The universe, in short, within which they habitually discourse, is that of equations with real coefficients. These implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments.

It is worthy of inquiry whether almost all identities are not really limited to an implied sphere of meaning. When we make such a plain statement as 'Gold is malleable' we obviously speak of gold only in its solid state; when we say that 'Mercury is a liquid metal' we must be understood to exclude the frozen condition to which it may be reduced in the Arctic regions. Even when we take such a fundamental law of nature as All substances gravitate,' we must mean by substance, material substance, not including that basis of heat, light and electrical undulations which occupies space and possesses many mechanical properties, but not gravity. The proposition then is really of the form

Material substance Material gravitating substance.

=

To De Morgan is due the remark, that we do usually think and argue in a limited universe or sphere of notions even when it is not expressly stated f.

Negative Propositions.

In every act of intellect, as we have seen, we are engaged with a certain degree of identity or difference between certain things or sensations compared together. Hitherto I have treated only of identities; and yet it might seem that the relation of difference must be infinitely more common than that of likeness. One thing may resemble a great many other things, but then it differs from all remaining things in the world. Difference or diversity may almost be said to constitute life, being to thought what motion is to a river. The

f 'Syllabus of a Proposed System of Logic,' §§ 122, 123.

very perception of an object involves its discrimination. from all other objects. But we may nevertheless be said to detect resemblance as often as we detect difference. We cannot, in fact, assert the existence of a difference, without at the same time implying the existence of an agreement.

If I compare mercury, for instance, with other metals, and decide that it is not solid, here is a difference between mercury and solid things, expressed in a negative proposition; but there must be implied, at the same time, an agreement between mercury and the other substances which are not solid. As it is impossible in the alphabet to separate the vowels from the consonants without at the same time separating the consonants from the vowels, so I cannot select as the object of thought solid things, without thereby throwing together into another class all things which are not solid. The very fact of not possessing a quality, constitutes a new quality or circumstance which may equally be the ground of judgment and classification. In this point of view, agreement and difference are ever the two sides of the same act of intellect, and it becomes equally possible to express the same judgment in the one or other aspect.

Between affirmation and negation there is accordingly a perfect balance or equilibrium. Every affirmative proposition implies a negative one, and vice versa. It is even a matter of indifference, in a logical point of view, whether a positive or negative term be used to denote a given quality and the class of things possessing it. If the ordinary state of man's body be called good health, then in other circumstances he is said not to be in good health; but we might equally describe him in the latter state as sickly, and in his normal condition he would be not sickly. Animal and vegetable substances are now called organic, so that the other substances, forming an immensely greater

part of the globe, are described negatively as inorganic. But we might, with at least equal logical correctness, have described the preponderating class of substances as mineral, and then vegetable and animal substances would have been non-mineral.

It is plain that any positive term, and its corresponding negative divide between them the whole universe of thought: whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Suppose that the things or classes represented by A and B are found to differ, we may indicate the result of the judgment by the notation (see p. 20)

A B.

~

But we may now represent the same judgment by the assertion that A agrees with those things which differ from B, or that A agrees with the not-B's. Using our notation for negative terms (see p. 17), we obtain

A = Ab

as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition:

There may also be several other classes of negative propositions, of which no notice was taken in the old logic. We may have cases where all A's are not-B's, and at the same time all not-B's are A's; there may, in short, be a simple identity between A and not-B, which may be expressed in the form

A = b.

An example of this form would be

Conductors of electricity = non-electrics.

We shall also frequently have to deal as results of

deduction, with simple, partial, or limited identities between negative terms, in the forms

a=b, a=ab, aC=bC.

It would be equally possible to represent affirmative propositions in the negative form. Thus Iron is solid,' might be expressed as 'Iron is not not-solid,' or ' Iron is not fluid'; or, taking A and b for the terms 'iron,' and 'not-solid,' the form would be

But there are very strong reasons why we should employ all propositions in their affirmative form. All inference proceeds by the substitution of equivalents, and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner. As will be more fully shown, we can infer in a negative proposition, but not by it. Difference is incapable of becoming the ground of inference; it is only the implied agreement with other differing objects, which admits of deduction; and it will always be found advantageous to employ propositions in the form which exhibits clearly all the implied agreements.

Conversion of Propositions.

The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition of the subject and predicate in such a way as to obtain a new proposition which will be equally true with the original. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium

chloride. If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, 'I will give anybody their choice of these two truths, which amount to the same thing; He who loves himself best is the honestest man; or, The honestest man loves himself bests. Scotus Erigena exactly expresses this reciprocal character of identity in saying, 'There are not two studies, one of philosophy and the other of religion; true philosophy is true religion, and true religion is true philosophy.'

A mathematician would not think it worth mention that if xy then also y=x. He would not consider these to be two equations at all, but one same equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity there is no need for succession in order; each is simultaneously equal and identical to the other. These remarks will hold true equally of logical and mathematical identity; so that I shall consider the two forms

A=B and B = A

to express exactly the same identity differently written. All need for rules of conversion disappears, and there will be no single proposition in the system which may not be written with either term foremost. Thus A = AB is the same as AB=A, AB= AC as AC=AB, and so on.

The same remarks are partially true of differences or inequalities, which are also reciprocal to the extent that one thing cannot differ from a second without the second differing from the first. Mars differs in colour from

g Chesterfield's Letters, 8vo, 1744; vol. i. p. 302.