would call this an intuitive thought, not that it is merely an intuition, but that it is an intuition constituted into the type of a class of objects; it is, in fact, an intuition and a thought. This is the highest and best form of an act of conception, and is that towards which, on all occasions, we ought as much as possible to strive.

In the second place, we may take the symbol or term which denotes the concept or notion, and rest satisfied with it, without fully realising the contents of the notion-unfolding them before the mind. This term, from its application and associations, designates equally any one of a class of individual objects, and only the individuals of that class. Whatever, accordingly, we think as applicable to the symbol or involved in the symbolical knowledge, we regard as applicable to any one and to all of the individuals which it represents. We have an illustration of intuitive thought in the case of Geometry. Here our reasonings refer to an individual diagram, regarded simply as representing all the possible figures of the class to which it belongs. We have an example of symbolical thought in the case of Algebra, where the process of investigation is carried on entirely by means of symbols, representative, it may be, of a quantity which, during the process, is regarded by us as entirely unknown or indefinite. In Algebra, for example, to quote a case, you may take the division of unity into any two parts. Here it is shown that the difference of their squares is equal to the difference of the parts themselves. It does not matter what the numbers are. Letters will represent them. This is a universal law or formula which is worked out, in total unconsciousness of definite pictures or images attached to the

terms.

§ 133. This distinction of symbolical and intuitive knowledge has a very wide and important application. There are cases in which symbolical thinking is an absolute necessity. Think of the difference between the idea of a figure of 1000 sides, and that of a triangle or figure of three sides. The latter we are able quite well definitely to imagine, to picture. The other we cannot; but we know what it means. And how so? As appears to me simply by repeating units, which we know or can picture. Five and ten we can picture, 100 we can hardly; but we can realise the 100 through the five or ten.

As we go on to 500, to 1000, the thought grows more dim as a picture, yet our knowledge is exact enough, because we go on forming units of which the larger number is composed. When I am told that the distance of the sun from the earth is 92,400,000 miles, or that the mean distance of Uranus from the sun is 1,754,000,000 miles, I confess that I cannot picture either of these distances to my imagination. I cannot make what is called an intuitive thought of it, yet I know it in a symbolical and even definite manner. In the same way, when I am told that light travels at the rate of 186,000 miles per second, and thus traverses the distance from the sun to the earth in eight minutes, I have but a symbolical or unpicturable knowledge; yet it is all the knowledge I can have in the case. We must be content to think those numbers through the repetition of picturable units merely. We may picture or construe to the imagination so many units-say five, ten, fifteen, twenty; but after that, each of these sums is itself regarded as a unit, and thus becomes the basis of a higher calculation or concept. And there is no reason why twenty units should be regarded as less a unit than one. The twenty is virtually one,-one as against everything less or more than itself, a true unit; and we may thus add or repeat this unit, as much as any smaller unit we know. Algebra all through is very much this kind of knowledge; geometry, as I have said, is not so; for at each step we have the picture of a figure before us. For this reason, algebraic training is not so good a mental discipline as geometry; and both are inferior as means of culture to the study of the sciences of intuition, or of fact and probability.

§ 134. It is possible to carry on long trains of reasoning in this the symbolical method. In fact, it is the most usual of all methods. But it is this circumstance which mainly allows contradiction and absurdity to escape us, which otherwise we should at once detect. It explains, indeed, how so much is written and accepted as true, which, nevertheless, we are totally unable to conceive, or even render intelligible. When contradictory propositions are stated in terms, whose meaning we fully apprehend, the contradiction at once flashes on the mind. This would be the case always, if each of our terms were fully and definitely understood. But as we use terms symbolically, we may and do employ

terms of contradictory import, form these into propositions and reasonings, accept the conclusion as valid, without being at all aware of any incongruity. Yet when our reasoning encloses a contradiction, however cloaked or concealed, the whole process is absolutely null; it is, in a word, nonsensical or meaningless. To accept the meaningless for the meaning, non-sense for sense, is one marked danger of purely symbolical thinking. A frequent use of definition, and the substitution of intuitive for symbolical thinking, are our main safeguards against contradiction and confusion in any discussion.

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CHAPTER XII.

THE LAWS OF THOUGHT:

IDENTITY-NON-CONTRADICTION—

EXCLUDED MIDDLE-DETERMINING REASON.

§ 135. If there be in thought form essential and universal, this must depend on law necessary in thinking. If, whenever we think, or in whatever we call thinking, there is a type to which the act of thinking conforms, in order to its very existence, then this type must depend on a law, that is, a rule so uniform and general as to amount to universality. The matter of our thinking varies indefinitely; rules of generality may apply to it; difference does not destroy the matter of thought. Variation from form destroys form,-destroys, in fact, thought itself. Hence the law which regulates this unchanging form must itself be an unchanging law,—dependent, that is, on the very nature of the thinking subject,necessary, universal, and thus essential to the very being and act of thinking.

§ 136. The unchanging character of the form of thought proves the necessary character of the law of thought; this, again, proves the unchanging character of the form of thought. We may either say that thought as form is necessary, unchanging, universal; or that the law of thought is so. The form is the concrete embodiment of the law; the law is the abstract statement of the form.

§ 137. The laws of thought are usually divided into the contingent and the necessary; but the latter alone are the proper laws of thought. We may think successively in various spheres of knowledge, or of various objects. Where the objects of thought differ, the laws or conditions of our thinking them differ also. Thus we may think a state of

consciousness; we do so as in time, as contrasted with a past state, and as void of dimensions. We may think an object of sense,-quality or percept. This we think not only as now or in time an object of thought, but as in a particular space related to an object or objects in co-adjacent spaces. It is contingent whether we think the sensation or say the sun-dial: and therefore the conditions under which we think in each case are in so far contingent. These may metaphysically or really become necessary to the thought regarded as the thought of the given object; but there being no necessity for our thinking the determinate object, there is no absolute or universal necessity of the condition upon our thought. These are therefore for thought itself contingent laws or conditions. They apply only if we happen to think of certain or determinate objects.

But the laws proper of thought are necessary laws. In other words, thought of any object is impossible apart from them. They are the laws of thought as thought. Whatever be the object we think, we must think it as identical with itself, as in absolute contrast to its contradictory correlative, and that on pain of the annihilation of the thought itself. Apart from the contingent conditions of thinking, certain acts of thought would not be; apart from the necessary conditions of thinking, no act of thought would be. The laws of thought thus imply a certain abstraction from objects. To them the object is as to its real nature or characters indifferent. Some object there must be in order that the law may be manifested in exercise. But any object is all that is needed. They bear the same relation to the objects of experience which the laws of universal grammar bear to the words of different languages. They contain the intelligible forms of the objects, as the principles of universal grammar embody the possible combinations of the words which constitute intelligible, that is, possible speech.

§ 138. At the same time these laws are inaccurately described as independent of all experience. They are not so, either as to their known origin,-the possibility even of their conception by us, or of their realisation in our consciousness; for this always supposes some instance, either given in experience or created in the interest of pure thought by the imagination. They are independent of experience only in

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