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. We may indicate this identity with a part of the larger class in various ways; as for instance,
In ordinary language the verbs is and are express mere inclusion more often than not. 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
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,
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.