The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
his genial friend. When this result has been reached, he may safely go on to the next Chapter.
CHAPTER II.
COUNTERS.
Let us agree that a Red Counter, placed within a Cell, shall mean “This Cell is occupied” (i.e. “There is at least one Thing in it”).
Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean “The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are.” Hence it may be understood to mean “At least one of these two Cells is occupied: possibly both are.”
Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be “sitting on the fence.” This phrase exactly describes the condition of the Red Counter.
Let us also agree that a Grey Counter, placed within a Cell, shall mean “This Cell is empty” (i.e. “There is nothing in it”).
[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]
CHAPTER III.
REPRESENTATION OF PROPOSITIONS.
§ 1.
Introductory.
Henceforwards, in stating such Propositions as “Some x-Things exist” or “No x-Things are y-Things”, I shall omit the word “Things”, which the Reader can supply for himself, and shall write them as “Some x exist” or “No x are y”.
[Note that the word “Things” is here used with a special meaning, as explained at p. 23.]
A Proposition, containing only one of the Letters used as Symbols for Attributes, is said to be ‘Uniliteral’.
[For example, “Some x exist”, “No y′ exist”, &c.]
A Proposition, containing two Letters, is said to be ‘Biliteral’.
[For example, “Some xy′ exist”, “No x′ are y”, &c.]
A Proposition is said to be ‘in terms of’ the Letters it contains, whether with or without accents.
[Thus, “Some xy′ exist”, “No x′ are y”, &c., are said to be in terms of x and y.]
§ 2.
Representation of Propositions of Existence.
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,