Читать книгу Selected Mathematical Works: Symbolic Logic + The Game of Logic + Feeding the Mind: by Charles Lutwidge Dodgson, alias Lewis Carroll - Lewis Carroll - Страница 46

Let us take, first, the Proposition “Some x exist”.

[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”.]

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”.

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

We have seen that the Proposition “No x exist” may be represented by placing two Grey Counters in the North Half, one in each Cell.

We have also seen that these two Grey Counters, taken separately, represent the two Propositions “No xy exist” and “No xy′ exist”.

Hence we see that the Proposition “No x exist” is a Double Proposition, and is equivalent to the two Propositions “No xy exist” and “No xy′ exist”.

[In the “books” example, this Proposition would be “No old books exist”.

Hence this is a Double Proposition, and is equivalent to the two Propositions “No old English books exist” and “No old foreign books exist”.]