Of course any other Things would have done just as well as Cakes.  We might make Propositions about “a Universe of Lizards”, or even “a Universe of Hornets”. (Wouldn’t that be a charming Universe to live in?)

So far, then, we have learned that

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|  1  |     |
|     |     |
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means “some x and y,” i.e. “some new are nice.”

I think you will see without further explanation, that

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|     |  1  |
|     |     |
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means “some x are y’,” i.e. “some new are not-nice.”

Now let us put a grey counter into No. 5, and ask ourselves the meaning of

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|  0  |     |
|     |     |
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This tells us that the x y-compartment is empty, which we may express by “no x are y”, or, “no new Cakes are nice”.  This is the second of the three Propositions at the head of this Section.

In the same way,

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|     |  0  |
|     |     |
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would mean “no x are y’,” or, “no new Cakes are not-nice.”

What would you make of this, I wonder?

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|  1  |  1  |
|     |     |
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I hope you will not have much trouble in making out that this represents a double Proposition:  namely, “some x are y, and some are y’,” i.e. “some new are nice, and some are not-nice.”

The following is a little harder, perhaps: 

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|  0  |  0  |
|     |     |
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This means “no x are y, and none are y’,” 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 x”, how would you do it?

This will puzzle you a little, I expect.  Evidently you must put a red counter somewhere in the 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.

What, then, are you to do?  I think the best way out of the difficulty is to place the red counter on the division-line between the xy-compartment and the xy’-compartment.  This I shall represent (as I always put ‘1’ where you are to put a red counter) by the diagram