[Note that this Proposition is (as explained at [p. 12]) equivalent to “Some existing Things are x-Things.”]
This tells us that there is at least one Thing in the North Half; that is, that the North Half is occupied. And this we can evidently represent by placing a Red Counter (here represented by a dotted circle) on the partition which divides the North Half.
[In the “books” example, this Proposition would be “Some old books exist”.]
Similarly we may represent the three similar Propositions “Some x′ exist”, “Some y exist”, and “Some y′ exist”.
[The Reader should make out all these for himself. In the “books” example, these Propositions would be “Some new books exist”, &c.]
Let us take, next, the Proposition “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”.]