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it doubtless does in negative points. But there is a rare property of mind which consists in penetrating the disguise of variety and seizing the common elements of sameness; and it is this property which furnishes the true measure of intellect. The name of" intellect" expresses the interlacing of the general and the single, which is the peculiar province of mind. To cogitate is the Latin coagitare, resting on a like metaphor. Logic, also, is but another name for the same process, the peculiar work of reason; for Xoyos is derived from Xeyev, which like the Latin legere meant originally to gather. Plato said of this unifying power, that if he met the man who could detect the one in the many, he would follow him as a god.

Laws of Identity and Difference.

At the base of all thought and science must lie the laws which express the very nature and conditions of the discriminating and identifying powers of mind. These are the so-called Fundamental Laws of Thought, usually stated as follows:

1. The Law of Identity.

Whatever is, is.

2. The Law of Contradiction.

and not be.

A thing cannot both be

3. The Law of Duality. A thing must either be or

not be.

The first of these statements may perhaps be regarded as a description of identity itself, if so fundamental a notion can admit of description. A thing at any moment is perfectly identical with itself, and, if any person were unaware of the meaning of the word "identity," we could not better describe it than by such an example.

The second law points out that contradictory attributes can never be joined together. The same object may vary in its different parts; here it may be black, and there white; at one time it may be hard and at another time

1 Max Müller, Lectures on the Science of Language, Second Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view of the etymological meaning of "intellect" is given above on the authority of Professor Max Müller. It seems to be opposed to the ordinary opinion, according to which the Latin intelligere means to choose between, to see a difference between, to discriminate, instead of to unite.

soft; but at the same time and place an attribute cannot be both present and absent. Aristotle truly described this law as the first of all axioms-one of which we need not seek for any demonstration. All truths cannot be proved, otherwise there would be an endless chain of demonstration ; and it is in self-evident truths like this that we find the simplest foundations.

The third of these laws completes the other two. It asserts that at every step there are two possible alternatives presence or absence, affirmation or negation. Hence I propose to name this law the Law of Duality, for it gives to all the formulæ of reasoning a dual character. It asserts also that between presence and absence, existence and non-existence, affirmation and negation, there is no third alternative. As Aristotle said, there can be no mean between opposite assertions: we must either affirm or deny. Hence the inconvenient name by which it has been known-The Law of Excluded Middle.

It may be allowed that these laws are not three independent and distinct laws; they rather express three different aspects of the same truth, and each law doubtless presupposes and implies the other two. But it has not hitherto been found possible to state these characters of identity and difference in less than the threefold formula. The reader may perhaps desire some information as to the mode in which these laws have been stated, or the way in which they have been regarded, by philosophers in different ages of the world. Abundant information on this and many other points of logical history will be found in Ueberweg's System of Logic, of which an excellent translation has been published by Professor T. M. Lindsay (see pp. 228-281).

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The Nature of the Laws of Identity and Difference.

I must at least allude to the profoundly difficult question concerning the nature and authority of these Laws of Identity and Difference. Are they Laws of Thought or Laws of Things? Do they belong to mind or to material nature? On the one hand it may be said that science is a purely mental existence, and must therefore conform to the laws of that which formed it. Science is in the mind and

not in the things, and the properties of mind are therefore all important. It is true that these laws are verified in the observation of the exterior world; and it would seem that they might have been gathered and proved by generalisation, had they not already been in our possession. But on the other hand, it may well be urged that we cannot prove these laws by any process of reasoning or observation, because the laws themselves are presupposed, as Leibnitz acutely remarked, in the very notion of a proof. They are the prior conditions of all thought and all knowledge, and even to question their truth is to allow them true. Hartley ingeniously refined upon this argument, remarking that if the fundamental laws of logic be not certain, there must exist a logic of a second order whereby we may determine the degree of uncertainty: if the second logic be not certain, there must be a third; and so on ad infinitum. Thus we must suppose either that absolutely certain laws of thought exist, or that there is no such thing as certainty whatever.

Logicians, indeed, appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always possible, and of not unfrequent occurrence. The Laws

of Thought are often called necessary laws, that is, laws which cannot but be obeyed. Yet as a matter of fact, who is there that does not often fail to obey them? They are the laws which the mind ought to obey rather than what it always does obey. Our thoughts cannot be the criterion of truth, for we often have to acknowledge mistakes in arguments of moderate complexity, and we sometimes only discover our mistakes by collision between our expectations and the events of objective nature.

Mr. Herbert Spencer holds that the laws of logic are objective laws, and he regards the mind as being in a state of constant education, each act of false reasoning or miscalculation leading to results which are likely to prevent similar mistakes from being again committed. I am quite inclined to accept such ingenious views; but at the same time it is necessary to distinguish between the accumulation of knowledge, and the constitution of the mind which allows of the acquisition of knowledge. Before the mind can perceive or reason at all it must have 1 Hartley on Man, vol. i. p. 359.

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Principles of Psychology, Second Ed., vol. ii. p. 86.

the conditions of thought impressed upon it. Before a mistake can be committed, the mind must clearly distinguish the mistaken conclusion from all other assertions. Are not the Laws of Identity and Difference the prior conditions of all consciousness and all existence? Must they not hold true, alike of things material and immaterial? and if so, can we say that they are only subjectively true or objectively true? I am inclined, in short, to regard them as true both "in the nature of thought and things," as I expressed it in my first logical essay;1 and I hold that they belong to the common basis of all existence. But this is one of the most difficult questions of psychology and metaphysics which can be raised, and it is hardly one for the logician to decide. As the mathematician does not inquire into the nature of unity and plurality, but develops the formal laws of plurality, so the logician, as I conceive, must assume the truth of the Laws of Identity and Difference, and occupy himself in developing the variety of forms of reasoning in which their truth may be manifested.

Again, I need hardly dwell upon the question whether logic treats of language, notions, or things. As reasonably might we debate whether a mathematician treats of symbols, quantities, or things. A mathematician certainly does treat of symbols, but only as the instruments whereby to facilitate his reasoning concerning quantities; and as the axioms and rules of mathematical science must be verified in concrete objects in order that the calculations founded upon them may have any validity or utility, it follows that the ultimate objects of mathematical science are the things themselves. In like manner I conceive that the logician treats of language so far as it is essential for the embodiment and exhibition of thought. Even if reasoning can take place in the inner consciousness of man without the use of any signs, which is doubtful, at any rate it cannot become the subject of discussion until by some system of material signs it is manifested to other persons. The logician then uses words and symbols as instruments of reasoning, and leaves the nature and peculiarities of language to the grammarian. But signs again must

1 Pure Logic, or the Logic of Quality apart from Quantity, 1864, pp. 10, 16, 22, 29, 36, &c.

correspond to the thoughts and things expressed, in order that they shall serve their intended purpose. We may therefore say that logic treats ultimately of thoughts and things, and immediately of the signs which stand for them. Sigus, thoughts, and exterior objects may be regarded as parallel and analogous series of phenomena, and to treat any one of the three series is equivalent to treating either of the other series.

The Process of Inference.

The fundamental action of our reasoning faculties consists in inferring or carrying to a new instance of a phenomenon whatever we have previously known of its like, analogue, equivalent or equal. Sameness or identity. presents itself in all degrees, and is known under various names; but the great rule of inference embraces all degrees, and affirms that so far as there exists sameness, identity or likeness, what is true of one thing will be true of the other. The great difficulty doubtless consists in ascertaining that there does exist a sufficient degree of likeness or sameness to warrant an intended inference; and it will be our main task to investigate the conditions under which reasoning is valid. In this place I wish to point out that there is something common to all acts of inference, however different their apparent forms. The one same rule lends itself to the most diverse applications.

The simplest possible case of inference, perhaps, occurs in the use of a pattern, example, or, as it is commonly called, a sample. To prove the exact similarity of two) portions of commodity, we need not bring one portion beside the other. It is sufficient that we take a sample which exactly represents the texture, appearance, and general nature of one portion, and according as this sample agrees or not with the other, so will the two portions of commodity agree or differ. Whatever is true. as regards the colour, texture, density, material of the sample will be true of the goods themselves. In such cases likeness of quality is the condition of inference.

Exactly the same mode of reasoning holds true of magnitude and figure. To compare the sizes of two objects, we need not lay them beside each other, A

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