were it not absolutely impossible. It is evident that the first terms we wished to define would require previous terms to serve for their explanation, and similarly the first propositions we wished to prove, would presuppose other propositions preceding them in our knowledge; and thus it is clear that we should never arrive at the first terms or first propositions.

"Accordingly in pushing our researches further and. further, we arrive necessarily at primitive words which we cannot define, and at principles so clear, that we cannot, find any principles more clear to prove them by. Thus it appears that men are naturally and inevitably incapable of treating any science whatever in a perfect method; but it does not thence follow that we ought to abandon every kind of method......The most perfect method available to men consists not in defining everything and demonstrating everything, nor in defining nothing and demonstrating nothing, but in pursuing the middle course of not defining things which are clear and understood by all persons, but of defining all others; and of not proving truths known to all persons, but of proving all others. From this method they equally err who undertake to define and prove everything, and they who neglect to do it in things which are not self-evident.”

It is made plain in this admirable passage that we can never by using words avoid an ultimate appeal to things, because each definition of a word must require one or more other words, which also will require definition, and so on ad infinitum. Nor must we ever return back upon the words already defined; for if we define A by B, and B by C, and C by D, and then D by A, we commit what may be called a circulus in definiendo; a most serious fallacy, which might lead us to suppose that we know the nature of A, B, C, and D, when we really know nothing about them.

Pascal's views of the geometrical method were clearly summed up in the following rules, inserted by him in the Port Royal Logic*.

I. To admit no terms in the least obscure or equivocal without defining them.

2. To employ in the definitions only terms perfectly known or already explained.

3. To demand as axioms only truths perfectly evident.

4. To prove all propositions which are at all obscure, by employing in their proof only the definitions which have preceded, or the axioms which have been accorded, or the propositions which have been already demonstrated, or the construction of the thing itself which is in dispute, when there may be any operation to perform.

5. Never to abuse the equivocation of terms by failing to substitute for them, mentally, the definitions which restrict and explain them.

The reader will easily see that these rules are much more easy to lay down than to observe, since even geometers are not agreed as to the simplest axioms to assume, or the best definitions to make. There are many different opinions as to the true definition of parallel lines, and the simplest assumptions concerning their nature; and how much greater must be the difficulty of observing Pascal's rules with confidence in less certain branches of science. Next after Geometry, Mechanics is perhaps the most perfect science, yet the best authorities have been far from agreeing as to the exact definitions of such notions as force, mass, moment, power, inertia, and the most different opinions are still held as to the simplest axioms by which the law of the composition of forces may be proved. Nevertheless if we steadily bear in mind, in

* Mr Spencer Baynes' Translation, p. 317.

studying each science, the necessity of defining every term as far as possible, and proving each proposition which can be proved by a simpler one, we shall do much to clear away error and confusion.

I also wish to give here the rules proposed by the celebrated Descartes for guiding the reason in the attainment of truth. They are as follows::

I.

Never to accept anything as true, which we do not clearly know to be so; that is to say, carefully to avoid haste or prejudice, and to comprise nothing more in our judgments than what presents itself so clearly and distinctly to the mind that we cannot have any room to doubt it.

2. To divide each difficulty we examine into as many parts as possible, or as may be required for resolving it.

3. To conduct our thoughts in an orderly manner, commencing with the most simple and easily known objects, in order to ascend by degrees to the knowledge of the most complex.

4. To make in every case enumerations so complete, and reviews so wide, that we may be sure of omitting nothing.

These rules were first stated by Descartes in his admirable Discourse on Method, in which he gives his reflections on the right mode of conducting the reason, and searching for truth in any of the sciences. This little treatise is easily to be obtained in the original French, and has also been translated into English by Mr Veitch*. The reader can be strongly advised to study it. Always to observe the rules of Descartes and Pascal, or to know whether we in every case observe them properly, is im

Published at Edinburgh in 1850.

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 Understanding, which contains admirable remarks on the acquirement of exact and logical habits of thought.

LESSON XIV.

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:

I. The Law of Identity. Whatever is, is.

2.

The Law of Contradiction. Nothing can both be and not be.

3. The Law of Excluded Middle. Everything must

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 all 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).