being as follows:--

(1) "No xy exist" = "No x are y" = "No y are x".
(2) "No xy' exist" = "No x are y'" = "No y' are x".
(3) "No x'y exist" = "No x' are y" = "No y are x'".
(4) "No x'y' exist" = "No x' are y'" = "No y' are x'".

Let us take, next, the Proposition "All x are y".

We know (see p. 17) that this is a _Double_ Proposition, and equivalent
to the _two_ Propositions "Some x are y" and "No x are y'", each of
which we already know how to represent.

.-------.
|(.)|( )|
|---|---|
| | |
.-------.

[Note that the _Subject_ of the given Proposition settles which
_Half_ we are to use; and that its _Predicate_ settles in which
_portion_ of that Half we are to place the Red Counter.]

pg034
TABLE II.

.-----------------------------------------------------.
| | .-------. | | .-------. |
| | | (.) | | | |( )|( )| |
| Some x exist | |---|---| | No x exist | |---|---| |
| | | | | | | | | | |
| | .-------. | | .-------. |
|---------------|-----------|-------------|-----------|
| | .-------. | | .-------. |
| | | | | | | | | | |
| Some x' exist | |---|---| | No x' exist | |---|---| |
| | | (.) | | | |( )|( )| |
| | .-------. | | .-------. |
|---------------|-----------|-------------|-----------|
| | .-------. | | .-------. |
| | | | | | | |( )| | |
| Some y exist | |(.)|---| | No y exist | |---|---| |
| | | | | | | |( )| | |
| | .-------. | | .-------. |
|---------------|-----------|-------------|-----------|
| | .-------. | | .-------. |
| | | | | | | | |( )| |
| Some y' exist | |---|(.)| | No y' exist | |---|---| |
| | | | | | | | |( )| |
| | .-------. | | .-------. |
.-----------------------------------------------------.

Similarly we may represent the seven similar Propositions "All x are
y'", "All x' are y"