Complete Works of Lewis Carroll (143 page)

Representation of Propositions of Relation.

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

This tells us that at least
one
Thing, in the
North
Half, is also in the
West
Half.
Hence it must be in the space
common
to them, that is, in the
North-West Cell
.
Hence the North-West Cell is
occupied
.
And this we can represent by placing a
Red
Counter in it.

[Note that the
Subject
of the Proposition settles which
Half
we are to use; and that the
Predicate
settles in which
portion
of it we are to place the Red Counter.

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

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

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

Let us take, next, the Proposition “Some
y
are
x
”.

This tells us that at least
one
Thing, in the
West
Half, is also in the
North
Half.
Hence it must be in the space
common
to them, that is, in the
North-West Cell
.
Hence 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 English books are old”.]

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

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

We see that this
one
Diagram has now served to represent no less than
three
Propositions, viz.

(1) “Some
xy
exist;

(2)   Some
x
are
y
;

(3)   Some
y
are
x
”.

Hence these three Propositions are equivalent.

[In the “books” example, these Propositions would be

(1) “Some old English books exist;

(2)   Some old books are English;

(3)   Some English books are old”.]

The two equivalent Propositions, “Some
x
are
y
” and “Some
y
are
x
”, are said to be ‘
Converse
’ to each other; and the Process, of changing one into the other, is called ‘
Converting
’, or ‘
Conversion
’.

[For example, if we were told to convert the Proposition

“Some apples are not ripe,”

we should first choose our Univ.
(say “fruit”), and then complete the Proposition, by supplying the Substantive “fruit” in the Predicate, so that it would be

“Some apples are not-ripe fruit”;

and we should then convert it by interchanging its Terms, so that it would be

“Some not-ripe fruit are apples”.]

Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of
four
Trios being as follows:—

(1) “Some
xy
exist” = “Some
x
are
y
” = “Some
y
are
x
”.

(2) “Some
xy
′
exist” = “Some
x
are
y
′
” = “Some
y
′
are
x
”.

(3) “Some
x
′y
exist” = “Some
x
′
are
y
” = “Some
y
are
x
′
”.

(4) “Some
x
′y′
exist” = “Some
x
′
are
y
′
” = “Some
y
′
are
x
′
”.

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

This tell us that no Thing, in the
North
Half, is also in the
West
Half.
Hence there is
nothing
in the space
common
to them, that is, in the
North-West Cell
.
Hence 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 books are English”.]

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

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

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

This tells us that no Thing, in the
West
Half, is also in the
North
Half.
Hence there is
nothing
in the space
common
to them, that is, in the
North-West Cell
.
That is, 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 English books are old”.]

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