# Page:Carroll - Game of Logic.djvu/24

8
[Ch. I.
NEW LAMPS FOR OLD.

What would you make of this, I wonder?

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I hope you will not have much trouble in making out that this represents a double Proposition: namely, "some ${\displaystyle x}$ are ${\displaystyle y}$, and some are ${\displaystyle y^{\prime }}$" i. e. "some new are nice, and some are not-nice."

The following is a little harder, perhaps:—

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This means "no ${\displaystyle x}$ are ${\displaystyle y}$, and none are ${\displaystyle y^{\prime }}$" i. e. "no new are nice, and none are not-nice": which leads to the rather curious result that "no new exist," i.e. "no Cakes are new." This is because "nice" and "not-nice" make what we call an 'exhaustive' division of the class "new Cakes": i. e. between them, they exhaust the whole class, so that all the new Cakes, that exist, must be found in one or the other of them.

And now suppose you had to represent, with counters, the contradictory to "no Cakes are new", which would be "some Cakes are new", or, putting letters for words, "some Cakes are ${\displaystyle x}$", how would you do it?

This will puzzle you a little, I expect. Evidently you must put a red counter somewhere in the ${\displaystyle x}$-half of the cupboard, since you know there are some new Cakes. But you must not put it into the left-hand compartment, since you do not know them to be nice: nor may you put it into the right-hand one, since you do not know them to be not-nice.