Read Complete Works of Lewis Carroll Online
Authors: Lewis Carroll
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.
16.
No y' exist.
i.e.
Nobody is unhealthy.
4.
Smaller Diagram.
Propositions represented.
__________
———- ———-
| 1 | | | | |
1.
|—-|—-| 2.
|—-|—-|
| 0 | | | 1 | |
———- ———-
———- ———-
| | | | | 1 |
3.
|—-|—-| 4.
|—-|—-|
| | 0 | | | |
———- ———-
———- ———-
| | 1 | | | |
5.
|—-|—-| 6.
|—-|—-|
| | | | 0 | |
———- ———-
———- ———-
| | | | | |
7.
|—-|—-| 8.
|—-|—-|
| | 1 | | 0 | 1 |
———- ———-
———- ———-
| | | | | |
9.
|—-|-1-| 10.
|—-|—-|
| | | | 0 | 0 |
———- ———-