Partial Identities.

A second highly important kind of proposition is that which I propose to call a partial identity. When we say that "All mammalia are vertebrata," we do not mean that mammalian animals are identical with vertebrate animals, but only that the mammalia form a part of the class vertebrata. Such a proposition was regarded in the old logic as asserting the inclusion of one class in another, or of an object in a class. It was called a universal affirmative proposition, because the attribute vertebrate was affirmed of the whole subject mammalia; but the attribute was said to be undistributed, because not all vertebrata were of necessity involved in the proposition. Aristotle, overlooking the importance of simple identities, and indeed almost denying their existence, unfortunately founded his system upon the notion of inclusion in a class, instead of adopting the basis of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, in place of the vastly more general rule that what is true of a class or thing is true of the like. Thus he not only reduced logic to a fragment of its proper self, but destroyed the deep analogies which bind together logical and mathematical reasoning. Hence a crowd of defects, difficulties and errors which will long disfigure the first and simplest of the sciences.

It is surely evident that the relation of inclusion rests upon the relation of identity. Mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates. Cabinet Ministers are included almost always in the class Members of Parliament, because they are identical with some who sit in Parliament. may indicate this identity with a part of the larger class in various ways; as for instance,

Mammalia part of the vertebrata.

=

Diatomaceæ a class of plants.

Cabinet Ministers = some members of Parliament.

Iron a metal.

=

In ordinary language the verbs inclusion more often than not.

We

is and are express mere Men are mortals, means

that men form a part of the class mortal; but great confusion exists between this sense of the verb and that in which it expresses identity, as in "The sun is the centre of the planetary system." The introduction of the indefinite article a often expresses partiality; when we say "Iron is a metal" we clearly mean that iron is one only of several metals.

Certain recent logicians have proposed to avoid the indefiniteness in question by what is called the Quantification of the Predicate, and they have generally used the little word some to show that only a part of the predicate is identical with the subject. Some is an indeterminate adjective; it implies unknown qualities by which we might select the part in question if the qualities were known, but it gives no hint as to their nature. I might make use of such an indeterminate sign to express partial identities in this work. Thus, taking the special symbol V = Some, the general form of a partial identity would be A = VB, and in Boole's Logic expressions of the kind were much used. But I believe that indeterminate symbols only introduce complexity, and destroy the beauty and simple universality of the system which may be created without their use. A vague word like some is only used in ordinary language by ellipsis, and to avoid the trouble of attaining accuracy. We can always employ more definite expressions if we like; but when once the indefinite some is introduced we cannot replace it by the special description. We do not know whether some colour is red, yellow, blue, or what it is; but on the other hand red colour is certainly some colour.

Throughout this system of logic I shall dispense with such indefinite expressions; and this can readily be done by substituting one of the other terms. To express the proposition "All A's are some B's" I shall not use the form A = VB, but

A = AB.

This formula states that the class A is identical with the class AB; and as the latter must be a part at least of the class B, it implies the inclusion of the class A in that of B. We might represent our former example thus,

Mammalia

=

Mammalian vertebrata.

This proposition asserts identity between a part (or it may

be the whole) of the vertebrata and the mammalia. If it is asked What part? the proposition affords no answer, except that it is the part which is mammalian; but the assertion "mammalia = some vertebrata" tells us no more.

It is quite likely that some readers will think this mode of representing the universal affirmative proposition artificial and complicated. I will not undertake to convince them of the opposite at this point of my exposition. Justification for it will be found, not so much in the immediate treatment of this proposition, as in the general harmony which it will enable us to disclose between all parts of reasoning. I have no doubt that this is the critical difficulty in the relation of logical to other forms of reasoning. Grant this mode of denoting that "all A's are B's," and I fear no further difficulties; refuse it, and we find want of analogy and endless anomaly in every direction. It is on general grounds that I hope to show overwhelming reasons for seeking to reduce every kind of proposition to the form of an identity.

I may add that not a few logicians have accepted this view of the universal affirmative proposition. Leibnitz, in his Difficultates Quædam Logica, adopts it, saying, "Omne A est B; id est æquivalent AB et A, seu A non B est nonens." Boole employed the logical equation x = xy concurrently with a = vy; and Spalding1 distinctly says that the proposition "all metals are minerals" might be described as an assertion of partial identity between the two classes. Hence the name which I have adopted for the proposition.

x

Limited Identities.

An important class of propositions have the form

AB = AC,

expressing the identity of the class AB with the class AC. In other words, "Within the sphere of the class A, all the B's are all the C's;" or again, "The B's and C's, which are A's, are identical." But it will be observed that nothing is asserted concerning things which are outside of the class A; and thus the identity is of limited extent. It is the proposition B = C limited to the sphere of things called A.

1 Encyclopædia Britannica, Eighth Ed. art. Logic, sect. 37, note. 8vo reprint, p. 79.

Thus we may say, with some approximation to truth, that "Large plants are plants devoid of locomotive power."

A barrister may make numbers of most general statements concerning the relations of persons and things in the course of an argument, but it is of course to be understood that he speaks only of persons and things under the English Law. Even mathematicians make statements which are not true with absolute generality. They say that imaginary roots enter into equations by pairs; but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients.1 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.

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.2

It is worthy of inquiry whether 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 wonderful mechanical properties, but not gravity. The proposition then is really of the form

Material substance = Material gravitating substance.

Negative Propositions.

In every act of intellect we are engaged with a certain identity or difference between things or sensations compared together. Hitherto I have treated only of identities; and yet it might seem that the relation of difference must be

1 De Morgan On the Root of any Function. Cambridge Philoophical Transactions, 1867, vol xi. p. 25.

2 Syllabus of a proposed System of Logic, §§ 122, 123.

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. Diversity may almost be said to constitute life, being to thought what motion is to a river. The 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 to separate the vowels of the alphabet 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 which may 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 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 a 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-mincral.