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Our ingenious American cousins have invented a phrase to express the position of a man who wants to join one or the other of two parties—­such as their two parties ‘Democrats’ and ’Republicans’—­but can’t make up his mind which.  Such a man is said to be “sitting on the fence.”  Now that is exactly the position of the red counter you have just placed on the division-line.  He likes the look of No. 5, and he likes the look of No. 6, and he doesn’t know which to jump down into.  So there he sits astride, silly fellow, dangling his legs, one on each side of the fence!

Now I am going to give you a much harder one to make out.  What does this mean?

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This is clearly a double Proposition.  It tells us not only that “some x are y,” but also the “no x are not y.”  Hence the result is “All x are y,” i.e. “all new Cakes are nice”, which is the last of the three Propositions at the head of this Section.

We see, then, that the Universal Proposition

“All new Cakes are nice”

consists of two Propositions taken together, namely,

“Some new Cakes are nice,”
and “No new Cakes are not-nice.”

In the same way

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would mean “all x are y’ “, that is,

“All new Cakes are not-nice.”

Now what would you make of such a Proposition as “The Cake you have given me is nice”?  Is it Particular or Universal?

“Particular, of course,” you readily reply.  “One single Cake is hardly worth calling ‘some,’ even.”

No, my dear impulsive Reader, it is ‘Universal’.  Remember that, few as they are (and I grant you they couldn’t well be fewer), they are (or rather ‘it is’) all that you have given me!  Thus, if (leaving ‘red’ out of the question) I divide my Universe of Cakes into two classes—­the Cakes you have given me (to which I assign the upper half of the cupboard), and those you haven’t given me (which are to go below)—­I find the lower half fairly full, and the upper one as nearly as possible empty.  And then, when I am told to put an upright division into each half, keeping the nice Cakes to the left, and the not-nice ones to the right, I begin by carefully collecting all the Cakes you have given me (saying to myself, from time to time, “Generous creature!  How shall I ever repay such kindness?"), and piling them up in the left-hand compartment.  And it doesn’t take long to do it!

Here is another Universal Proposition for you.  “Barzillai Beckalegg is an honest man.”  That means “All the Barzillai Beckaleggs, that I am now considering, are honest men.” (You think I invented that name, now don’t you?  But I didn’t.  It’s on a carrier’s cart, somewhere down in Cornwall.)