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".]
pg032
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
w