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. ·-------· |( )| | |---|---| | | | ·-------· [In the "books" example, this Proposition would be "No old English books exist".] Similarly we may represent the three similar Propositions "No xy' exist", "No x'y exist", and "No x'y' exist".