The Game of Logic by Lewis Carroll (funny books to read .txt) 📕

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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| 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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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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| 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 EXAUST 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

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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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| 1 | 0 |

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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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| 0 | 1 |

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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.)

This kind of Universal Proposition(where the Subject is a single Thing) is called an ‘INDIVIDUAL’ Proposition.

Now let us take “NICE Cakes” as the Subject of Proposition: that is, let us fix our thoughts on the LEFT-HAND half of the cupboard, where all the Cakes have attribute y, that is, “nice.”

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Suppose we find it marked like this:— | |

| 1 |

What would that tell us? | |

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I hope that it is not necessary, after explaining the HORIZONTAL oblong so fully, to spend much time over the UPRIGHT one. I hope you will see, for yourself, that this means “some y are x”, that is,

 

“Some nice Cakes are new.”

 

“But,” you will say, “we have had this case before. You put a red counter into No. 5, and you told us it meant ‘some new Cakes are nice’; and NOW you tell us that it means ‘some NICE Cakes are NEW’! Can it mean BOTH?”

The question is a very thoughtful one, and does you GREAT credit, dear Reader! It DOES mean both. If you choose to take x (that is, “new Cakes”) as your Subject, and to regard No. 5 as part of a HORIZONTAL oblong, you may read it “some x are y”, that is, “some new Cakes are nice”: but, if you choose to take y (that is, “nice Cake”) as your Subject, and to regard No. 5 as part of an UPRIGHT oblong, THEN you may read it “some y are x”, that is, “some nice Cakes are new”. They are merely two different ways of expressing the very same truth.

Without more words, I will simply set down the other ways in which this upright oblong might be marked, adding the meaning in each case. By comparing them with the various cases of the horizontal oblong, you will, I hope, be able to understand them clearly.

You will find it a good plan to examine yourself on this table, by covering up first one column and then the other, and ‘dodging about’, as the children say.

Also you will do well to write out for yourself two other tables—one for the LOWER half of the cupboard, and the other for its RIGHT-HAND half.

And now I think we have said all we need to say about the smaller Diagram, and may go on to the larger one.

 

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Symbols. | Meanings. _______________|_________________________________

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| | | Some y are x’;

| | | i.e. Some nice are not-new.

–— |

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| 1 | |

| | |

–— |

|

–— |

| | | No y are x;

| 0 | | i.e. No nice are new.

| | |

–— | [Observe that this is merely another way of

| | | expressing “No new are nice.”]

| | |

| | |

–— |

|

–— |

| | |

| | | No y are x’;

| | | i.e. No nice are not-new.

–— |

| | |

| 0 | |

| | |

–— |

|

–— |

| | |

| 1 | | Some y are x, and some are x’;

 

| | | i.e. Some nice are new, and some are

–— | not-new.

| | |

| 1 | |

| | |

–— |

|

–— |

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| 0 | | No y are x, and none are x’; i.e. No y

| | | exist;

–— | i.e. No Cakes are nice.

| | |

| 0 | |

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–— |

|

–— |

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| 1 | | All y are x;

| | | i.e. All nice are new.

–— |

| | |

| 0 | |

| | |

–— |

|

–— |

| | |

| 0 | | All y are x’;

| | | i.e. All nice are not-new.

–— |

| | |

| 1 | |

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–— | _______________|_________________________________

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