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 when, after marking them on the larger Diagram, you try to transfer the marks to the smaller. You will take its four compartments, one by one, and ask, for each in turn, "What mark can I place HERE?"; and in EVERY one the answer will be "No information!", showing that there is NO CONCLUSION AT ALL. For instance,
"All soldiers are brave;
Some Englishmen are brave.
&there4 Some Englishmen are soldiers."
looks uncommonly LIKE a Syllogism, and might easily take in a less experienced Logician. But YOU are not to be caught by such a trick! You would simply set out the Premisses, and would then calmly remark "Fallacious PREMISSES!": you wouldn't condescend to ask what CONCLUSION the writer professed to draw—knowing that, WHATEVER it is, it MUST be wrong. You would be just as safe as that wise mother was, who said "Mary, just go up to the nursery, and see what Baby's doing, AND TELL HIM NOT TO DO IT!"
The other kind of Fallacy—'Fallacious Conclusion'—you will not detect till you have marked BOTH Diagrams, and have read off the correct Conclusion, and have compared it with the Conclusion which the writer has drawn.
But mind, you mustn't say "FALLACIOUS Conclusion," simply because it is not IDENTICAL with the correct one: it may be a PART of the correct Conclusion, and so be quite correct, AS FAR AS IT GOES. In this case you would merely remark, with a pitying smile, "DEFECTIVE Conclusion!" Suppose, of example, you were to meet with this Syllogism:—
"All unselfish people are generous;
No misers are generous.
&there4 No misers are unselfish."
the Premisses of which might be thus expressed in letters:—
"All x' are m;
No y are m."
Here the correct Conclusion would be "All x' are y'" (that is, "All unselfish people are not misers"), while the Conclusion, drawn by the writer, is "No y are x'," (which is the same as "No x' are y," and so is PART of "All x' are y'.") Here you would simply say "DEFECTIVE Conclusion!" The same thing would happen, if you were in a confectioner's shop, and if a little boy were to come in, put down twopence, and march off triumphantly with a single penny-bun. You would shake your head mournfully, and would remark "Defective Conclusion! Poor little chap!" And perhaps you would ask the young lady behind the counter whether she would let YOU eat the bun, which the little boy had paid for and left behind him: and perhaps SHE would reply "Sha'n't!"
But if, in the above example, the writer had drawn the Conclusion "All misers are selfish" (that is, "All y are x"), this would be going BEYOND his legitimate rights (since it would assert the EXISTENCE of y, which is not contained in the Premisses), and you would very properly say "Fallacious Conclusion!"
Now, when you read other treatises on Logic, you will meet with
various kinds of (so-called) 'Fallacies' which are by no means
ALWAYS so. For example, if you were to put before one of these
Logicians the Pair of Premisses
"No honest men cheat;
No dishonest men are trustworthy."
and were to ask him what Conclusion followed, he would probably say "None at all! Your Premisses offend against TWO distinct Rules, and are as fallacious as they can well be!" Then suppose you were bold enough to say "The Conclusion is 'No men who cheat are trustworthy'," I fear your Logical friend would turn away hastily—perhaps angry, perhaps only scornful: in any case, the result would be unpleasant. I ADVISE YOU NOT TO TRY THE EXPERIMENT!
"But why is this?" you will say. "Do you mean to tell us that all these Logicians are wrong?" Far from it, dear Reader! From THEIR point of view, they are perfectly right. But they do not include, in their system, anything like ALL the possible forms of Syllogisms.
They have a sort of nervous dread of Attributes beginning with a negative particle. For example, such Propositions as "All not-x are y," "No x are not-y," are quite outside their system. And thus, having (from sheer nervousness) excluded a quantity of very useful forms, they have made rules which, though quite applicable to the few forms which they allow of, are no use at all when you consider all possible forms.
Let us not quarrel with them, dear Reader! There is room enough in the world for both of us. Let us quietly take our broader system: and, if they choose to shut their eyes to all these useful forms, and to say "They are not Syllogisms at all!" we can but stand aside, and let them Rush upon their Fate! There is scarcely anything of yours, upon which it is so dangerous to Rush, as your Fate. You may Rush upon your Potato-beds, or your Strawberry-beds, without doing much harm: you may even Rush upon your Balcony (unless it is a new house, built by contract, and with no clerk of the works) and may survive the foolhardy enterprise: but if you once Rush upon your FATE—why, you must take the consequences!
CHAPTER II.
CROSS QUESTIONS.
"The Man in the Wilderness asked of me
'How many strawberries grow in the sea?'"
__________
1. Elementary.
1. What is an 'Attribute'? Give examples.
2. When is it good sense to put "is" or "are" between two names? Give examples.
3. When is it NOT good sense? Give examples.
4. When it is NOT good sense, what is the simplest agreement to make, in order to make good sense?
5. Explain 'Proposition', 'Term', 'Subject', and 'Predicate'. Give examples.
6. What are 'Particular' and 'Universal' Propositions? Give examples.
7. Give a rule for knowing, when we look at the smaller Diagram, what Attributes belong to the things in each compartment.
8. What does "some" mean in Logic? [See pp. 55, 6]
9. In what sense do we use the word 'Universe' in this Game?
10. What is a 'Double' Proposition? Give examples.
11. When is a class of Things said to be 'exhaustively' divided? Give examples.
12. Explain the phrase "sitting on the fence."
13. What two partial Propositions make up, when taken together, "all x are y"?
14. What are 'Individual' Propositions? Give examples.
15. What kinds of Propositions imply, in this Game, the EXISTENCE of their Subjects?
16. When a Propo
sition contains more than two Attributes, these Attributes may in some cases be re-arranged, and shifted from one Term to the other. In what cases may this be done? Give examples.
__________
Break up each of the following into two partial
Propositions:
17. All tigers are fierce.
18. All hard-boiled eggs are unwholesome.
19. I am happy.
20. John is not at home.
__________
[See pp. 56, 7]
21. Give a rule for knowing, when we look at the larger Diagram, what Attributes belong to the Things contained in each compartment.
22. Explain 'Premisses', 'Conclusion', and 'Syllogism'. Give examples.
23. Explain the phrases 'Middle Term' and 'Middle Terms'.
24. In marking a pair of Premisses on the larger Diagram, why is it best to mark NEGATIVE Propositions before AFFIRMATIVE ones?
25. Why is it of no consequence to us, as Logicians, whether the Premisses are true or false?
26. How can we work Syllogisms in which we are told that "some x are y" is to be understood to mean "the Attribute x, y are COMPATIBLE", and "no x are y" to mean "the Attributes x, y are INCOMPATIBLE"?
27. What are the two kinds of 'Fallacies'?
28. How may we detect 'Fallacious Premisses'?
29. How may we detect a 'Fallacious Conclusion'?
30. Sometimes the Conclusion, offered to us, is not identical with the correct Conclusion, and yet cannot be fairly called 'Fallacious'. When does this happen? And what name may we give to such a Conclusion?
[See pp. 57-59]
2. Half of Smaller Diagram.
Propositions to be represented.
—————-
| | |
| x |
| | |
—y——-y'-
__________
1. Some x are not-y.
2. All x are not-y.
3. Some x are y, and some are not-y.
4. No x exist.
5. Some x exist.
6. No x are not-y.
7. Some x are not-y, and some x exist.
Complete Works of Lewis Carroll Page 119