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

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,

that every case of the latter is excluded from at least one case, and it may be from every case, of the former.

On all the views, 'B' is always taken in its entire extent, ‘A’ always in a part, and sometimes also in the whole of its extent. This fact is, what is meant by saying that the predicate of an O proposition is distributed and the subject undistributed.

§ 5. Recapitulation.-Representing 'A' and 'B,' the subject and the predicate of a proposition, by two circles, and the copula, by the mutual position or relation of the two circles, A is represented by the two diagrams (1) and (2),

and O by the three diagrams (8), (9), and (10).

On a comparison of these diagrams, it will be seen that (1) and (6), (2) and (7), (3) and (10), (4) and (8), (5) and (9) are identical, and that there are altogether five fundamental diagrams. To help the memory of the student, these five diagrams are given below in a definite order :

1st.

2nd.

3rd.

4th.

5th.

AB

ACABA B A B C

These diagrams will be henceforth called the 1st, 2nd, 3rd, 4th, and 5th respectively, and the student is advised to remember their respective numbers. A is represented by the 1st and 2nd, E by the 4th, I by the 1st, 2nd, 3rd, and 5th, and O by the 3rd, 4th, and 5th.

The subject of A is distributed, and the predicate undistributed. Both the subject and predicate of E are distributed. Both the subject and predicate of I are undistributed. The predicate of O is distributed, and the subject undistributed. That is, only universal propositions distribute their subjects, and only negative propositions distribute their predicates.

§ 6. Exercises on the meaning and representation of propositions by diagrams.

I. Show how the four propositional forms-viz., A, E, I, and O -may be represented by diagrams.

II. Draw the five fundamental diagrams representing all propositions in their proper order, and state which of them represent A, which E, which I, and which O respectively.

III. Which of the four propositional forms-A, E, I, and O—may be represented by the 1st, which by the 2nd, which by the 3rd, which by the 4th, and which by the 5th diagram?

IV. Name the diagrams which represent A, E, I, and O respectively.

V. Represent each of the following propositions by its appropriate diagrams, and state its meaning according to the various theories of predication and of the import of propositions:

1. All men are rational.

2. All men are fallible.

3. Some men are rich.

4. Some elements are not metals.

5. Rain is produced by clouds.
6. Some plants have flowers.

7. All material bodies are extended.

8. No man is perfect.

9. All metals are elements.

10. All sensations are feelings.

11. Material bodies gravitate.

12. Silver is white.

13. Water boils at 100° C. under a pressure of 760 m.m.

14. Heat expands bodies.

15. Friction produces heat.