considered such propositions purely accidental, and came to the extraordinary conclusion, that 'Singulars cannot be predicated of other terms.'

Propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole; as when we say—

'The Queen, Lords, and Commons - The Legislature

of the United Kingdom.'

=

When Blackstone asserts, The only true and natural foundation of society are the wants and fears of individuals,' we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society. But many propositions which might seem to be collective are but groups of singular propositions or identities. When we say 'Potassium and sodium are the metallic bases of potash and soda,' we obviously

mean

Potassium Metallic base of potash;

=

Sodium Metallic base of soda.

It is the work of grammatical analysis to separate the various propositions often combined in a single sentence. Logic cannot be properly required to interpret the forms and devices of language, but to treat the meaning or information when clearly exhibited.

Partial Identities.

However numerous and important may be propositions expressing simple identity of one term or class with another, there is an almost equally important kind of proposition 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 mammalian 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, in place of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, instead 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 a 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— Mammalia = part of the vertebrata

Diatoms = species of plants.

Cabinet Ministers = some Members of Parliament.
Iron a metal.

In ordinary language the verbs is or 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 seems to express partiality, as when we say 'Iron is a metal' we clearly mean one only of several metals.

Certain eminent 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 they were known, but it gives no hint as to their nature. I might make extensive 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 freely used. But I find 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 substitute for it 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; as is also yellow, blue, &c.

Throughout this system of logic I shall usually dispense with all 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.

c Elementary Lessons in Logic,' p. 183. 'Substitution of Similars,'

P. 7.

E

This formula expresses 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. Thus we might represent our former example thus

Mammalia = Mammalian vertebrata.

This proposition asserts identity between a part 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 may think this mode of representing the universal affirmative proposition of the old logic artificial and complicated. I will not undertake to convince them of the opposite at this point of the system. My justification for it will be found, not in the immediate treatment of this proposition, but in the general harmony which it enables us to discover between all parts of reasoning. I have no doubt that this is the point of 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 complication in every direction. For instance -Aristotle, in accepting inclusion of class in class as the fundamental relation of logic, was at once obliged to ignore the existence of the very extensive and allimportant class of propositions denoting the similarity of one thing with another. 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 previous logicians have accepted this view of the universal affirmative proposition. Boole often employed this mode of expression, and

d

Spalding a 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.

Limited Identities.

A highly important class of propositions have the general form

AB= AC,

expressing the identity of the class AB with the class AC. In other words, Within the sphere of the class of things A, all the B's are all the C's,' or '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 the class A. Thus if we say 'Plants are devoid of locomotive power,' we must limit the statement to large plants, since minute microscopic plants often have very remarkable powers of motion. When we say 'Metals possess metallic lustre,' we mean in their uncombined state.

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.e

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

e De Morgan 'On the Root of any Function.' Cambridge Philosophical Transactions, 1867, vol. xi. p. 25.