Complete Works of Lewis Carroll (152 page)

   All
m
′
are
x
′
  

TABLE VIII.

   All
y
are
m
  

   All
y
are
m
′
  

   All
y
′
are
m
  

   All
y
′
are
m
′
  

   All
m
are
y
  

   All
m
are
y
′
  

   All
m
′
are
y
  

   All
m
′
are
y
′
  

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
.

a

b

(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

(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
.

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