Complete Works of Lewis Carroll (153 page)

(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

[Work Examples §
1
, 9–12 (p.
97); §
2
, 1–20 (p.
98).]

CHAPTER IV.

INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.

The problem before us is, given a marked Triliteral Diagram, to ascertain
what
Propositions of Relation, in terms of
x
and
y
, are represented on it.

The best plan, for a
beginner
, is to draw a
Biliteral
Diagram alongside of it, and to transfer, from the one to the other, all the information he can.
He can then read off, from the Biliteral Diagram, the required Propositions.
After a little practice, he will be able to dispense with the Biliteral Diagram, and to read off the result from the Triliteral Diagram itself.

To
transfer
the information, observe the following Rules:—

(1) Examine the N.W.
Quarter of the Triliteral Diagram.

(2) If it contains a “I”, in
either
Cell, it is certainly
occupied
, and you may mark the N.W.
Quarter of the Biliteral Diagram with a “I”.

(3) If it contains
two
“O”s, one in
each
Cell, it is certainly
empty
, and you may mark the N.W.
Quarter of the Biliteral Diagram with a “O”.

(4) Deal in the same way with the N.E., the S.W., and the S.E.
Quarter.

[Let us take, as examples, the results of the four Examples worked in the previous Chapters.

(1)

In the N.W.
Quarter, only
one
of the two Cells is marked as
empty
: so we do not know whether the N.W.
Quarter of the Biliteral Diagram is
occupied
or
empty
: so we cannot mark it.

In the N.E.
Quarter, we find
two
“O”s: so
this
Quarter is certainly
empty
; and we mark it so on the Biliteral Diagram.

In the S.W.
Quarter, we have no information
at all
.

In the S.E.
Quarter, we have not enough to use.

We may read off the result as “No
x
are
y
′
”, or “No
y
′
are
x
,” whichever we prefer.

(2)

In the N.W.
Quarter, we have not enough information to use.

In the N.E.
Quarter, we find a “I”.
This shows us that it is
occupied
: so we may mark the N.E.
Quarter on the Biliteral Diagram with a “I”.

In the S.W.
Quarter, we have not enough information to use.

In the S.E.
Quarter, we have none at all.

We may read off the result as “Some
x
are
y
′
”, or “Some
y
′
are
x
”, whichever we prefer.

(3)

In the N.W.
Quarter, we have
no
information.
(The “I”, sitting on the fence, is of no use to us until we know on
which
side he means to jump down!)

In the N.E.
Quarter, we have not enough information to use.

Neither have we in the S.W.
Quarter.

The S.E.
Quarter is the only one that yields enough information to use.
It is certainly
empty
: so we mark it as such on the Biliteral Diagram.

We may read off the results as “No
x
′
are
y
′
”, or “No
y
′
are
x
′
”, whichever we prefer.

(4)

The N.W.
Quarter is
occupied
, in spite of the “O” in the Outer Cell.
So we mark it with a “I” on the Biliteral Diagram.

The N.E.
Quarter yields no information.