Complete Works of Lewis Carroll (149 page)

xy
′m′

〃

Outer

〃

x
′ym

South-

West

〃

Inner

〃

x
′ym′

〃

Outer

〃

x
′y′m

〃

East

〃

Inner

〃

x
′y′m′

〃

Outer

〃

CHAPTER II.

REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m.

§ 1.

Representation of Propositions of Existence in terms of x and m, or of y and m.

Let us take, first, the Proposition “Some
xm
exist”.

[Note that the
full
meaning of this Proposition is (as explained at p.
12) “Some existing Things are
xm
-Things”.]

This tells us that there is at least
one
Thing in the Inner portion of the North Half; that is, that this Compartment is
occupied
.
And this we can evidently represent by placing a
Red
Counter on the partition which divides it.

[In the “books” example, this Proposition would mean “Some old bound books exist” (or “There are some old bound books”).]

Similarly we may represent the seven similar Propositions, “Some
xm
′
exist”, “Some
x
′m
exist”, “Some
x
′m′
exist”, “Some
ym
exist”, “Some
ym
′
exist”, “Some
y
′m
exist”, and “Some
y
′m′
exist”.

Let us take, next, the Proposition “No
xm
exist”.

This tells us that there is
nothing
in the Inner portion of the North Half; that is, that this Compartment is
empty
.
And this we can represent by placing
two Grey
Counters in it, one in each Cell.

Similarly we may represent the seven similar Propositions, in terms of
x
and
m
, or of
y
and
m
, viz.
“No
xm
′
exist”, “No
x
′m
exist”, &c.

These sixteen Propositions of Existence are the only ones that we shall have to represent on this Diagram.

§ 2.

Representation of Propositions of Relation in terms of x and m, or of y and m.

Let us take, first, the Pair of Converse Propositions

“Some
x
are
m
” = “Some
m
are
x
.”

We know that each of these is equivalent to the Proposition of Existence “Some
xm
exist”, which we already know how to represent.

Similarly for the seven similar Pairs, in terms of
x
and
m
, or of
y
and
m
.

Let us take, next, the Pair of Converse Propositions

“No
x
are
m
” = “No
m
are
x
.”

We know that each of these is equivalent to the Proposition of Existence “No
xm
exist”, which we already know how to represent.

Similarly for the seven similar Pairs, in terms of
x
and
m
, or of
y
and
m
.

Let us take, next, the Proposition “All
x
are
m
.”

We know (see p.
18) that this is a
Double
Proposition, and equivalent to the
two
Propositions “Some
x
are
m
” and “No
x
are
m
′
”, each of which we already know how to represent.

Similarly for the fifteen similar Propositions, in terms of
x
and
m
, or of
y
and
m
.

These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.

The Reader should now get his genial friend to question him on the following four Tables.

The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor,
e.g.
“No
y
′
are
m
”, “Some
xm
′
exist”, &c., &c.