'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 Proposition 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.
__________
Taking x="judges"; y="just";
8. No judges are just.
9. Some judges are unjust.
10. All judges are just.
__________
Taking x="plums"; y="wholesome";
11. Some plums are wholesome.
12. There are no wholesome plums.
13. Plums are some of them wholesome, and some not.
14. All plums are unwholesome.
[See pp. 59, 60]
- - -
| |
| x
| |
| - y - |
| |
| x'
| |
- - -
__________
Taking y="diligent students"; x="successful";
15. No diligent students are unsuccessful.
16. All diligent students are successful.
17. No students are diligent.
18. There are some diligent, but unsuccessful, students.
19. Some students are diligent.
[See pp. 60, 1]
3. Half of Smaller Diagram.
Symbols to be interpreted.
__________
- - - - - -
| | |
| x |
| | |
- y - - -y'-
__________
- - - - - - - -
| | | | | |
1. | | 0 | 2. | 0 | 0 |
| | | | | |
- - - - - - - -
- - - - - - - -
| | | | | |
3. | - | 4. | 0 | 1 |
| | | | | |
- - - - - - - -
__________
Taking x="good riddles"; y="hard";
- - - - - - - -
| | | | | |
5. | 1 | | 6. | 1 | 0 |
| | | | | |
- - - - - - - -
- - - - - - - -
| | | | | |
7. | 0 | 0 | 8. | 0 | |
| | | | | |
- - - - - - - -
__________
[See pp. 61, 2]
Taking x="lobster"; y="selfish";
- - - - - - - -
| | | | | |
9. | | 1 | 10. | 0 | |
| | | | | |
- - - - - - - -
- - - - - - - -
| | | | | |
11. | 0 | 1 | 12. | 1 | 1 |
| | | | | |
- - - - - - - -
__________
- - -
| |
x |
| |
| - y'-|
| |
x' |
| |
- - -
Taking y="healthy people"; x="happy";
- - - - - - - -
| 0 | | | | 1 | | 0 |
13. | - -| 14. |-1-| 15. | - -| 16. | - -|
| 1 | | | | 1 | | |
- - - - - - - -
[See p. 62]
4. Smaller Diagram.
Propositions to be represented.
- - - - - -
| | |
| x |
| - y - | - y'-|
| x' |
| | |
- - - - - -
__________
1. All y are x.
2. Some y are not-x.
3. No not-x are not-y.
4. Some x are not-y.
5. Some not-y are x.
6. No not-x are y.
7. Some not-x are not-y.
8. All not-x are not-y.
9. Some not-y exist.
10. No not-x exist.
11. Some y are x, and some are not-x.
12. All x are y, and all not-y are not-x.
[See pp. 62, 3]
Taking "nations" as Universe; x="civilised";
y="warlike";
13. No uncivilised nation is warlike.
14. All unwarlike nations are uncivilised.
15. Some nations are unwarlike.
16. All warlike nations are civilised, and all civilised nations
are warlike.
17. No nation is uncivilised.
__________
Taking "crocodiles" as Universe; x="hungry"; and
y="amiable";
18. All hungry crocodiles are unamiable.
19. No crocodiles are amiable when hungry.
20. Some crocodiles, when not hungry, are amiable; but some
are not.
21. No crocodiles are amiable, and some are hungry.
22. All crocodiles, when not hungry, are amiable; and all
unamiable crocodiles are hungry.
23. Some hungry crocodiles are amiable, and some that are
not hungry are unamiable.
[See pp. 63, 4]
5. Smaller Diagram.
Symbols to be interpreted.
__________
- - - - - -
| | |
| x |
| - y - | - y'-|
| x' |
| | |
- - - - - -
__________
- - - - - - - -
| | | | | |
1. | - -| - -| 2. | - -| - -|
| 1 | | | | 0 |
- - - - - - - -
- - - - - - - -
| | 1 | | | |
3. | - -| - -| 4. | - -| - -|
| | 0 | | 0 | 0 |
- - - - - - - -
__________
Taking "houses" as Universe; x="built of brick"; and
y="two-storied"; interpret
- - - - - - - -
| 0 | | | | |
5. | - -| - -| 6. | - -| - -|
| 0 | | | - |
- - - - - -| - -
- - - - - - - -
| | 0 | | | |
7. | - -| - -| 8. | - -| - -|
| | | | 0 | 1 |
- - - - - - - -
[See p. 65]
Taking "boys" as Universe; x="fat"; and y="active";
interpret
- - - - - - - -
| 1 | 1 | | | 0 |
9. | - -| - -| 10. | - -| - -|
| | | | | 1 |
- - - - - - - -
- - - - - - - -
| 0 | 1 | | 1 | |
11. | - -| - -| 12. | - -| - -|
| | 0 | | 0 | 1 |
- - - - - - - -
__________
Taking "cats" as Universe; x="green-eyed"; and
y="good-tempered"; interpret
- - - - - - - -
| 0 | 0 | | | 1 |
13. | - -| - -| 14. | - -| - -|
| | 0 | | 1 | |
- - - - - - - -
- - - - - - - -
| 1 | | | 0 | 1 |
15. | - -| - -| 16. | - -| - -|
| | 0 | | 1 | 0 |
- - - - - - - -
[See pp. 65, 6]
6. Larger Diagram.
Propositions to be represented.
__________
- - - - - -
| | |
| - x - |
| | | | |
| - y - m - y'-|
| | | | |
| - x'- |
| | |
- - - - - -
__________
1. No x are m.
2. Some y are m'.
3. All m are x'.
4. No m' are y'.
5. No m are x; All y are m.
6. Some x are m; No y are m.
7. All m are x'; No m are y.
8. No x' are m; No y' are m'.
[See pp. 67,8]
Taking "rabbits" as Universe; m="greedy"; x="old"; and
y="black"; represent
9. No old rabbits are greedy.
10. Some not-greedy rabbits are black.
11. All white rabbits are free from greediness.
12. All greedy rabbits are young.
13. No old rabbits are greedy; All black rabbits are greedy.
14. All rabbits, that are not greedy, are black; No old
rabbits are free from greediness.
__________
Taking "birds" as Universe; m="that sing loud"; x="well-fed";
and y="happy"; represent
15. All well-fed birds sing loud; No birds, that sing loud,
are unhappy.
16. All birds, that do not sing loud, are unhappy; No well-fed
birds fail to sing loud.
__________
Taking "persons" as Universe; m="in the house"; x="John";
and y="having a tooth-ache"; represent
17. John is in the house; Everybody in the house is suffering
from tooth-ache.
18. There is no one in the house but John; Nobody, out of
the house, has a tooth-ache.
__________
[See pp. 68-70]
Taking "persons" as Universe; m="I"; x="that has taken a
walk"; y="that feels better"; represent
19. I have been out for a walk; I feel much better.
__________
Choosing your own 'Universe' &c., represent
20. I sent him to bring me a kitten; He brought me a kettle
by mistake.
[See pp. 70, 1]
7. Both Diagrams to be employed.
__________
- - - - - -
| | | - - - - - -
| - x - | | | |
| | | | | | x |
| - y - m - y'-| | - y - | - y'-|
| | | | | | x' |
| - x'- | | | |
| | | - - - - - -
- - - - - -
__________
N.B. In each Question, a small Diagram should be drawn, for x and
y only, and marked in accordance with the given large Diagram: and
then as many Propositions as possible, for x and y, should be read
off from this small Diagram.
- - - - - - - - - - - -
|0 | | | | |
| - | - | | - | - |
| |0 | 0| | | |0 | 1| |
1. | - | - | - | - | 2. | - | - | - | - |
| |1 | | | | |0 | | |
| - | - | | - | - |
|0 | | | | |
- - - - - - - - - - - -
[See p. 72]
- - - - - - - - - - - -
| | | | | 0|
| - | - | | - | - |
| |0 | 0| | | | | | |
3. | - | - | - | - | 4. | - | - | - | - |
| |1 | 0| | | |0 | | |
| - | - | | - | - |
| | | | | 0|
- - - - - - - - - - - -
__________
Mark, in a large Diagram, the following pairs of Propositions from
the preceding Section: then mark a small Diagram in accordance with
it, &c.
5. No. 13. [see p. 49] 9. No. 17.
6. No. 14. 10. No. 18.
7. No. 15. 11. No. 19. [see p. 50]
8. No. 16. 12. No. 20.
__________
Mark, on a large Diagram, the following Pairs of Propositions: then
mark a small Diagram, &c. These are, in fact, Pairs of PREMISSES
for Syllogisms: and the results, read off from the small Diagram,
are the CONCLUSIONS.
13. No exciting books suit feverish patients; Unexciting
books make one drowsy.
14. Some, who deserve the fair, get their deserts; None
but the brave deserve the fair.
15. No children are patient; No impatient person can sit
still.
[See pp. 72-5]
16. All pigs are fat; No skeletons are fat.
17. No monkeys are soldiers; All monkeys are mischievous.
18. None of my cousins are just; No judges are unjust.
19. Some days are rainy; Rainy days are tiresome.
20. All medicine is nasty; Senna is a medicine.
21. Some Jews are rich; All Patagonians are Gentiles.
22. All teetotalers like sugar; No nightingale drinks wine.
23. No muffins are wholesome; All buns are unwholesome.
24. No fat creatures run well; Some greyhounds run well.
25. All soldiers march; Some youths are not soldiers.
26. Sugar is sweet; Salt is not sweet.
27. Some eggs are hard-boiled; No eggs are uncrackable.
28. There are no Jews in the house; There are no Gentiles
in the garden.
[See pp. 75-82]
29. All battles are noisy; What makes no noise may escape
notice.
30. No Jews are mad; All Rabbis are Jews.
31. There are no fish that cannot swim; Some skates are
fish.
32. All passionate people are unreasonable; Some orators
are passionate.
[See pp. 82-84]
CHAPTER III.
CROOKED ANSWERS.
"I answered him, as I thought good,
'As many as red-herrings grow in the wood'."
__________
1. Elementary.
1. Whatever can be "attributed to", that is "said to belong to",
a Thing, is called an 'Attribute'. For example, "baked", which
can (frequently) be attributed to "Buns", and "beautiful", which
can (seldom) be attributed to "Babies".
2. When they are the Names of two Things (for example, "these
Pigs are fat Animals"), or of two Attributes (for example, "pink
is light red").
3. When one is the Name of a Thing, and the other the Name of an
Attribute (for example, "these Pigs are pink"), since a Thing cannot
actually BE an Attribute.
4. That the Substantive shall be supposed to be repeated at the
end of the sentence (for example, "these Pigs are pink (Pigs)").
5. A 'Proposition' is a sentence stating that some, or none, or all,
of the Things belonging to a certain class, called the 'Subject',
are also Things belonging to a certain other class, called the
'Predicate'. For example, "some new Cakes are not nice", that is
(written in full) "some new Cakes are not nice Cakes"; where the
class "new Cakes" is the Subject, and the class "not-nice Cakes"
is the Predicate.
6. A Proposition, stating that SOME of the Things belonging to
its Subject are so-and-so, is called 'Particular'. For example,
"some new Cakes are nice", "some new Cakes are not nice."
A Proposition, stating that NONE of the Things belonging to its
Subject, or that ALL of them, are so-and-so, is called 'Universal'.
For example, "no new Cakes are nice", "all new Cakes are not nice".
7. The Things in each compartment possess TWO Attributes, whose
symbols will be found written on two of the EDGES of that compartment.
8. "One or more."
9. As a name of the class of Things to which the whole Diagram is
assigned.
10. A Proposition containing two statements. For example, "some
new Cakes are nice and some are not-nice."
11. When the whole class, thus divided, is "exhausted" among the
sets into which it is divided, there being no member of it which
does not belong to some one of them. For example, the class "new
Cakes" is "exhaustively" divided into "nice" and "not-nice" since
EVERY new Cake must be one or the other.
12. When a man cannot make up his mind which of two parties he
will join, he is said to be "sitting on the fence" - not being able
to decide on which side he will jump down.
13. "Some x are y" and "no x are y'".
14. A Proposition, whose Subject is a single Thing, is called
'Individual'. For example, "I am happy", "John is not at home".
These are Universal Propositions, being the same as "all the I's
that exist are happy", "ALL the Johns, that I am now considering,
are not at home".
15. Propositions beginning with "some" or "all".
16. When they begin with "some" or "no". For example, "some
abc are def" may be re-arranged as "some bf are acde", each being
equivalent to "some abcdef exist".
17. Some tigers are fierce, No tigers are not-fierce.
18. Some hard-boiled eggs are unwholesome, No hard-boiled
eggs are wholesome.
19. Some I's are happy, No I's are unhappy.
20. Some Johns are not at home, No Johns are at home.
21. The Things, in each compartment of the larger Diagram, possess
THREE Attributes, whose symbols will be found written at three of
the CORNERS of the compartment (except in the case of m', which is
not actually inserted in the Diagram, but is SUPPOSED to stand at
each of its four outer corners).
22. If the Universe of Things be divided with regard to three
different Attributes; and if two Propositions be given, containing
two different couples of these Attributes; and if from these we
can prove a third Proposition, containing the two Attributes that
have not yet occurred together; the given Propositions are called
'the Premisses', the third one 'the Conclusion', and the whole set
'a Syllogism'. For example, the Premisses might be "no m are x'"
and "all m' are y"; and it might be possible to prove from them
a Conclusion containing x and y.
23. If an Attribute occurs in both Premisses, the Term containing
it is called 'the Middle Term'. For example, if the Premisses are
"some m are x" and "no m are y'", the class of "m-Things" is 'the
Middle Term.'
If an Attribute occurs in one Premiss, and its contradictory in the
other, the Terms containing them may be called 'the Middle Terms'.
For example, if the Premisses are "no m are x'" and "all m' are
y", the two classes of "m-Things" and "m'-Things" may be called
'the Middle Terms'.
24. Because they can be marked with CERTAINTY: whereas AFFIRMATIVE
Propositions (that is, those that begin with "some" or "all")
sometimes require us to place a red counter 'sitting on a fence'.
25. Because the only question we are concerned with is whether the
Conclusion FOLLOWS LOGICALLY from the Premisses, so that, if THEY
were true, IT also would be true.
26. By understanding a red counter to mean "this compartment CAN
be occupied", and a grey one to mean "this compartment CANNOT be
occupied" or "this compartment MUST be empty".
27. 'Fallacious Premisses' and 'Fallacious Conclusion'.
28. By finding, when we try to transfer marks from the larger
Diagram to the smaller, that there is 'no information' for any of
its four compartments.
29. By finding the correct Conclusion, and then observing that
the Conclusion, offered to us, is neither identical with it nor a
part of it.
30. When the offered Conclusion is PART of the correct Conclusion.
In this case, we may call it a 'Defective Conclusion'.
2. Half of Smaller Diagram.
Propositions represented.
__________
- - - - - - - -
| | | | | |
1. | | 1 | 2. | 0 | 1 |
| | | | | |
- - - - - - - -
- - - - - - - -
| | | | | |
3. | 1 | 1 | 4. | 0 | 0 |
| | | | | |
- - - - - - - -
- - - - - - - -
| | | | | |
5. | 1 | 6. | | 0 |
| | | | | |
- - - - - - - -
- - - -
| | |
7. | 1 | 1 | It might be thought that the proper
| | |
- - - - - - - -
| | |
Diagram would be | 1 1 |, in order to express "some
| | |
- - - -
x exist": but this is really contained in "some x are y'."
To put a red counter on the division-line would only tell
us "ONE OF THE compartments is occupied", which we
know already, in knowing that ONE is occupied.
- - - -
| | |
8. No x are y. i.e. | 0 | |
| | |
- - - -
- - - -
| | |
9. Some x are y'. i.e. | | 1 |
| | |
- - - -
- - - -
| | |
10. All x are y. i.e. | 1 | 0 |
| | |
- - - -
- - - -
| | |
11. Some x are y. i.e. | 1 | |
| | |
- - - -
- - - -
| | |
12. No x are y. i.e. | 0 | |
| | |
- - - -
- - - -
| | |
13. Some x are y, and some are y'. i.e. | 1 | 1 |
| | |
- - - -
- - - -
| | |
14. All x are y'. i.e. | 0 | 1 |
| | |
- - - -
- -
| |
15. No y are x'. i.e. | - -|
| 0 |
- -
- -
| 1 |
16. All y are x. i.e. | - -|
| 0 |
- -
- -
| 0 |
17. No y exist. i.e. | - -|
| 0 |
- -
- -
| |
18. Some y are x'. i.e. | - -|
| 1 |
- -
- -
| |
15. Some y exist. i.e. |-1-|
| |
- -
3. Half of Smaller Diagram.
Symbols interpreted.
__________
1. No x are y'.
2. No x exist.
3. Some x exist.
4. All x are y'.
5. Some x are y. i.e. Some good riddles are hard.
6. All x are y. i.e. All good riddles are hard.
7. No x exist. i.e. No riddles are good.
8. No x are y. i.e. No good riddles are hard.
9. Some x are y'. i.e. Some lobsters are unselfish.
10. No x are y. i.e. No lobsters are selfish.
11. All x are y'. i.e. All lobsters are unselfish.
12. Some x are y, and some are y'. i.e. Some lobsters are
selfish, and some are unselfish.
13. All y' are x'. i.e. All invalids are unhappy.
14. Some y' exist. i.e. Some people are unhealthy.
15. Some y' are x, and some are x'. i.e. Some invalids are
happy, and some are unhappy.