_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 Sec. =1=, 9-12 (p. 97); Sec. =2=, 1-20 (p. 98).]
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