The Game of Logic by Lewis Carroll (best books under 200 pages .TXT) 📖

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 must 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 “mThings” is the Middle Term; and, in the second case, the two classes of “mThings” 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