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".
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(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)
·---------------·
|(O) | (O)|
| ·---|---· |
| | |(O)| |
|---|---|---|---|
| | |(O)| |
| ·---|---· |
| | |
·---------------·

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.

·-------·
| |(O)|
|---|---|
| | |
·-------·

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)
·---------------·
| | |
| ·---|---· |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| ·---|---· |
| | |
·---------------·

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".

·-------·
| |(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.
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(3)
·---------------·
| | |
| ·---|---· |
| | |(O)| |
|---|(I)|---|---|
| | |(O)| |
| ·---|---· |
|(O) | (O)|
·---------------·

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.

·-------·
| | |
|---|---|
| |(O)|
·-------·

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)
·---------------·
|(O) | |
| ·---|---· |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| ·---|---· |
|(O) | |
·---------------·

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.

·-------·
|(I)| |
|---|---|
| | |
·-------·

The S.W. Quarter is certainly _empty_. So we mark it as such on
the Biliteral Diagram.

·-------·
|(I)| |
|---|---|
|(O)| |
·-------·

The S.E. Quarter does not yield enough information to use.

We read off the result as "All y are x."]

[Review Tables V, VI (pp. 46, 47). Work Examples § =1=, 13-16
(p. 97); § =2=, 21-32 (p. 98); § =3=, 1-20 (p. 99).]

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