Chapter III

_REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM._

The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits "I" and "O", instead of using the Board and Counters: he may put a "I" to represent a _Red_ Counter (this may be interpreted to mean "There is at least _one_ Thing here"), and a "O" to represent a _Grey_ Counter (this may be interpreted to mean "There is _nothing_ here").

The Pair of Propositions, that we shall have to represent, will always be, one in terms of x and m, and the other in terms of y and m.

When we have to represent a Proposition beginning with "All", we break it up into the _two_ Propositions to which it is equivalent.

When we have to represent, on the same Diagram, Propositions, of which some begin with "Some" and others with "No", we represent the _negative_ ones _first_. This will sometimes save us from having to put a "I" "on a fence" and afterwards having to shift it into a Cell.

[Let us work a few examples.

(1)

"No x are m';
No y' are m".

Let us first represent "No x are m'". This gives us Diagram a.

Then, representing "No y' are m" on the same Diagram, we get
Diagram b.
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a b
·---------------· ·---------------·
|(O) | (O)| |(O) | (O)|
| ·---|---· | | ·---|---· |
| | | | | | | |(O)| |
|---|---|---|---| |---|---|---|---|
| | | | | | | |(O)| |
| ·---|---· | | ·---|---· |
| | | | | |
·---------------· ·---------------·

(2)

"Some m are x;
No m are y".

If, neglecting the Rule, we were begin with "Some m are x", we
should get Diagram a.

And if we were then to take "No m are y", which tells us that
the Inner N.W. Cell is _empty_, we should be obliged to take the
"I" off the fence (as it no longer has the choice of _two_
Cells), and to put it into the Inner N.E. Cell, as in Diagram c.

This trouble may be saved by beginning with "No m are y", as in
Diagram b.

And _now_, when we take "Some m are x", there is no fence to sit
on! The "I" has to go, at once, into the N.E. Cell, as in
Diagram c.

a b c
·---------------· ·---------------· ·---------------·
| | | | | | | | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | (I) | | | |(O)| | | | |(O)|(I)| |
|---|---|---|---| |---|---|---|---| |---|---|---|---|
| | | | | | |(O)| | | | |(O)| | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | | |
·---------------· ·---------------· ·---------------·

(3)

"No x' are m';
All m are y".

Here we begin by breaking up the Second into the two
Propositions to which it is equivalent. Thus we have _three_
Propositions to represent, viz.--

(1) "No x' are m';
(2) Some m are y;
(3) No m are y'".

These we will take in the order 1, 3, 2.

First we take No. (1), viz. "No x' are m'". This gives us
Diagram a.
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Adding to this, No. (3), viz. "No m are y'", we get Diagram b.

This time the "I", representing No. (2), viz. "Some m are y,"
has to sit on the fence, as there is no "O" to order it off!
This gives us Diagram c.

a b c
·---------------· ·---------------· ·---------------·
| | | | | | | | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | |(O)| | | | |(O)| |
|---|---|---|---| |---|---|---|---| |---|(I)|---|---|
| | | | | | | |(O)| | | | |(O)| |
| ·---|---· | | ·---|---· | | ·---|---· |
|(O) | (O)| |(O) | (O)| |(O) | (O)|
·---------------· ·---------------· ·---------------·

(4)

"All m are x;
All y are m".

Here we break up _both_ Propositions, and thus get _four_ to
represent, viz.--

(1) "Some m are x;
(2) No m are x';
(3) Some y are m;
(4) No y are m'".

These we will take in the order 2, 4, 1, 3.

First we take No. (2), viz. "No m are x'". This gives us Diagram
a.

To this we add No. (4), viz. "No y are m'", and thus get Diagram
b.

If we were to add to this No. (1), viz. "Some m are x", we
should have to put the "I" on a fence: so let us try No. (3)
instead, viz. "Some y are m". This gives us Diagram c.

And now there is no need to trouble about No. (1), as it would
not add anything to our information to put a "I" on the fence.
The Diagram _already_ tells us that "Some m are x".]

a b c
·---------------· ·---------------· ·---------------·
| | | |(O) | | |(O) | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | | | | | |(I)| | |
|---|---|---|---| |---|---|---|---| |---|---|---|---|
| |(O)|(O)| | | |(O)|(O)| | | |(O)|(O)| |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | |(O) | | |(O) | |
·---------------· ·---------------· ·---------------·

[Work Examples § =1=, 9-12 (p. 97); § =2=, 1-20 (p. 98).]

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