for it gives to all the formula of reasoning a dual character. It asserts also that between presence or absence, existence or non-existence, affirmation or 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 somewhat inconvenient name by which it has been generally knownThe Law of Excluded Middle.

It may be held 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 three-fold 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 Mr. T. M. Lindsay. I must confess however that the history of logical doctrines has seemed to me one of the most confusing and least beneficial studies in which a person can engage; and over-abundant attention perhaps has been paid to it by Hamilton, Mansel, and many German logicians.

The Nature and Authority of the Laws of Identity and Difference.

I must at least allude to the profoundly difficult question concerning the nature and authority of these

d Ueberweg's 'System of Logic,' transl. by Lindsay, London, 1871, pp. 228-281.

Laws of Identity or 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 o.

Logicians, indeed, appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always likely to occur. 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 hows which the mind ought to obey rather than what hways does obey. Our thoughts cannot be the orion of truth, for we often have to acknowledge

Hartley on Man, vol. i. p. 359.

mistakes in arguments of very moderate complexity, and we sometimes only discover our mistakes by a collision between our mental 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 experience, 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 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 essays, and I hold that they belong to the common basis of all existence. But this is one of the most profound and 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 developes the formal laws of plurality, so the logician, as I conceive, must assume the truth of the Laws of

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

'Pure Logic, or the Logic of Quality apart from Quantity,' London (Stanford), 1864, pp. 10, 16, 22, 29, 36, &c.

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, 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 existing language to the grammarian. But signs again must 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. Signs, thoughts and exterior objects may be regarded as parallel and analogous series of phenomena, and to treat one series is equivalent to treating either of the other series h

h See also 'Elementary Lessons in Logic,' Second Ed., p. 10.

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 of reasoning 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 the inference 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 cut 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 size of two objects, we need not lay them alongside each other. A