert Spencer and his disciples, who also accepted the same theories, but followed them out to their legitimate conclusion – a substantially atheistic one.

      Hamilton in this was himself but a follower of Kant, who brought this law to support his celebrated “antinomies of the human understanding,” warnings set up to all metaphysical explorers to keep off of holy ground.

      On another construction of it, one which sought to escape the dilemma of the contradictories by confining them to matters of the understanding, Hegel and Schelling believed they had gained the open field. They taught that in the highest domain of thought, there where it deals with questions of pure reason, the unity and limits which must be observed in matters of the understanding and which give validity to this third law, do not obtain. This view has been closely criticized, and, I think, with justice. Pretending to deal with matters of pure reason, it constantly though surreptitiously proceeds on the methods of applied logic; its conclusions are as fallacious logically as they are experimentally. The laws of thought are formal, and are as binding in transcendental subjects as in those which concern phenomena.

      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, x2=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.”15

      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.

      A thought and its privative alone – that is, a quality and its negative – cannot lead to a more comprehensive thought. It is devoid of relation and barren. In pure logic this is always the case, and must be so. In concrete thought it may be otherwise. There are certain propositions in which the negative is a reciprocal quality, quite as positive as that which it is set over against. The members of such a proposition are what are called “true contraries.” To whatever they apply as qualities, they leave no middle ground. If a thing is not one of them, it is the other. There is no third possibility. An object is either red or not red; if not red, it may be one of many colors. But if we say that all laws are either concrete or abstract, then we know that a law not concrete has all the properties of one which is abstract. We must examine, then, this third law of thought in its applied forms in order to understand its correct use.

      It will be observed that there is an assumption of space or time in many propositions having the form of the excluded middle. They are only true under given conditions. “All gold is fusible or not,” means that some is fusible at the time. If all gold be already fused, it does not hold good. This distinction was noted by Kant in his discrimination between synthetic judgments, which assume other conditions; and analytic judgments, which look only at the members of the proposition.

      Only the latter satisfy the formal law, for the proposition must not look outside of itself for its completion. Most analytic propositions cannot extend our knowledge beyond their immediate statement. If A is either B or not B, and it is shown not to be B, it is left uncertain what A may be. The class of propositions referred to do more than this, inasmuch as they present alternative conceptions, mutually exhaustive, each the privative of the other. Of these two contraries, the one always evokes the other; neither can be thought except in relation to the other. They do not arise from the dichotomic process of classification, but from the polar relations of things. Their relation is not in the mind but in themselves, a real externality. The distinction between such as spring from the former and the latter is the most important question in philosophy.

      To illustrate by examples, we familiarly speak of heat and cold, and to say a body is not hot is as much as to say it is cold. But every physicist knows that cold is merely a diminution of heat, not a distinct form of force. The absolute zero may be reached by the abstraction of all heat, and then the cold cannot increase. So, life and death are not true contraries, for the latter is not anything real but a mere privative, a quantitative diminution of the former, growing less to an absolute zero where it is wholly lost.

      Thus it is easy to see that the Unconditioned exists only as a part of the idea of the Conditioned, the Unknowable as the foil of the Knowable; and the erecting of these mere privatives, these negatives, these shadows, into substances and realities, and then setting them up as impassable barriers to human thought, is one of the worst pieces of work that metaphysics has been guilty of.

      The like does not hold in true contrasts. Each of them has an existence as a positive, and is never lost in a zero of the other. The one is always thought in relation to the other. Examples of these are subject and object, absolute and relative, mind and matter, person and consciousness, time and space. When any one of these is thought, the other is assumed. It is vain to attempt their separation. Thus those philosophers who assert that all knowledge is relative, are forced to maintain this assertion, to wit, All knowledge is relative, is nevertheless absolute, and thus they falsify their own position. So also, those others who say all mind is a property of matter, assume in this sentence the reality of an idea apart from matter. Some have argued that space and time can be conceived independently of each other; but their experiments to show it do not bear repetition.

      All true contraries are universals. A universal concept is one of “maximum extension,” as logicians say, that is, it is without limit. The logical limitation of such a universal is not its negation, but its contrary, which is itself also a universal. The synthesis of the two can be in theory only, yet yields a real product. To illustrate this by a geometrical example, a straight line produced indefinitely is, logically considered, a universal. Its antithesis or true contrary is not a crooked line, as might be supposed, but the straight line which runs at right angles to it. Their synthesis is not the line which bisects their angle but that formed by these contraries continually uniting, that is, the arc of a circle, the genesis of which is theoretically the union of two such lines. Again, time can only be measured by space, space by time; they are true universals and contraries; their synthesis is motion, a conception which requires them both and is completed by them. Or again, the philosophical extremes of downright materialism and idealism are each wholly true, yet but half the truth. The insoluble enigmas that either meets in standing alone are kindred to those which puzzled the old philosophers in the sophisms relating to motion, as, for instance,

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