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possible, but it must nevertheless be valuable to know at what we ought to aim.
Read Locke's brief Essay on the Conduct of the Un
derstanding, which contains admirable remarks on the acquirement of exact and logical habits of thought.
THE LAWS OF THOUGHT. BEFORE the reader proceeds to the lessons which treat of the most common forms of reasoning, known as the syllogism, it is desirable that he should give a careful attention to the very simple laws of thought on which all reasoning must ultimately depend. These laws describe the very simplest truths, in which all people must agree, and which at the same time apply to all notions which we can conceive. It is impossible to think correctly and avoid evident self-contradiction unless we observe what are called the Three Primary Laws of Thought, which may be stated as follows :
The Law of Identity: Whatever is, is.
either be or not be. Though these laws when thus stated may seem absurdly obvious, and were ridiculed by Locke and others on that account, I have found that students are seldom able to see at first their full meaning and importance. It will be pointed out in Lesson XXIII. that logicians have
overlooked until recent years the very simple way in which ali arguments may be explained when these self-evident laws are granted; and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key.
The first of the laws may be regarded as the best definition we can give of identity or sameness. Could any one be ignorant of the meaning of the word Identity, it would be sufficient to inform him that everything is identical with itself.
The second law however is the one which requires more consideration. Its meaning is that nothing can have at the same time and at the same place contradictory and inconsistent qualities. A piece of paper may be blackened in one part, while it is white in other parts; or it may be white at one time, and afterwards become black; but we cannot conceive that it should be both white and black at the same place and time. A door after being open may be shut, but it cannot at once be shut and open. Water may feel warm to one hand and cold to another hand, but it cannot be both warm and cold to the same hand. No quality can both be present and absent at the same time; and this seems to be the most simple and general truth which we can assert of all things. It is the very nature of existence that a thing cannot be otherwise than it is; and it may be safely said that all fallacy and error arise from unwittingly reasoning in a way inconsistent with this law. All statements or inferences which imply a combination of contradictory qualities must be taken as impossible and false, and the breaking of this law is the mark of their being false. It can easily be shewn that if Iron be a metal, and every metal an element, Iron must be an element, or it can be nothing at all, since it would combine qualities which are inconsistent (see Lesson XXIII).
The Law of Excluded Middle is much less self-evident than either of the two preceding ones, and the reader will not perhaps see at the first moment that it is equally important and necessary with them. Its meaning may be best explained by saying that it is impossible to mention any thing and any quality or circumstance, without allowing that the quality or circumstance either belongs to the thing or does not belong. The name of the law expresses the fact that there is no third or middle course; the answer must be Yes or No. Let the thing be rock and the quality hard; then rock must be either hard or not-hard. Gold must be either white or not white; a line must be either straight or not straight; an action must be either virtuous or not virtuous. Indeed when we know nothing of the terms used we may nevertheless make assertions concerning them in accordance with this law. The reader may not know and in fact chemists may not really know with certainty, whether vanadium is a metal or not a metal, but any one knows that it must be one or the other. Some readers may not know what a cycloid is or what an isochronous curve is; but they must know that a cycloid is either an isochronous curve or it is not an isochronous curve.
This law of excluded middle is not so evident but that plausible objections may be suggested to it. Rock, it may be urged, is not always either hard or soft, for it
may he half way between, little hard and a little soft at the same time. This objection points to a distinction which is of great logical importance, and when neglected often leads to fallacy. The law of excluded middle affirmed nothing about hard and soft, but only referred to hard and not-hard; if the reader chooses to substitute soft for not-hard he falls into a serious confusion between opposite terms and contradictory terms. It is quite possible that a thing may be neither hard nor soft, being half way
between; but in that case it cannot be fairly called hard, so that the law holds true. Similarly water must be either warm or not-warm, but it does not follow that it must be warm or cold. The alternative not-warm evidently includes all cases in which it is cold besides cases where it is of a medium temperature, so that we should call it neither warm nor cold. We must thus carefully distinguish questions of degree or quantity from those of simple logical fact. In cases where a thing or quality may exist to a greater or less extent there are many alternatives. Warm water, for, instance may have any temperature from 700 perhaps up to 120°. Exactly the same question occurs in cases of geometrical reasoning; for Euclid in his Elements frequently argues from the selfevident truth that any line must be either greater than, equal to, or less than any other line. While there are only two alternatives to choose from in logic there are three in Mathematics; thus one line, compared with another, may be
[ greater......... ..... greater) .
......less Another and even more plausible objection may be raised to the third law of thought in this way. Virtue being the thing proposed, and triangular the quality, the Law of Excluded Middle enables us at once to assert that virtue is either triangular or not-triangular. At first sight it might seem false and absurd to say that an immaterial . notion such as virtue should be either triangular or not, because it has nothing in common with those material substances occupying space to which the notion of figure belongs. But the absurdity would arise, not from any falseness in the law, but from misinterpretation of the expression not-triangular. If in saying that a thing is
“not triangular” we are taken to imply that it has some figure though not a triangular figure, then of course the expression cannot be applied to virtue or anything immaterial. In strict logic however no such implied meaning is to be allowed, and not-triangular will include both things which have figure other than triangular, as well as things which have not the properties of figure at all; and it is in the latter meaning that it is applicable to an immaterial thing.
These three laws then being universally and necessarily true to whatever things they are applied, become the foundation of reasoning. All acts of reasoning proceed from certain judgments, and the act of judgment logic consists in comparing two things or ideas together and let discovering whether they agree or differ, that is to say whether they are identical in any qualities. The laws of thought inform us of the very nature of this identity with which all thought is concerned. But in the operation of discourse or reasoning we need certain additional laws, or axioms, or self-evident truths, which may be thus stated :
Two terms agreeing with one and the same third term agree with each other.
Two terms of which one agrees and the other does not agree with one and the same third term, do not agree with each other.
These self-evident truths are commonly called the Canons or Fundamental Principles of Syllogism, and they are true whatever may be the kind of agreement in question. The example we formerly used (p. 3) of the agreement of the terms “Most useful metal” and “cheapest metal” with the third common term “ Iron," was but an instance of the first Canon, and the agreement consisted in complete identity. In the case of the “Earth,” the "Planets," and "Bodies revolving in elliptic orbits,”