Complete Works of Lewis Carroll (147 page)

CHAPTER IV.

INTERPRETATION OF BILITERAL DIAGRAM WHEN MARKED WITH COUNTERS.

The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.

As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.

First, let us suppose that we find a
Red
Counter placed in the North-West Cell.

We know that this represents each of the Trio of equivalent Propositions

“Some
xy
exist” = “Some
x
are
y
” = “Some
y
are
x
”.

Similarly we may interpret a
Red
Counter, when placed in the North-East, or South-West, or South-East Cell.

Next, let us suppose that we find a
Grey
Counter placed in the North-West Cell.

We know that this represents each of the Trio of equivalent Propositions

“No
xy
exist” = “No
x
are
y
” = “No
y
are
x
”.

Similarly we may interpret a
Grey
Counter, when placed in the North-East, or South-West, or South-East Cell.

Next, let us suppose that we find a
Red
Counter placed on the partition which divides the North Half.

We know that this represents the Proposition “Some
x
exist.”

Similarly we may interpret a
Red
Counter, when placed on the partition which divides the South, or West, or East Half.

Next, let us suppose that we find
two Red
Counters placed in the North Half, one in each Cell.

We know that this represents the
Double
Proposition “Some
x
are
y
and some are
y
′
”.

Similarly we may interpret
two Red
Counters, when placed in the South, or West, or East Half.

Next, let us suppose that we find
two Grey
Counters placed in the North Half, one in each Cell.

We know that this represents the Proposition “No
x
exist”.

Similarly we may interpret
two Grey
Counters, when placed in the South, or West, or East Half.

Lastly, let us suppose that we find a
Red
and a
Grey
Counter placed in the North Half, the
Red
in the North-
West
Cell, and the
Grey
in the North-
East
Cell.

We know that this represents the Proposition, “All
x
are
y
”.

[Note that the
Half
, occupied by the two Counters, settles what is to be the
Subject
of the Proposition, and that the
Cell
, occupied by the
Red
Counter, settles what is to be its
Predicate
.]

Similarly we may interpret a
Red
and a
Grey
counter, when placed in any one of the seven similar positions

Red in North-East, Grey in North-West;

Red in South-West, Grey in South-East;

Red in South-East, Grey in South-West;

Red in North-West, Grey in South-West;

Red in South-West, Grey in North-West;

Red in North-East, Grey in South-East;

Red in South-East, Grey in North-East.

Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not only
represent
Propositions, but also
interpret
Diagrams when marked with Counters.

The Questions and Answers should be like this:—

Q.
Represent “No
x
′
are
y
′
.”

A.
Grey Counter in S.E.
Cell.

Q.
Interpret Red Counter on E.
partition.

A.
“Some
y
′
exist.”

Q.
Represent “All
y
′
are
x
.”

A.
Red in N.E.
Cell; Grey in S.E.

Q.
Interpret Grey Counter in S.W.
Cell.

A.
“No
x
′y
exist” = “No
x
′
are
y
” = “No
y
are
x
′
”.

&c., &c.

At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.

[Work Examples §
1
, 5–8 (p.
97).]

BOOK IV.

THE TRILITERAL DIAGRAM.