class="x-ebookmaker-pageno" title="[25]"/>This he did by showing that the laws of correct thinking can be expressed in algebraic notation, and, thus expressed, will be subject to all the mathematical laws of an algebra whose symbols bear the uniform value of unity or nought (1 or 0)—a limitation required by the fact that pure logic deals in notions of quality only, not of quantity.

This mathematical form of logic was foreseen by Kant when he declared that all mathematical reasoning derives its validity from the logical laws; but no one before Professor Boole had succeeded in reaching the notation which subordinated these two divisions of abstract thought to the same formal types. His labors have not yet borne fruit in proportion to their value, and they are, I believe, comparatively little known. But in the future they will be regarded as epochal in the science of mind. They make us to see the same law governing mind and matter, thought and extension.

Not the least important result thus achieved was in emphasizing the contrast between the natural laws of mental association, and the laws of thinking which are the foundation of the syllogism.

By attending to this distinction we are enabled to keep the form and the matter of thought well apart—a neglect to do which, or rather a studied attempt to ignore which, is the radical error of the logic devised by Hegel, as I shall show more fully a little later.

All applied logic, inductive as well as deductive, is based on formal logic, and this in turn on the “laws of thought,” or rather of thinking. These are strictly regulative or abstract, and differ altogether from the natural laws of thought, such as those of similarity, contiguity and harmony, as well as from the rules of applied logic, such as those of agreement and difference. The fundamental laws of thinking are three in number, and their bearing on all the higher questions of religious philosophy is so immediate that their consideration becomes of the last moment in such a study as this. They are called the laws of Determination, Limitation and Excluded Middle.

The first affirms that every object thought about must be conceived as itself, and not as some other thing. “A is A,” or “x = x,” is its formal expression. This teaches us that whatever we think of, must be thought as one or a unity. It is important, however, to note that this does not mean a mathematical unit, but a logical one, that is, identity and not contrast. So true is this that in mathematical logic the only value which can satisfy the formula is a concept which does not admit of increase, to wit, a Universal.

From this necessity of conceiving a thought under unity has arisen the interesting tendency, so frequently observable even in early times, to speak of the universe as one whole, the το παν of the Greek philosophers; and also the monotheistic leaning of all thinkers, no matter what their creed, who have attained very general conceptions. Furthermore, the strong liability of confounding this speculative or logical unity with the concrete notion of individuality, or mathematical unity, has been, as I shall show hereafter, a fruitful source of error in both religious and metaphysical theories. Pure logic deals with quality only, not with quantity.

The second law is that of Limitation. As the first is sometimes called that of Affirmation, so this is called that of Negation. It prescribes that a thing is not that which it is not. Its formula is, “A is not not-A.” If this seems trivial, it is because it is so familiar.

These two laws are two aspects of the same law. The old maxim is, omnis determinatio est negatio; a quality can rise into cognition only by being limited by that which it is not. It is not a comparison of two thoughts, however, nor does it limit the quality itself. For the negative is not a thought, and the quality is not in suo genere finita, to use an expression of the old logicians; it is limited not by itself but by that which it is not. These are not idle distinctions, as will soon appear.

The third law comes into play when two thoughts are associated and compared. There is qualitative identity, or there is not. A is either B or not B. An animal is either a man or not a man. There is no middle class between the two to which it can be assigned. Superficial truism as this appears, we have now come upon the very battle ground of the philosophies. This is the famous “Law of the contradictories and excluded middle,” on the construction of which the whole fabric of religious dogma, and I may add of the higher metaphysics, must depend. “One of the principal retarding causes of philosophy,” remarks Professor Ferrier, “has been the want of a clear and developed doctrine of the contradictory.”28-1 The want is as old as the days of Heraclitus of Ephesus, and lent to his subtle paradoxes that obscurity which has not yet been wholly removed.

Founding his arguments on one construction of this law, expressed in the maxim, “The conceivable lies 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