The Game of Logic, Lewis Carroll [the best ebook reader for android txt] 📗

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.

As to No. 7, we are in the same condition as with No. 5—we find it PARTLY ‘empty’, but we do not know whether the other part is empty or occupied: so we dare not mark this Square.

And as to No. 8, we have simply no information at all.

The result is

––-

| | 1 |

|–|–|

| | |

––-

Our ‘Conclusion’, then, must be got out of the rather meager piece of information that there is a red counter in the xy’-Square. Hence our Conclusion is “some x are y’ “, i.e. “some new Cakes are not-nice (Cakes)”: or, if you prefer to take y’ as your Subject, “some not-nice Cakes are new (Cakes)”; but the other looks neatest.

We will now write out the whole Syllogism, putting the symbol &there4[*] for “therefore”, and omitting “Cakes”, for the sake of brevity, at the end of each Proposition.

[*][NOTE from Brett: The use of “&there4” is a rather arbitrary selection. There is no font available in general practice which renders the “therefore” symbol correction (three dots in a triangular formation). This can be done, however, in HTML, so if this document is read in a browser, then the symbol will be properly recognized. This is a poor man’s excuse.]

 

“Some new Cakes are unwholesome;

No nice Cakes are unwholesome

&there4 Some new Cakes are not-nice.”

And you have now worked out, successfully, your first ‘SYLLOGISM’. Permit me to congratulate you, and to express the hope that it is but the beginning of a long and glorious series of similar victories!

We will work out one other Syllogism—a rather harder one than the last—and then, I think, you may be safely left to play the Game by yourself, or (better) with any friend whom you can find, that is able and willing to take a share in the sport.

Let us see what we can make of the two Premisses—

 

“All Dragons are uncanny;

All Scotchmen are canny.”

 

Remember, I don’t guarantee the Premisses to be FACTS. In the first place, I never even saw a Dragon: and, in the second place, it isn’t of the slightest consequence to us, as LOGICIANS, whether our Premisses are true or false: all WE have to do is to make out whether they LEAD LOGICALLY TO THE CONCLUSION, so that, if THEY were true, IT would be true also.

You see, we must give up the “Cakes” now, or our cupboard will be of no use to us. We must take, as our ‘Universe’, some class of things which will include Dragons and Scotchmen: shall we say ‘Animals’? And, as “canny” is evidently the Attribute belonging to the ‘Middle Terms’, we will let m stand for “canny”, x for “Dragons”, and y for “Scotchmen”. So that our two Premisses are, in full,

 

“All Dragon-Animals are uncanny (Animals);

All Scotchman-Animals are canny (Animals).”

 

And these may be expressed, using letters for words, thus:—

 

“All x are m’;

All y are m.”

 

The first Premiss consists, as you already know, of two parts:—

 

“Some x are m’,”

and “No x are m.”

 

And the second also consists of two parts:—

 

“Some y are m,”

and “No y are m’.”

 

Let us take the negative portions first.

We have, then, to mark, on the larger Diagram, first, “no x are m”, and secondly, “no y are m’”. I think you will see, without further explanation, that the two results, separately, are

–––— –––—

| | | |0 | |

| —|— | | —|— |

| |0 | 0| | | | | | |

|—|—|—|—| |—|—|—|—|

| | | | | | | | | |

| —|— | | —|— |

| | | |0 | |

–––— –––—

and that these two, when combined, give us

–––—

|0 | |

| —|— |

| |0 | 0| |

|—|—|—|—|

| | | | |

| —|— |

|0 | |

–––—

We have now to mark the two positive portions, “some x are m’” and “some y are m”.

The only two compartments, available for Things which are xm’, are No. 9 and No. 10. Of these, No. 9 is already marked as ‘empty’; so our red counter must go into No. 10.

Similarly, the only two, available for ym, are No. 11 and No. 13. Of these, No. 11 is already marked as ‘empty’; so our red counter MUST go into No. 13.

The final result is

–––—

|0 | 1|

| —|— |

| |0 | 0| |

|—|—|—|—|

| |1 | | |

| —|— |

|0 | |

–––—

And now how much of this information can usefully be transferred to the smaller Diagram?

Let us take its four compartments, one by one.

As to No. 5? This, we see, is wholly ‘empty’. (So mark it with a grey counter.)

As to No. 6? This, we see, is ‘occupied’. (So mark it with a red counter.)

 

As to No. 7? Ditto, ditto.

As to No. 8? No information.

The smaller Diagram is now pretty liberally marked:—

––-

| 0 | 1 |

|–|–|

| 1 | |

––-

And now what Conclusion can we read off from this? Well, it is impossible to pack such abundant information into ONE Proposition: we shall have to indulge in TWO, this time.

First, by taking x as Subject, we get “all x are y’”, that is,

 

“All Dragons are not-Scotchmen”:

 

secondly, by taking y as Subject, we get “all y are x’”, that is,

 

“All Scotchmen are not-Dragons”.

 

Let us now write out, all together, our two Premisses and our brace of Conclusions.

 

“All Dragons are uncanny;

All Scotchmen are canny.

&there4 All Dragons are not-Scotchmen;

All Scotchmen are not-Dragons.”

 

Let me mention, in conclusion, that you may perhaps meet with logical treatises in which it is not assumed that any Thing EXISTS at all, by “some x are y” is understood to mean “the Attributes x, y are COMPATIBLE, so that a Thing can have both at once”, and “no x are y” to mean “the Attributes x, y are INCOMPATIBLE, so that nothing can have both at once”.

In such treatises, Propositions have quite different meanings from what they have in our ‘Game of Logic’, and it will be well to understand exactly what the difference is.

First take “some x are y”. Here WE understand “are” to mean “are, as an actual FACT”—which of course implies that some x-Things EXIST. But THEY (the writers of these other treatises) only understand “are” to mean “CAN be”, which does not at all imply that any EXIST. So they mean LESS than we do: our meaning includes theirs (for of course “some x ARE y” includes “some x CAN BE y”), but theirs does NOT include ours. For example, “some Welsh hippopotami are heavy” would be TRUE, according to these writers (since the Attributes “Welsh” and “heavy” are quite COMPATIBLE in a hippopotamus), but it would be FALSE in our Game (since there are no Welsh hippopotami to BE heavy).

Secondly, take “no x are y”. Here WE only understand “are” to mean “are, as an actual FACT”—which does not at all imply that no x CAN be y. But THEY understand the Proposition to mean, not only that none ARE y, but that none CAN POSSIBLY be y. So they mean more than we do: their meaning includes ours (for of course “no x CAN be y” includes “no x ARE y”), but ours does NOT include theirs. For example, “no Policemen are eight feet high” would be TRUE in our Game (since, as an actual fact, no such splendid specimens are ever found), but it would be FALSE, according to these writers (since the Attributes “belonging to the Police Force” and “eight feet high” are quite COMPATIBLE: there is nothing to PREVENT a Policeman from growing to that height, if sufficiently rubbed with Rowland’s Macassar Oil—which said to make HAIR grow, when rubbed on hair, and so of course will make a POLICEMAN grow, when rubbed on a Policeman).

Thirdly, take “all x are y”, which consists of the two partial Propositions “some x are y” and “no x are y’”. Here, of course, the treatises mean LESS than we do in the FIRST part, and more than we do in the SECOND. But the two operations don’t balance each other—any more than you can console a man, for having knocked down one of his chimneys, by giving him an extra door-step.

If you meet with Syllogisms of this kind, you may work them, quite easily, by the system I have given you: you have only to make ‘are’ mean ‘are CAPABLE of being’, and all will go smoothly. For “some x are y” will become “some x are capable of being y”, that is, “the Attributes x, y are COMPATIBLE”. And “no x are y” will become “no x are capable of being y”, that is, “the Attributes x, y are INCOMPATIBLE”. And, of course, “all x are y” will become “some x are capable of being y, and none are capable of being y’”, that is, “the Attributes x, y are COMPATIBLE, and the Attributes x, y’ are INCOMPATIBLE.” In using the Diagrams for this system, you must understand a red counter to mean “there may POSSIBLY be something in this compartment,” and a grey one to mean “there cannot POSSIBLY be anything in this compartment.”

 

3. Fallacies.

 

And so you think, do you, that the chief use of Logic, in real life, is to deduce Conclusions from workable Premisses, and to satisfy yourself that the Conclusions, deduced by other people, are correct? I only wish it were! Society would be much less liable to panics and other delusions, and POLITICAL life, especially, would be a totally different thing, if even a majority of the arguments, that scattered broadcast over the world, were correct! But it is all the other way, I fear. For ONE workable Pair of Premisses (I mean a Pair that lead to a logical Conclusion) that you meet with in reading your newspaper or magazine, you will probably find FIVE that lead to no Conclusion at all: and, even when the Premisses ARE workable, for ONE instance, where the writer draws a correct Conclusion, there are probably TEN where he draws an incorrect one.

In the first case, you may say “the PREMISSES are fallacious”: in the second, “the CONCLUSION is fallacious.”

 

The chief use you will find, in such Logical skill as this Game may teach you, will be in detecting ‘FALLACIES’ of these two kinds.

The first kind of Fallacy—‘Fallacious Premisses’—you will detect