No x exist".

This tells us that there is _nothing_ in the North Half; that is, that
the North Half is _empty_; that is, that the North-West Cell and the
North-East Cell are both of them _empty_. And this we can represent by
placing _two Grey_ Counters in the North Half, one in each Cell.

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

[The Reader may perhaps think that it would be enough to place a
_Grey_ Counter on the partition in the North Half, and that,
just as a _Red_ Counter, so placed, would mean "This Half is
_occupied_", so a _Grey_ one would mean "This Half is _empty_".

This, however, would be a mistake. We have seen that a _Red_
Counter, so placed, would mean "At least _one_ of these two
Cells is occupied: possibly _both_ are." Hence a _Grey_ one
would merely mean "At least _one_ of these two Cells is empty:
possibly _both_ are". But what we have to represent is, that
both Cells are _certainly_ empty: and this can only be done by
placing a _Grey_ Counter in _each_ of them.

In the "books" example, this Proposition would be "No old books
exist".]
pg029
Similarly we may represent the three similar Propositions "No x' exist",
"No y exist", and "No y' exist".

[The Reader should make out all these for himself. In the
"books" example, these three Propositions would be "No new books
exist", &c.]

Let us take, next, the Proposition "Some xy exist".

This tells us that there is at least _one_ Thing in the North-West Cell;
that is, that 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 old
English books exist".]

Similarly we may represent the three similar Propositions "Some xy'
exist", "Some x'y exist", and "Some x'y' exist".

[The Reader should make out all these for himself. In the
"books" example, these three Propositions would be "Some old
foreign books exist", &c.]

Let us take, next, the Proposition "No xy exist".

This tells us that there is _nothing_ in the North-West Cell; that is,
that the North-West Cell is _empty_. And this we can represent by
placing a _Grey_ Counter in it.

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