modes in which thought begins to express itself with accuracy 1."

According to this view, all relations expressed by propositions may be reduced to the single type of the relation of subject and attribute. The subject of a proposition may be anything that can possess an attribute or attributes. It may be a substance, a phenomenon, or an attribute. The predicate of a proposition is an attribute; and even when the predicate is a concrete term, the term is interpreted in its connotation (or comprehension).

This view of Proposition's does not ignore the relations of .space and time, of cause and effect, of resemblance and difference, expressed by many propositions; but it holds that, for logical purposes, they may all be reduced to the relation of subject and attribute. Some Logicians holding this view so far as a certain class of propositions, namely, those expressing the relation of substance and attribute, are concerned, maintain that the other relations, such as those of time and space, of cause and effect, of resemblance and difference, can not, or should not, be reduced to the single type of subject and attribute. According to them, there are different classes of propositions founded upon different categories of thought and giving rise to distinct types of reasoning2.

1 Symbolic Logic, p. 4.

2 The relation of subject and attribute is also called the relation of substance and attribute. It is not necessary to inquire here into the nature of this relation, or into the meaning of Subject, Substance, Thing, or Attribute, or to discuss the question as to whether an attribute possessing attributes becomes a substance (or thing), or remains an attribute. For the Predicative view, it is sufficient if propositions expressing other relations can, in some way, be understood to express the relation of subject and attribute; and this may be done in the following manner:-The proposition "A is equal to B," for example, expressing the relation of Equality, means, according to this view, that the attribute of being equal to B is possessed by A, whether A and B be things or attributes; the proposition "A is the cause of B," expressing the relation of Cause and Effect, means, according to this

(2) The Denotative View, in which both A and B are taken in denotation (or extension). This view includes (a) Hobbes' View, (b) the Class View, in which the class or group of things denoted by A is included in the class or group of things denoted by B, and (c) the Equational View, in which the things denoted by A are the same as those denoted by B.

(3) The Connotative or Attributive View, in which both A and B are taken in connotation, and the relation expressed by the proposition is variable and depends on the nature of A and B. It does not reduce all relations expressed by propositions to one single type. It recognizes different fundamental relations, and distinct modes of reasoning arising from those relations. Thus, according to this view, there are Mathematical reasonings founded upon the relations of Equality or Inequality, Inductive reasonings founded upon the relations of succession or of cause and effect, besides Reasonings which are founded upon the relation of co-existence of attributes, which takes the place of the relation of substance and attribute.

(4) The Denotative-Connotative View, in which A and B are taken both in denotation (or extension) and in connotation (or comprehension), and the relation of A and B is a twofold one. Hamilton, for instance, holds that when both A and B are taken in extension, A is contained in B, and that when both A and B are taken in comprehension, B is contained in A.

There is another point on which Logicians differ in their views of the Proposition. It is connected with the different views which they take of Logic as a science. The different views of the Proposition arising from difference on this point may be noted as follows:

(1) The Conceptualist or Subjective View, in which both A and B are concepts, not necessarily corresponding to really existing things, but true of possible things, that is, of things that may be realised in Thought.

view, that the attribute of being the cause of B is possessed by A whatever A and B may be.

(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.

A

A B

B

§ 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 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.

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