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
—————-
| | |
| -1- |
| | |
—————-
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?
—————-
| | |
| 1 | 0 |
| | |
—————-
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
—————-
| | |
| 0 | 1 |
| | |
—————-
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."
——-
Suppose we find it marked like this:— | |
| 1 |
What would that tell us? | |
——-
| |
| |
| |
——-
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.
_________________________________________________ | Symbols. | Meanings. _______________|_________________________________ ——- | | | | | | | Some y are x'; | | | i.e. Some nice are not-new. ——- | | | | | 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 | | | | | ——- | | ——- | | | | | 0 | | No y are x, and none are x'; i.e. No y | | | exist; ——- | i.e. No Cakes are nice. | | | | 0 | | | | | ——- | | ——- | | | | | 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 | | | | | ——- | _______________|_________________________________
This may be taken to be a cupboard divided in the same way as the last, but ALSO divided into two portions, for the Attribute m. Let us give to m the meaning "wholesome": and let us suppose that all WHOLESOME Cakes are placed INSIDE the central Square, and all the UNWHOLESOME ones OUTSIDE it, that is, in one or other of the four queer-shaped OUTER compartments.
We see that, just as, in the smaller Diagram, the Cakes in each compartment had TWO Attributes, so, here, the Cakes in each compartment have THREE Attributes: and, just as the letters, representing the TWO Attributes, were written on the EDGES of the compartment, so, here, they are written at the CORNERS. (Observe that m' is supposed to be written at each of the four outer corners.) So that we can tell in a moment, by looking at a compartment, what three Attributes belong to the Things in it. For instance, take No. 12. Here we find x, y', m, at the corners: so we know that the Cakes in it, if there are any, have the triple Attribute, 'xy'm', that is, "new, not-nice, and wholesome." Again, take No. 16. Here we find, at the corners, x', y', m': so the Cakes in it are "not-new, not-nice, and unwholesome." (Remarkably untempting Cakes!)
It would take far too long to go through all the Propositions, containing x and y, x and m, and y and m which can be represented on this diagram (there are ninety-six altogether, so I am sure you will excuse me!) and I m
ust content myself with doing two or three, as specimens. You will do well to work out a lot more for yourself.
Taking the upper half by itself, so that our Subject is "new Cakes", how are we to represent "no new Cakes are wholesome"?
This is, writing letters for words, "no x are m." Now this tells us that none of the Cakes, belonging to the upper half of the cupboard, are to be found INSIDE the central Square: that is, the two compartments, No. 11 and No. 12, are EMPTY. And this, of course, is represented by
—————————- | | | | _____|_____ | | | | | | | | 0 | 0 | | | | | | | —————————-
And now how are we to represent the contradictory Proposition "SOME x are m"? This is a difficulty I have already considered. I think the best way is to place a red counter ON THE DIVISION-LINE between No. 11 and No. 12, and to understand this to mean that ONE of the two compartments is 'occupied,' but that we do not at present know WHICH. This I shall represent thus:—
—————————- | | | | _____|_____ | | | | | | | | -1- | | | | | | | —————————-
Now let us express "all x are m."
This consists, we know, of TWO Propositions,
"Some x are m,"
and "No x are m'."
Let us express the negative part first. This tells us that none of the Cakes, belonging to the upper half of the cupboard, are to be found OUTSIDE the central Square: that is, the two compartments, No. 9 and No. 10, are EMPTY. This, of course, is represented by
—————————- | 0 | 0 | | _____|_____ | | | | | | | | | | | | | | | | —————————-
But we have yet to represent "Some x are m." This tells us that there are SOME Cakes in the oblong consisting of No. 11 and No. 12: so we place our red counter, as in the previous example, on the division-line between No. 11 and No. 12, and the result is
—————————- | 0 | 0 | | _____|_____ | | | | | | | | -1- | | | | | | | —————————-
Now let us try one or two interpretations.
What are we to make of this, with regard to x and y?
—————————- | | 0 | | _____|_____ | | | | | | | | 1 | 0 | | | | | | | —————————-
This tells us, with regard to the xy'-Square, that it is wholly 'empty', since BOTH compartments are so marked. With regard to the xy-Square, it tells us that it is 'occupied'. True, it is only ONE compartment of it that is so marked; but that is quite enough, whether the other be 'occupied' or 'empty', to settle the fact that there is SOMETHING in the Square.
If, then, we transfer our marks to the smaller Diagram, so as to get rid of the m-subdivisions, we have a right to mark it
—————-
| | |
| 1 | 0 |
| | |
—————-
which means, you know, "all x are y."
The result would have been exactly the same, if the given oblong had been marked thus:—
—————————- | 1 | 0 | | _____|_____ | | | | | | | | | 0 | | | | | | | —————————-
Once more: how shall we interpret this, with regard to x and y?
—————————- | 0 | 1 | | _____|_____ | | | | | | | | | | | | | | | | —————————-
This tells us, as to the xy-Square, that ONE of its compartments is 'empty'. But this information is quite useless, as there is no mark in the OTHER compartment. If the other compartment happened to be 'empty' too, the Square would be 'empty': and, if it happened to be 'occupied', the Square would be 'occupied'. So, as we do not know WHICH is the case, we can say nothing about THIS Square.
The other Square, the xy'-Square, we know (as in the previous example) to be 'occupied'.
If, then, we transfer our marks to the smaller Diagram, we get merely this:—
—————-
| | |
| | 1 |
| | |
—————-
which means, you know, "some x are y'."
These principles may be applied to all the other
oblongs. For instance, to represent
"all y' are m'" we should mark the ———-
RIGHT-HAND UPRIGHT OBLONG (the one | |
that has the attribute y') thus:— |—- |
| 0 | |
|—-|-1-|
| 0 | |
|—- |
| |
———-
and, if we were told to interpret the lower half of the cupboard, marked as follows, with regard to x and y,
—————————- | | | | | | | | 0 | | | | | | | | ——-|——- | | 1 | 0 | —————————-
we should transfer it to the smaller Diagram thus,
—————-
| | |
| 1 | 0 |
| | |
—————-
and read it "all x' are y."
Two more remarks about Propositions need to be made.
One is that, in every Proposition beginning with "some" or "all", the ACTUAL EXISTENCE of the 'Subject' is asserted. If, for instance, I say "all misers are selfish," I mean that misers ACTUALLY EXIST. If I wished to avoid making this assertion, and merely to state the LAW that miserliness necessarily involves selfishness, I should say "no misers are unselfish" which does not assert that any misers exist at all, but merely that, if any DID exist, they WOULD be selfish.
The other is that, when a Proposition begins with "some" or "no", and contains more that two Attributes, these Attributes may be re-arranged, and shifted from one Term to the other, "ad libitum." For example, "some abc are def" may be re-arranged as "some bf are acde," each being equivalent to "some Things are abcdef". Again "No wise old men are rash and reckless gamblers" may be re-arranged as "No rash old gamblers are wise and reckless," each being equivalent to "No men are wise old rash reckless gamblers."
2. Syllogisms
Now suppose we divide our Universe of Things in three ways, with regard to three different Attributes. Out of these three Attributes, we may make up three different couples (for instance, if they were a, b, c, we might make up the three couples ab, ac, bc). Also suppose we have two Propositions given us, containing two of these three couples, and that from them we can prove a third Proposition containing the third couple. (For example, if we divide our Universe for m, x, and y; and if we have the two Propositions given us, "no m are x'" and "all m' are y", containing the two couples mx and my, it might be possible to prove from them a third Proposition, containing x and y.)
In such a case we call the given Propositions 'THE PREMISSES', the third one 'THE CONCLUSION' and the whole set 'A SYLLOGISM'.
Evidently, ONE of the Attributes must occur in both Premisses; or else one must occur in ONE Premiss, and its CONTRADICTORY in the other.
In the first case (when, for example, the Premisses are "some m are x" and "no m are y'") the Term, which occurs twice, is called 'THE MIDDLE TERM', because it serves as a sort of link between the other two Terms.
In the second case (when, for example, the Premisses are "no m are x'" and "all m' are y") the two Terms, which contain these contradictory Attributes, may be called 'THE MIDDLE TERMS'.
Thus, in the first case, the class of "m-Things" is the Middle Term; and, in the second case, the two classes of "m-Things" and "m'-Things" are the Middle Terms.
The Attribute, which occurs in the Middle Term or Terms, disappears in the Conclusion, and is said to be "eliminated", which literally means "turned out of doors".
Now let us try to draw a Conclusion from the two Premisses—
"Some new Cakes are unwholesome;
No nice Cakes are unwholesome."
In order to express them with counters, we need to divide Cakes in THREE different ways, with regard to newness, to niceness, and to wholesomeness. For this we must use the larger Diagram, making x mean "new", y "nice", and m "wholesome". (Everything INSIDE the central Square is supposed to have the attribute m, and everything OUTSIDE it
the attribute m', i.e. "not-m".)
You had better adopt the rule to make m mean the Attribute which occurs in the MIDDLE Term or Terms. (I have chosen m as the symbol, because 'middle' begins with 'm'.)
Now, in representing the two Premisses, I prefer to begin with the NEGATIVE one (the one beginning with "no"), because GREY counters can always be placed with CERTAINTY, and will then help to fix the position of the red counters, which are sometimes a little uncertain where they will be most welcome.
Let us express, the "no nice Cakes are unwholesome (Cakes)", i.e. "no y-Cakes are m'-(Cakes)". This tells us that none of the Cakes belonging to the y-half of the cupboard are in its m'-compartments (i.e. the ones outside the central Square). Hence the two compartments, No. 9 and No. 15, are both 'EMPTY'; and we must place a grey counter in EACH of them, thus:—
—————- |0 | | | —|— | | | | | | |—|——-|—| | | | | | | —|— | |0 | | —————-
We have now to express the other Premiss, namely, "some new Cakes are unwholesome (Cakes)", i.e. "some x-Cakes are m'-(Cakes)". This tells us that some of the Cakes in the x-half of the cupboard are in its m'-compartments. Hence ONE of the two compartments, No. 9 and No. 10, is 'occupied': and, as we are not told in WHICH of these two compartments to place the red counter, the usual rule would be to lay it on the division-line between them: but, in this case, the other Premiss has settled the matter for us, by declaring No. 9 to be EMPTY. Hence the red counter has no choice, and MUST go into No. 10, thus:—
—————- |0 | 1| | —|— | | | | | | |—|——-|—| | | | | | | —|— | |0 | | —————-
And now what counters will this information enable us to place in the SMALLER Diagram, so as to get some Proposition involving x and y only, leaving out m? Let us take its four compartments, one by one.
First, No. 5. All we know about THIS is that its OUTER portion
is empty: but we know nothing about its inner portion. Thus the
Square MAY be empty, or it MAY have something in it. Who can tell?
So we dare not place ANY counter in this Square.
Secondly, what of No. 6? Here we are a little better off. We know that there is SOMETHING in it, for there is a red counter in its outer portion. It is true we do not know whether its inner portion is empty or occupied: but what does THAT matter? One solitary Cake, in one corner of the Square, is quite sufficient excuse for saying "THIS SQUARE IS OCCUPIED", and for marking it with a red counter.
Complete Works of Lewis Carroll Page 118
