(2) The Materialist or Objective View, in which both A and B are concepts corresponding to really existing things, and the relation of A and B is a relation of concepts corresponding to a relation of things; e.g. Ueberweg's view.

(3) There is another view which is usually identified with the second view, but which should be distinguished from it. I mean the view according to which A and B stand for really existing things, and the relation of A and B is a relation of things: e.g. Spencer's view.

Mill, in his Examination of Hamilton's Philosophy, holds the second view; but in his System of Logic he very nearly gives it up and passes on to the third view. Among English Logicians he seems to occupy an intermediate position between subjective or conceptualist Logicians, represented by Hamilton and Mansel, and objective Logicians, represented by Mr Spencer and Mr Carveth Read.

The difference between the second and the third view, is that, according to the former, the two terms of a proposition are two concepts corresponding to really existing things, while, according to the latter, the two terms are really existing things or phenomena themselves. The upholders of the third view do not seem to face the question as to how things or phenomena can be either the subject or the predicate of proposition, without being thought, that is, without being concepts. The upholders of the second view recognize this necessity and treat in Logic of the forms and relations of Thought as corresponding to the forms and relations of Things, while the upholders of the third view profess to treat of the forms and relations of things themselves1.

1 See Appendix E, "The Nature and Province of Objective Logic."

CHAPTER III.

THE MEANING AND REPRESENTATION OF A, E, I, O BY

DIAGRAMS.

A, 1.

A, 2.

§ 1. A STANDS for any Universal Affirmative proposition of the type 'All A is B.' It may be represented by the two diagrams, A, 1, and A, 2. According to the ordinary or predicative view of propositions, the meaning of A is that the attribute connoted by 'B' belongs to all the things or objects denoted by 'A,' and

A

A B

the implication is that it may or may not belong to any other things. The diagrams represent this, thus,—the circle A stands for the things denoted by the term A, and the circle B for the cases in which the attribute connoted by the term B occurs; the first diagram shows that these cases are more numerous than the things, and the second shows that the two are equal. The meaning of the proposition will be represented by one or other of the two diagrams.

According to the denotative view of propositions, the meaning of A is that the whole of the class denoted by the term A is included in the class denoted by the term B, or that the former is co-extensive with the latter. And this is shown by the diagrams,—in the first, the whole of the class A is a part of the class B, and in the second, the two classes coincide. The mean

ing of the proposition will be represented by one or other of the two diagrams.

According to the connotative view of propositions, the meaning of A is that the attribute connoted by 'B' accompanies the attribute connoted by 'A' in every case, that is, wherever the latter is, there the former is. The diagrams may be understood to represent this, thus,-the first shows that the cases in which the attribute connoted by A occurs are a part of, or are less numerous than, the cases in which the attribute connoted by B occurs; the second shows that the two classes of cases coincide or are equal in number.

Thus, on all the three views, A can be represented by these two diagrams. On each of them, the subject of A is always taken in its whole extent, while the predicate is always taken in a partial and sometimes also in its total extent. This is plainly the case on the first and second views. On the third, too, this is the case, because in all cases the attribute connoted by A is accompanied by the attribute connoted by B. This fact is what is meant by saying that, in an A proposition, the subject is distributed, and the predicate undistributed. By the extent of an attribute is meant the number of cases in which it occurs.

E.

§ 2. E stands for any Universal Negative proposition of the type 'No A is B.' It is represented by the following diagram. The meaning of the diagram is different on the different views of propositions.

O

A

B

On the first view, the circle A stands for the things denoted by

the term A; and the circle B for the cases in which the attribute connoted by the term B occurs; and the diagram shows that the one set is quite distinct from the other,—that the attribute connoted by B does not in any case belong to any of the things denoted by A.

On the second view, the two circles A, B stand for two classes denoted respectively by A and B; and the diagram shows that

the one class is entirely excluded from the other, that the things denoted by B are quite distinct from those denoted by A.

On the third view, the circle A stands for the cases in which the attribute connoted by A occurs, and the circle B for the cases in which the attribute connoted by B occurs; and the diagram shows that the two sets do not coincide, even in a single instance.

On all the three views, then, the diagram represents the meaning of an E proposition, and shows that both A and B are taken in their entire extent, or in all cases wherever they are found. This last fact is what is meant by saying that both the subject and the predicate of an E proposition are distributed.

§ 3. I stands for any Particular Affirmative proposition of the form 'Some A is B.' The meaning of 'some' in logical propositions, as we have already noted, is 'not none,'' at least one.' It does not mean a part only. Its universal and necessary meaning is, at least one; but it does not necessarily exclude the rest. It may mean 'many,' 'most,' 'nearly the whole,' and does not exclude 'the whole' or 'all.' In accordance with this signification of the word 'some,' the proposition 'Some A is B' is represented by the following four diagrams, each of which shows that at least one A is B.

On the first view the meaning of I is that at least one thing, and that, it may be, every thing, denoted by A, has the attribute connoted by B; and this is represented by the diagrams thus: each of them shows that at least one thing or a part of the things coincides with the cases, while two of them (I, 3 and I, 4) show also that the whole of A may coincide with B.

On the second view the meaning of I is that at least one thing, and that, it may be, every thing denoted by A, is included

in the class denoted by B; and this is, as in the preceding case, represented by the diagrams.

On the third view the meaning of I is that in at least one case, and that, it may be, in every case, in which the attribute connoted by A occurs, there occurs the attribute connoted by B; and this is, as in the preceding cases, represented by the diagrams.

On all the views, both the subject and the predicate are always taken in a partial extent, and sometimes also in the whole of their extent. This fact is what is meant by saying that both the subject and the predicate of an I proposition are undistributed.

§ 4. O stands for any Particular Negative proposition of the form 'Some A is not B.' In accordance with the logical meaning of the word 'some,' as given above, it is represented by the following three diagrams, each of which shows that at least one A is not B.

On the first view, the meaning of O is that at least one thing, and that, it may be, every thing, denoted by A, has not the

0000

A

attribute connoted by 'B,'-that all the cases in which the attribute occurs are excluded from at least one thing, and, it may be, from every thing, denoted by A.

On the second view the meaning is, that at least one thing, and that it may be every thing, denoted by 'A' does not belong to the class denoted by 'B'; that the whole of the latter class is excluded from at least one, and it may be from every, individual of the former.

On the third view the meaning is, that in at least one case, and that it may be in every case, in which the attribute connoted by 'A' occurs, the attribute connoted by 'B' does not occur,