Complete Works of Lewis Carroll (142 page)

CHAPTER II.

COUNTERS.

Let us agree that a
Red
Counter, placed within a Cell, shall mean “This Cell is
occupied
” (i.e.
“There is at least
one
Thing in it”).

Let us also agree that a
Red
Counter, placed on the partition between two Cells, shall mean “The Compartment, made up of these two Cells, is
occupied
; but it is not known
whereabouts
, in it, its occupants are.”
Hence it may be understood to mean “At least
one
of these two Cells is occupied: possibly
both
are.”

Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind
which
of two political parties he will join: such a man is said to be “
sitting on the fence
.”
This phrase exactly describes the condition of the Red Counter.

Let us also agree that a
Grey
Counter, placed within a Cell, shall mean “This Cell is
empty
” (i.e.
“There is
nothing
in it”).

[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]

CHAPTER III.

REPRESENTATION OF PROPOSITIONS.

§ 1.

Introductory.

Henceforwards, in stating such Propositions as “Some
x
-Things exist” or “No
x
-Things are
y
-Things”, I shall omit the word “Things”, which the Reader can supply for himself, and shall write them as “Some
x
exist” or “No
x
are
y
”.

[Note that the word “Things” is here used with a special meaning, as explained at p.
23.]

A Proposition, containing only
one
of the Letters used as Symbols for Attributes, is said to be ‘
Uniliteral
’.

[For example, “Some
x
exist”, “No
y
′
exist”, &c.]

A Proposition, containing
two
Letters, is said to be
‘Biliteral’
.

[For example, “Some
xy
′
exist”, “No
x
′
are
y
”, &c.]

A Proposition is said to be ‘
in terms of
’ the Letters it contains, whether with or without accents.

[Thus, “Some
xy
′
exist”, “No
x
′
are
y
”, &c., are said to be
in terms of
x
and
y
.]

§ 2.

Representation of Propositions of Existence.

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

[Note that this Proposition is (as explained at p.
12) equivalent to “Some existing Things are
x
-Things.”]

This tells us that there is at least
one
Thing in the North Half; that is, that the North Half is
occupied
.
And this we can evidently represent by placing a
Red
Counter (here represented by a
dotted
circle) on the partition which divides the North Half.

[In the “books” example, this Proposition would be “Some old books exist”.]

Similarly we may represent the three similar Propositions “Some
x
′
exist”, “Some
y
exist”, and “Some
y
′
exist”.

[The Reader should make out all these for himself.
In the “books” example, these Propositions would be “Some new books exist”, &c.]

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

This tells us that there is
nothing
in the North Half; that is, that the North Half is
empty
; that is, that the North-West Cell and the North-East Cell are both of them
empty
.
And this we can represent by placing
two Grey
Counters in the North Half, one in each Cell.

[The Reader may perhaps think that it would be enough to place a
Grey
Counter on the partition in the North Half, and that, just as a
Red
Counter, so placed, would mean “This Half is
occupied
”, so a
Grey
one would mean “This Half is
empty
”.

This, however, would be a mistake.
We have seen that a
Red
Counter, so placed, would mean “At least
one
of these two Cells is occupied: possibly
both
are.”
Hence a
Grey
one would merely mean “At least
one
of these two Cells is empty: possibly
both
are”.
But what we have to represent is, that both Cells are
certainly
empty: and this can only be done by placing a
Grey
Counter in
each
of them.

In the “books” example, this Proposition would be “No old books exist”.]

Similarly we may represent the three similar Propositions “No
x
′
exist”, “No
y
exist”, and “No
y
′
exist”.

[The Reader should make out all these for himself.
In the “books” example, these three Propositions would be “No new books exist”, &c.]

Let us take, next, the Proposition “Some xy exist”.

This tells us that there is at least
one
Thing in the North-West Cell; that is, that the North-West Cell is
occupied
.
And this we can represent by placing a
Red
Counter in it.

[In the “books” example, this Proposition would be “Some old English books exist”.]

Similarly we may represent the three similar Propositions “Some
xy
′
exist”, “Some
x
′y
exist”, and “Some
x
′y′
exist”.

[The Reader should make out all these for himself.
In the “books” example, these three Propositions would be “Some old foreign books exist”, &c.]

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

This tells us that there is
nothing
in the North-West Cell; that is, that the North-West Cell is
empty
.
And this we can represent by placing a
Grey
Counter in it.

[In the “books” example, this Proposition would be “No old English books exist”.]

Similarly we may represent the three similar Propositions “No
xy
′
exist”, “No
x
′y
exist”, and “No
x
′y′
exist”.

[The Reader should make out all these for himself.
In the “books” example, these three Propositions would be “No old foreign books exist”, &c.]

We have seen that the Proposition “No
x
exist” may be represented by placing
two Grey
Counters in the North Half, one in each Cell.

We have also seen that these two
Grey
Counters, taken
separately
, represent the two Propositions “No
xy
exist” and “No
xy
′
exist”.

Hence we see that the Proposition “No
x
exist” is a
Double
Proposition, and is equivalent to the
two
Propositions “No
xy
exist” and “No
xy
′
exist”.

[In the “books” example, this Proposition would be “No old books exist”.

Hence this is a
Double
Proposition, and is equivalent to the
two
Propositions “No old
English
books exist” and “No old
foreign
books exist”.]

§ 3.