40. No kitten with green eyes will play with a gorilla.
41. All my friends dine at the lower table.
42. My writing-desk is full of live scorpions.
43. No Mandarin ever reads Hogg’s poems.
44. Shakespeare was clever.
45. Rainbows are not worth writing odes to.
46. These Sorites-examples are difficult.
47. All my dreams come true.
48. All the English pictures here are painted in oils.
49. Donkeys are not easy to swallow.
50. Opium-eaters never wear white kid gloves.
51. A good husband always comes home for his tea.
52. Bathing-machines are never made of mother-of-pearl.
53. Rainy days are always cloudy.
54. No heavy fish is unkind to children.
55. No engine-driver lives on barley-sugar.
56. All the animals in the yard gnaw bones.
57. No badger can guess a conundrum.
58. No cheque of yours, received by me, is payable to order.
59. I cannot read any of Brown’s letters.
60. I always avoid a kangaroo.
CHAPTER III.
SOLUTIONS.
§ 1.
Propositions of Relation reduced to normal form.
SL1Solutions for § 1.
1. The Univ. is “persons.” The Individual “I” may be regarded as a Class, of persons, whose peculiar Attribute is “represented by the Name ‘I’”, and may be called the Class of “I’s”. It is evident that this Class cannot possibly contain more than one Member: hence the Sign of Quantity is “all”. The verb “have been” may be replaced by the phrase “are persons who have been”. The Proposition may be written thus:—
“All”
Sign of Quantity.
“I’s”
Subject.
“are”
Copula.
“persons who have been out for a walk”
Predicate.
or, more briefly,
“All | I’s | are | persons who have been out for a walk”.
2. The Univ. and the Subject are the same as in Ex. 1. The Proposition may be written
“All | I’s | are | persons who feel better”.
3. Univ. is “persons”. The Subject is evidently the Class of persons from which John is excluded; i.e. it is the Class containing all persons who are not “John”.
The Sign of Quantity is “no”.
The verb “has read” may be replaced by the phrase “are persons who have read”.
The Proposition may be written
“No | persons who are not ‘John’ | are | persons who have read the letter”.
4. Univ. is “persons”. The Subject is evidently the Class of persons whose only two Members are “you and I”.
Hence the Sign of Quantity is “no”.
The Proposition may be written
“No | Members of the Class ‘you and I’ | are | old persons”.
5. Univ. is “creatures”. The verb “run well” may be replaced by the phrase “are creatures that run well”.
The Proposition may be written
“No | fat creatures | are | creatures that run well”.
6. Univ. is “persons”. The Subject is evidently the Class of persons who are not brave.
The verb “deserve” may be replaced by the phrase “are deserving of”.
The Proposition may be written
“No | not-brave persons | are | persons deserving of the fair”.
7. Univ. is “persons”. The phrase “looks poetical” evidently belongs to the Predicate; and the Subject is the Class, of persons, whose peculiar Attribute is “not-pale”.
The Proposition may be written
“No | not-pale persons | are | persons who look poetical”.
8. Univ. is “persons”.
The Proposition may be written
“Some | judges | are | persons who lose their tempers”.
9. Univ. is “persons”. The phrase “never neglect” is merely a stronger form of the phrase “am a person who does not neglect”.
The Proposition may be written
“All | ‘I’s’ | are | persons who do not neglect important business”.
10. Univ. is “things”. The phrase “what is difficult” (i.e. “that which is difficult”) is equivalent to the phrase “all difficult things”.
The Proposition may be written
“All | difficult things | are | things that need attention”.
11. Univ. is “things”. The phrase “what is unwholesome” may be interpreted as in Ex. 10.
The Proposition may be written
“All | unwholesome things | are | things that should be avoided”.
12. Univ. is “laws”. The Predicate is evidently a Class whose peculiar Attribute is “relating to excise”.
The Proposition may be written
“All | laws passed last week | are | laws relating to excise”.
13. Univ. is “things”. The Subject is evidently the Class, of studies, whose peculiar Attribute is “logical”; hence the Sign of Quantity is “all”.
The Proposition may be written
“All | logical studies | are | things that puzzle me”.
14. Univ. is “persons”. The Subject is evidently “persons in the house”.
The Proposition may be written
“No | persons in the house | are | Jews”.
15. Univ. is “dishes”. The phrase “if not well-cooked” is equivalent to the Attribute “not well-cooked”.
The Proposition may be written
“Some | not well-cooked dishes | are | unwholesome dishes”.
16. Univ. is “books”. The phrase “make one drowsy” may be replaced by the phrase “are books that make one drowsy”.
The Sign of Quantity is evidently “all”.
The Proposition may be written
“All | unexciting books | are | books that make one drowsy”.
17. Univ. is “men”. The Subject is evidently “a man who knows what he’s about”; and the word “when” shows that the Proposition is asserted of every such man, i.e. of all such men. The verb “can” may be replaced by “are men who can”.
The Proposition may be written
“All | men who know what they’re about | are | men who can detect a sharper”.
18. The Univ. and the Subject are the same as in Ex. 4.
The Proposition may be written
“All | Members of the Class ‘you and I’ | are | persons who know what they’re about”.
19. Univ. is “persons”. The verb “wear” may be replaced by the phrase “are accustomed to wear”.
The Proposition may be written
“Some | bald persons | are | persons accustomed to wear wigs”.
20. Univ. is “persons”. The phrase “never talk” is merely a stronger form of “are persons who do not talk”.
The Proposition may be written
“All | fully occupied persons | are | persons who do not talk about their grievances”.
21. Univ. is “riddles”. The phrase “if they can be solved” is equivalent to the Attribute “that can be solved”.
The Proposition may be written
“No | riddles that can be solved | are | riddles that interest me”.
§ 2.
Method of Diagrams.
SL4-ASolutions for § 4, Nos. 1–12.
1.
No m are x′;
All m′ are y.
∴ No x′ are y′.
2.
No m′ are x;
Some m′ are y′.
∴ Some x are y′.
3.
All m′ are x;
All m′ are y′.
∴ Some x are y′.
4.
No x′ are m′;
All y′ are m.
There is no Conclusion.
5.
Some m are x′;
No y are m.
∴ Some x′ are y′.
6.
No x′ are m;
No m are y.
There is no Conclusion.
7.
No m are x′;
Some y′ are m.
∴ Some x are y′.
8.
All m′ are x′;
No m′ are y.
∴ Some x′ are y′.
9.
Some x′ are m′;
No m are y′.
There is no Conclusion.
10.
All x are m;
All y′ are m′.
∴ All x are y;
All y′ are x′.
11.
No m are x;
All y′ are m′.
There is no Conclusion.
12.
No x are m;
All y are m.
∴ All y are x′.
SL5-ASolutions for § 5, Nos. 1–12.
1. I have been out for a walk;
I am feeling better.
Univ. is “persons”; m = the Class of I’s; x = persons who have been out for a walk; y = persons who are feeling better.
All m are x;
All m are y.
∴ Some x are y.
i.e. Somebody, who has been out for a walk, is feeling better.
2. No one has read the letter but John;
No one, who has not read it, knows what it is about.
Univ. is “persons”; m = persons who have read the letter; x = the Class of Johns; y = persons who know what the letter is about.
No x′ are m;
No m′ are y.
∴ No x′ are y.
i.e. No one, but John, knows what the letter is about.
3. Those who are not old like walking;
You and I are young.
Univ. is “persons”; m = old; x = persons who like walking; y = you and I.
All m′ are x;
All y are m′.
∴ All y are x.
i.e. You and I like walking.
4. Your course is always honest;
Your course is always the best policy.
Univ. is “courses”; m = your; x = honest; y = courses which are the best policy.
All m are x;
All m are y.
∴ Some x are y.
i.e. Honesty is sometimes the best policy.
5. No fat creatures run well;
Some greyhounds run well.
Univ. is “creatures”; m = creatures that run well; x = fat; y = greyhounds.
No x are m;
Some y are m.
∴ Some y are x′.
i.e. Some greyhounds are not fat.
6. Some, who deserve the fair, get their deserts;
None but the brave deserve the fair.
Univ. is “persons”; m = persons who deserve the fair; x = persons who get their deserts; y = brave.
Some m are x;
No y′ are m.
∴ Some y are x.
i.e. Some brave persons get their deserts.
7. Some Jews are rich;
All Esquimaux are Gentiles.
Univ. is “persons”; m = Jews; x = rich; y = Esquimaux.
Some m are x;
All y are m′.
∴ Some x are y′.
i.e. Some rich persons are not Esquimaux.
8. Sugar-plums are sweet;
Some sweet things are liked by children.
Univ. is “things”; m = sweet; x = sugar-plums; y = things that are liked by children.
All x are m;
Some m are y.
There is no Conclusion.
9. John is in the house;
Everybody in the house is ill.
Univ. is “persons”; m = persons in the house; x = the Class of Johns; y = ill.
All x are m;
All m are y.
∴ All x are y.
i.e. John is ill.
10. Umbrellas are useful on a journey;
What is useless on a journey should be left behind.
Univ. is “things”; m = useful on a journey; x = umbrellas; y = things that should be left behind.
All x are m;
All m′ are y.
∴ Some x′ are y.
i.e. Some things, that are not umbrellas, should be left behind on a journey.
11. Audible music causes vibration in the air;
Inaudible music is not worth paying for.
Univ. is “music”; m = audible; x = music that causes vibration in the air; y = worth paying for.
All m are x;
All m′ are y′.
∴ No x′ are y.
i.e. No music is worth paying for, unless it causes vibration in the air.
12. Some holidays are rainy;
Rainy days are tiresome.
Univ. is “days”; m = rainy; x = holidays; y = tiresome.
Some x are m;
All m are y.
∴ Some x are y.
i.e. Some holidays are tiresome.
SL6-ASolutions for § 6, Nos. 1–10.
1.
Some x are m; No m are y′. Some x are y.
Hence proposed Conclusion is right.
2.
All x are m; No y are m′. No y are x′.
There is no Conclusion.
3.
Some x are m′; All y′ are m. Some x are y.
Hence proposed Conclusion is right.
4.
All x are m; No y are m. All x are y′.
Hence proposed Conclusion is right.
5.
Some m′ are x′; No m′ are y. Some x′ are y′.
Hence proposed Conclusion is right.
6.
No x′ are m; All y are m′. All y are x.
There is no Conclusion.
7.
Some m′ are x′; All y′ are m′. Some x′ are y′.
There is no Conclusion.
8.
No m′ are x′; All y′ are m′. All y′ are x.
Hence proposed Conclusion is right.
9.
Some m are x′; No m are y. Some x′ are y′.
Hence proposed Conclusion is right.
10.
All m′ are x′; All m are y. Some y are x′.
Hence proposed Conclusion is right.
SL7-ASolutions for § 7, Nos. 1–6.
1.
No doctors are enthusiastic;
You are enthusiastic.
You are not a doctor.
Univ. “persons”; m = enthusiastic; x = doctors; y = you.
No x are m;
All y are m.
All y are x′.
∴ All y are x′.
Hence proposed Conclusion is right.
2.
All dictionaries are useful;
Useful books are valuable.
Dictionaries are valuable.
Univ. “books”; m = useful; x = dictionaries; y = valuable.
All x are m;
All m are y.
All x are y.
∴ All x are y.
Hence proposed Conclusion is right.
3.
No misers are unselfish;
None but misers save egg-shells.
No unselfish people save egg-shells.
Univ. “people”; m = misers; x = selfish; y = people who save egg-shells.
No m are x′;
No m′ are y.
No x′ are y.
∴ No x′ are y.
Hence proposed Conclusion is right.
4.
Some epicures are ungenerous;
All my uncles are generous.
My uncles are not epicures.
Univ. “persons”; m = generous; x = epicures; y = my uncles.
Some x are m′.
All y are m.
All y are x′.
∴ Some x are y′.
Hence proposed Conclusion is wrong, the right one being “Some epic
ures are not uncles of mine.”
5.
Gold is heavy;
Nothing but gold will silence him.
Nothing light will silence him.
Univ. “things”; m = gold; x = heavy; y = able to silence him.
All m are x;
No m′ are y.
No x′ are y.
∴ No x′ are y.
Hence proposed Conclusion is right.
6.
Some healthy people are fat;
No unhealthy people are strong.
Complete Works of Lewis Carroll Page 111
