Now let us make another agreement--that a red counter in a compartment shall mean that it is 'OCCUPIED', that is, that there are SOME Cakes in it. (The word 'some,' in Logic, means 'one or more' so that a single Cake in a compartment would be quite enough reason for saying "there are SOME Cakes here"). Also let us agree that a grey counter in a compartment shall mean that it is 'EMPTY', that is that there are NO Cakes in it. In the following Diagrams, I shall put '1' (meaning 'one or more') where you are to put a RED counter, and '0' (meaning 'none') where you are to put a GREY one.
As the Subject of our Proposition is to be "new Cakes", we are only concerned, at present, with the UPPER half of the cupboard, where all the Cakes have the attribute x, that is, "new."
Now, fixing our attention on this upper half, suppose we found it marked like this,
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that is, with a red counter in No. 5. What would this tell us, with regard to the class of "new Cakes"?
Would it not tell us that there are SOME of them in the x y-compartment? That is, that some of them (besides having the Attribute x, which belongs to both compartments) have the Attribute y (that is, "nice"). This we might express by saying "some x-Cakes are y-(Cakes)", or, putting words instead of letters,
"Some new Cakes are nice (Cakes)",
or, in a shorter form,
"Some new Cakes are nice".
At last we have found out how to represent the first Proposition of this Section. If you have not CLEARLY understood all I have said, go no further, but read it over and over again, till you DO understand it. After that is once mastered, you will find all the rest quite easy.
It will save a little trouble, in doing the other Propositions, if we agree to leave out the word "Cakes" altogether. I find it convenient to call the whole class of Things, for which the cupboard is intended, the 'UNIVERSE.' Thus we might have begun this business by saying "Let us take a Universe of Cakes." (Sounds nice, doesn't it?)
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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means "some x and y," i.e. "some new are nice."
I think you will see without further explanation, that
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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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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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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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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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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?