9. Concl. wrong: right one is "Some tedious songs are not his."
10. Concl. right.
11. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
12. Concl. wrong: right one is "Some fierce creatures do not drink
coffee."
13. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
14. Concl. right.
15. Concl. wrong: right one is "Some shallow persons are not students."
16. No Concl. Fallacy of Like Eliminands not asserted to exist.
17. Concl. wrong: right one is "Some business, other than railways,
is unprofitable."
18. Concl. wrong: right one is "Some vain persons are not Professors."
19. Concl. right.
20. Concl. wrong: right one is "Wasps are not puppies."
21. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
22. No Concl. Same Fallacy.
23. Concl. right.
24. Concl. wrong: right one is "Some chocolate-creams are delicious."
25. No Concl. Fallacy of Like Eliminands not asserted to exist.
26. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
27. Concl. wrong: right one is "Some pillows are not pokers."
28. Concl. right.
29. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
30. No Concl. Fallacy of Like Eliminands not asserted to exist.
31. Concl. right.
32. No Concl. Fallacy of Like Eliminands not asserted to exist.
33. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
pg132
34. Concl. wrong: right one is "Some dreaded persons are not
begged to prolong their visits."
35. Concl. wrong: right one is "No man walks on neither."
36. Concl. right.
37. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
38. Concl. wrong: right one is "Some persons, dreaded by children,
are not emperors."
39. Concl. incomplete: the omitted portion is "Sugar is not salt."
40. Concl. right.
_Answers to Sec. 8._ AN8
1. a_{1}b_{0} + b_{1}a_{0}.
2. d_{1}a_{0}.
3. ac_{0}.
4. a_{1}d_{0}.
5. cd_{0}.
6. d_{1}c_{0}.
7. a'c_{0}.
8. c_{1}a'_{0}.
9. c'd_{0}.
10. b_{1}a_{0}.
11. d_{1}b_{0}.
12. a'd_{0}.
13. e_{1}b_{0}.
14. d_{1}e'_{0}.
15. e_{1}a'_{0}.
16. b'c_{0}.
17. a_{1}b_{0}.
18. d_{1}c_{0}.
19. a_{1}d_{0}.
20. ac_{0}.
21. de_{0}.
22. a_{1}b'_{0}.
23. h_{1}c_{0}.
24. e_{1}a_{0}.
25. e_{1}c'_{0}.
26. e_{1}c'_{0}.
27. hk'_{0}.
28. e_{1}d'_{0}.
29. l'a_{0}.
30. k_{1}b'_{0}.
_Answers to Sec. 9._ AN9
1. Babies cannot manage crocodiles.
2. _Your_ presents to me are not made of tin.
3. All my potatoes in this dish are old ones.
4. My servants never say "shpoonj."
5. My poultry are not officers.
6. None of _your_ sons are fit to serve on a jury.
7. No pencils of mine are sugar-plums.
8. Jenkins is inexperienced.
9. No comet has a curly tail.
10. No hedge-hog takes in the _Times_.
11. This dish is unwholesome.
12. My gardener is very old.
13. All humming-birds are small.
14. No one with a hooked nose ever fails to make money.
15. No gray ducks in this village wear lace collars.
16. No jug in this cupboard will hold water.
17. These apples were grown in the sun.
18. Puppies, that will not lie still, never care to do worsted work.
19. No name in this list is unmelodious.
20. No M.P. should ride in a donkey-race, unless he has perfect
self-command.
21. No goods in this shop, that are still on sale, may be carried away.
pg133
22. No acrobatic feat, which involves turning a quadruple somersault, is
ever attempted in a circus.
23. Guinea-pigs never really appreciate Beethoven.
24. No scentless flowers please me.
25. Showy talkers are not really well-informed.
26. None but red-haired boys learn Greek in this school.
27. Wedding-cake always disagrees with me.
28. Discussions, that go on while Tomkins is in the chair, endanger the
peacefulness of our Debating-Club.
29. All gluttons, who are children of mine, are unhealthy.
30. An egg of the Great Auk is not to be had for a song.
31. No books sold here have gilt edges, unless they are priced at 5s.
and upwards.
32. When you cut your finger, you will find Tincture of Calendula
useful.
33. _I_ have never come across a mermaid at sea.
34. All the romances in this library are well-written.
35. No bird in this aviary lives on mince-pies.
36. No plum-pudding, that has not been boiled in a cloth, can be
distinguished from soup.
37. All _your_ poems are uninteresting.
38. None of my peaches have been grown in a hot-house.
39. No pawnbroker is dishonest.
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.
pg134
CHAPTER III.
_SOLUTIONS._
Sec. 1.
_Propositions of Relation reduced to normal form._
_Solutions for Sec. 1._ SL1
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".
pg135
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".
pg136
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".
Sec. 2.
_Method of Diagrams._
_Solutions for Sec. 4, Nos. 1-12._ SL4-A
.---------------. .-------.
1. No m are x'; | | (O)| | | |
All m' are y. | .---|---. | |---|---|
| | | | | | |(O)|
|(I)|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. No x' are y'.
| | (O)|
.---------------.
pg137
.---------------. .-------.
2. No m' are x; |(O) | (O)| | | |
Some m' are y'. | .---|---. | |---|---|
| | | | | | |(I)|
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. Some x are y'.
| | (I)|
.---------------.
.---------------. .-------.
3. All m' are x; |(O) | (I)| | |(I)|
All m' are y'. | .---|---. | |---|---|
| | | | | | | |
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. Some x are y'.
|(O) | (O)|
.---------------.
.---------------.
4. No x' are m'; | | (O)|
All y' are m. | .---|---. |
| | | | |
|---|---|(I)|---| There is no Conclusion.
| | | | |
| .---|---. |
|(O) | (O)|
.---------------.
.---------------. .-------.
5. Some m are x'; | | | | | |
No y are m. | .---|---. | |---|---|
| |(O)| | | | |(I)|
|---|---|---|---| .-------.
| |(O)|(I)| |
| .---|---. | .'. Some x' are y'.
| | |
.---------------.
.---------------.
6. No x' are m; | | |
No m are y. | .---|---. |
| |(O)| | |
|---|---|---|---| There is no Conclusion.
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
.---------------. .-------.
7. No m are x'; | | | | |(I)|
Some y' are m. | .---|---. | |---|---|
| | |(I)| | | | |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. Some x are y'.
| | |
.---------------.
.---------------. .-------.
8. All m' are x'; |(O) | (O)| | | |
No m' are y. | .---|---. | |---|---|
| | | | | | |(I)|
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. Some x' are y'.
|(O) | (I)|
.---------------.
pg138
.---------------.
9. Some x' are m'; | | |
No m are y'. | .---|---. |
| | |(O)| |
|---|---|---|---| There is no Conclusion.
| | |(O)| |
| .---|---. |
| (I) |
.---------------.
.---------------. .-------.
10. All x are m; |(O) | (O)| |(I)|(O)|
All y' are m'. | .---|---. | |---|---|
| |(I)|(O)| | | |(I)|
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | .'. All x are y;
| | (I)| All y' are x'.
.---------------.
.---------------.
11. No m are x; | | |
All y' are m'. | .---|---. |
| |(O)|(O)| |
|---|---|---|(I)| There is no Conclusion.
| | |(O)| |
| .---|---. |
| | |
.---------------.
.---------------. .-------.
12. No x are m; |(O) | | |(O)| |
All y are m. | .---|---. | |---|---|
| |(O)|(O)| | |(I)| |
|---|---|---|---| .-------.
| |(I)| | |
| .---|---. | .'. All y are x'.
|(O) | |
.---------------.
_Solutions for Sec. 5, Nos. 1-12._ SL5-A
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; | | | |(I)| |
All m are y. | .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. Some x are y.
| | |
.---------------.
i.e. Somebody, who has been out for a walk, is feeling better.
pg139
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; |(O) | | | | |
No m' are y. | .---|---. | |---|---|
| | | | | |(O)| |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. No x' are y.
|(O) | |
.---------------.
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; |(I) | | |(I)| |
All y are m'. | .---|---. | |---|---|
| |(O)| | | |(O)| |
|---|---|---|---| .-------.
| |(O)| | |
| .---|---. | .'. All y are x.
|(O) | (O)|
.---------------.
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; | | | |(I)| |
All m are y. | .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. 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. | .---|---. | |---|---|
| |(O)|(O)| | |(I)| |
|---|---|---|---| .-------.
| |(I)| | |
| .---|---. | .'. Some y are x'.
| | |
.---------------.
i.e. Some greyhounds are not fat.
pg140
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; | | | |(I)| |
No y' are m. | .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | .'. 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; | | | | |(I)|
All y are m'. | .---|---. | |---|---|
| |(O)|(I)| | | | |
|(I)|---|---|---| .-------.
| |(O)| | |
| .---|---. | .'. 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; |(O) | (O)|
Some m are y. | .---|---. |
| | (I) | |
|---|(I)|---|---|
| | | | |
| .---|---. |
| | |
.---------------.
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; |(O) | (O)| |(I)|(O)|
All m are y. | .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | .'. All x are y.
| | |
.---------------.
i.e. John is ill.
pg141
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; |(O) | (O)| | | |
All m' are y. | .---|---. | |---|---|
| | (I) | | |(I)| |
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. Some x' are y.
|(I) | (O)|
.---------------.
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; |(O) | | | | |
All m' are y'. | .---|---. | |---|---|
| | (I) | | |(O)| |
|---|---|---|(I)| .-------.
| |(O)|(O)| |
| .---|---. | .'. No x' are y.
|(O) | |
.---------------.
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; | | | |(I)| |
All m are y. | .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | .'. Some x are y.
| | |
.---------------.
i.e. Some holidays are tiresome.
_Solutions for Sec. 6, Nos. 1-10._ SL6-A
1.
Some x are m; No m are y'. Some x are y.
.---------------. .-------.
| | | |(I)| |
| .---|---. | |---|---|
| |(I)|(O)| | | | |
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | Hence proposed Conclusion is right.
| | |
.---------------.
pg142
2.
All x are m; No y are m'. No y are x'.
.---------------.
|(O) | (O)|
| .---|---. |
| | (I) | |
|---|---|---|---| There is no Conclusion.
| | | | |
| .---|---. |
|(O) | |
.---------------.
3.
Some x are m'; All y' are m. Some x are y.
.---------------. .-------.
|(I) | (O)| |(I)| |
| .---|---. | |---|---|
| | | | | | | |
|---|---|(I)|---| .-------.
| | | | |
| .---|---. | Hence proposed Conclusion is right.
| | (O)|
.---------------.
4.
All x are m; No y are m. All x are y'.
.---------------. .-------.
|(O) | (O)| |(O)|(I)|
| .---|---. | |---|---|
| |(O)|(I)| | | | |
|---|---|---|---| .-------.
| |(O)| | |
| .---|---. | Hence proposed Conclusion is right.
| | |
.---------------.
5.
Some m' are x'; No m' are y. Some x' are y'.
.---------------. .-------.
|(O) | | | | |
| .---|---. | |---|---|
| | | | | | |(I)|
|---|---|---|---| .-------.
| | | | |
| .---|---. | Hence proposed Conclusion is right.
|(O) | (I)|
.---------------.
6.
No x' are m; All y are m'. All y are x.
.---------------.
| | |
| .---|---. |
| |(O)| | |
|(I)|---|---|---| There is no Conclusion.
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
pg143
7.
Some m' are x'; All y' are m'. Some x' are y'.
.---------------.
| | |
| .---|---. |
| | |(O)| |
|---|---|---|(I)| There is no Conclusion.
| | |(O)| |
| .---|---. |
| (I) |
.---------------.
8.
No m' are x'; All y' are m'. All y' are x.
.---------------. .-------.
| | (I)| | |(I)|
| .---|---. | |---|---|
| | |(O)| | | |(O)|
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | Hence proposed Conclusion is right.
|(O) | (O)|
.---------------.
9.
Some m are x'; No m are y. Some x' are y'.
.---------------. .-------.
| | | | | |
| .---|---. | |---|---|
| |(O)| | | | |(I)|
|---|---|---|---| .-------.
| |(O)|(I)| |
| .---|---. | Hence proposed Conclusion is right.
| | |
.---------------.
10.
All m' are x'; All m are y. Some y are x'.
.---------------. .-------.
|(O) | (O)| | | |
| .---|---. | |---|---|
| | | | | |(I)| |
|---|---|---|---| .-------.
| | | | |
| .---|---. | Hence proposed Conclusion is right.
|(I) | (O)|
.---------------.
pg144
_Solutions for Sec. 7, Nos. 1-6._ SL7-A
1.
No doctors are enthusiastic;
You are enthusiastic.
You are not a doctor.
Univ. "persons"; m = enthusiastic; x = doctors; y = you.
.---------------. .-------.
|(O) | | |(O)| |
| .---|---. | |---|---|
No x are m; | |(O)|(O)| | |(I)| |
All y are m. |---|---|---|---| .-------.
All y are x'. | |(I)| | |
| .---|---. | .'. All y are x'.
|(O) | |
.---------------.
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.
.---------------. .-------.
|(O) | (O)| |(I)|(O)|
| .---|---. | |---|---|
All x are m; | |(I)|(O)| | | | |
All m are y. |---|---|---|---| .-------.
All x are y. | | |(O)| |
| .---|---. | .'. 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.
.---------------. .-------.
|(O) | | | | |
| .---|---. | |---|---|
No m are x'; | | | | | |(O)| |
No m' are y. |---|---|---|---| .-------.
No x' are y. | |(O)|(O)| |
| .---|---. | .'. No x' are y.
|(O) | |
.---------------.
Hence proposed Conclusion is right.
pg145
4.
Some epicures are ungenerous;
All my uncles are generous.
My uncles are not epicures.
Univ. "persons"; m = generous; x = epicures; y = my uncles.
.---------------. .-------.
|(O) | (I)| | |(I)|
| .---|---. | |---|---|
Some x are m'. | | | | | | | |
All y are m. |---|(I)|---|---| .-------.
All y are x'. | | | | |
| .---|---. | .'. Some x are y'.
|(O) | |
.---------------.
Hence proposed Conclusion is wrong, the right one being "Some epicures
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.
.---------------. .-------.
|(O) | | | | |
| .---|---. | |---|---|
All m are x; | | (I) | | |(O)| |
No m' are y. |---|---|---|---| .-------.
No x' are y. | |(O)|(O)| |
| .---|---. | .'. No x' are y.
|(O) | |
.---------------.
Hence proposed Conclusion is right.
6.
Some healthy people are fat;
No unhealthy people are strong.
Some fat people are not strong.
Univ. "persons"; m = healthy; x = fat; y = strong.
.---------------.
|(O) | |
| .---|---. |
Some m are x; | | (I) | |
No m' are y. |---|---|---|---| There is no Conclusion.
Some x are y'.| | | | |
| .---|---. |
|(O) | |
.---------------.
pg146
Sec. 3.
_Method of Subscripts._
_Solutions for Sec. 4._ SL4-B
1. mx'_{0} + m'_{1}y'_{0} > x'y'_{0} [Fig. I.
i.e. "No x' are y'."
2. m'x_{0} + m'y'_{1} > x'y'_{1} [Fig. II.
i.e. "Some x' are y'."
3. m'_{1}x'_{0} + m'_{1}y_{0} > xy'_{1} [Fig. III.
i.e. "Some x are y'."
4. x'm'_{0} + y'_{1}m'_{0} > nothing.
[Fallacy of Like Eliminands
not asserted to exist.]
5. mx'_{1} + ym_{0} > x'y'_{1} [Fig. II.
i.e. "Some x' are y'."
6. x'm_{0} + my_{0} > nothing.
[Fallacy of Like Eliminands
not asserted to exist.]
7. mx'_{0} + y'm_{1} > xy'_{1} [Fig. II.
i.e. "Some x are y'."
8. m'_{1}x_{0} + m'y_{0} > x'y'_{1} [Fig. III.
i.e. "Some x' are y'."
9. x'm'_{1} + my_{0} > nothing.
[Fallacy of Unlike Eliminands
with an Entity-Premiss.]
10. x_{1}m'_{0} + y'_{1}m_{0} > x_{1}y'_{0} + y'_{1}x_{0} [Fig. I (b).
i.e. "All x are y, and all y' are x'."
11. mx_{0} + y'_{1}m_{0} > nothing.1
[Fallacy of Like Eliminands
not asserted to exist.]
12. xm_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
i.e. "All y are x'."
13. m'_{1}x'_{0} + ym_{0} > x'y_{0} [Fig. I.
i.e. "No x' are y."
14. m_{1}x'_{0} + m'_{1}y'_{0} > x'y'_{0} [Fig. I.
i.e. "No x' are y'."
15. xm_{0} + m'y_{0} > xy_{0} [Fig. I.
i.e. "No x are y."
16. x_{1}m_{0} + y_{1}m'_{0} > (x_{1}y_{0} + y_{1}x_{0}) [Fig. I (b).
i.e. "All x are y' and all y are x'."
17. xm_{0} + m'_{1}y'_{0} > xy'_{0} [Fig. I.
i.e. "No x are y'."
18. xm'_{0} + my_{0} > xy_{0} [Fig. I.
i.e. "No x are y."
19. m_{1}x'_{0} + m_{1}y_{0} > xy'_{1} [Fig. III.
i.e. "Some x are y'."
20. mx_{0} + m'_{1}y'_{0} > xy'_{0} [Fig. I.
i.e. "No x are y'."
21. x_{1}m'_{0} + m'y_{1} > x'y_{1} [Fig. II.
i.e. "Some x' are y."
22. xm_{1} + y_{1}m'_{0} > nothing.
[Fallacy of Unlike Eliminands
with an Entity-Premiss.]
23. m_{1}x'_{0} + ym_{1} > xy_{1} [Fig. II.
i.e. "Some x are y."
24. xm_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
i.e. "All y are x'."
25. mx'_{1} + my'_{0} > x'y_{1} [Fig. II.
i.e. "Some x' are y."
26. mx'_{0} + y_{1}m'_{0} > y_{1}x'_{0} [Fig. I (a).
i.e. "All y are x."
27. x_{1}m_{0} + y'_{1}m'_{0} > (x_{1}y'_{0} + y'_{1}x_{0}) [Fig. I (b).
i.e. "All x are y, and all y' are x'."
28. m_{1}x_{0} + my_{1} > x'y_{1} [Fig. II.
i.e. "Some x' are y."
29. mx_{0} + y_{1}m_{0} > nothing.
[Fallacy of Like Eliminands
not asserted to exist.]
30. x_{1}m_{0} + ym_{1} > x'y_{1} [Fig. II.
i.e. "Some y are x'."
31. x_{1}m'_{0} + y_{1}m'_{0} > nothing.
[Fallacy of Like Eliminands
not asserted to exist.]
pg147
32. xm'_{0} + m_{1}y'_{0} > xy'_{0} [Fig. I.
i.e. "No x are y'."
33. mx_{0} + my_{0} > nothing.
[Fallacy of Like Eliminands
not asserted to exist.]
34. mx'_{0} + ym_{1} > xy_{1} [Fig. II.
i.e. "Some x are y."
35. mx_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
i.e. "All y are x'."
36. m_{1}x_{0} + ym_{1} > x'y_{1} [Fig. II.
i.e. "Some x' are y."
37. m_{1}x'_{0} + ym_{0} > xy'_{1} [Fig. III.
i.e. "Some x are y'."
38. mx_{0} + m'y_{0} > xy_{0} [Fig. I.
i.e. "No x are y."
39. mx'_{1} + my_{0} > x'y'_{1} [Fig. II.
i.e. "Some x' are y'."
40. x'm_{0} + y'_{1}m'_{0} > y'_{1}x'_{0} [Fig. I (a).
i.e. "All y' are x."
41. x_{1}m_{0} + ym'_{0} > x_{1}y_{0} [Fig. I (a).
i.e. "All x are y'."
42. m'x_{0} + ym_{0} > xy_{0} [Fig. I.
i.e. "No x are y."
_Solutions for Sec. 5, Nos. 13-24._ SL5-B
13. No Frenchmen like plumpudding;
All Englishmen like plumpudding.
Univ. "men"; m = liking plumpudding; x = French; y = English.
xm_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
i.e. Englishmen are not Frenchmen.
14. No portrait of a lady, that makes her simper or scowl, is
satisfactory;
No photograph of a lady ever fails to make her simper or scowl.
Univ. "portraits of ladies"; m = making the subject simper or scowl;
x = satisfactory; y = photographic.
mx_{0} + ym'_{0} > xy_{0} [Fig. I.
i.e. No photograph of a lady is satisfactory.
15. All pale people are phlegmatic;
No one looks poetical unless he is pale.
Univ. "people"; m = pale; x = phlegmatic; y = looking poetical.
m_{1}x'_{0} + m'y_{0} > x'y_{0} [Fig. I.
i.e. No one looks poetical unless he is phlegmatic.
16. No old misers are cheerful;
Some old misers are thin.
Univ. "persons"; m = old misers; x = cheerful; y = thin.
mx_{0} + my_{1} > x'y_{1} [Fig. II.
i.e. Some thin persons are not cheerful.
17. No one, who exercises self-control, fails to keep his temper;
Some judges lose their tempers.
Univ. "persons"; m = keeping their tempers; x = exercising self-control;
y = judges.
xm'_{0} + ym'_{1} > x'y_{1} [Fig. II.
i.e. Some judges do not exercise self-control.
pg148
18. All pigs are fat;
Nothing that is fed on barley-water is fat.
Univ. is "things"; m = fat; x = pigs; y = fed on barley-water.
x_{1}m'_{0} + ym_{0} > x_{1}y_{0} [Fig. I (a).
i.e. Pigs are not fed on barley-water.
19. All rabbits, that are not greedy, are black;
No old rabbits are free from greediness.
Univ. is "rabbits"; m = greedy; x = black; y = old.
m'_{1}x'_{0} + ym'_{0} > xy'_{1} [Fig. III.
i.e. Some black rabbits are not old.
20. Some pictures are not first attempts;
No first attempts are really good.
Univ. is "things"; m = first attempts; x = pictures; y = really good.
xm'_{1} + my_{0} > nothing.
[Fallacy of Unlike Eliminands with an Entity-Premiss.]
21. I never neglect important business;
Your business is unimportant.
Univ. is "business"; m = important; x = neglected by me; y = your.
mx_{0} + y_{1}m_{0} > nothing.
[Fallacy of Like Eliminands not asserted to exist.]
22. Some lessons are difficult;
What is difficult needs attention.
Univ. is "things"; m = difficult; x = lessons; y = needing attention.
xm_{1} + m_{1}y'_{0} > xy_{1} [Fig. II.
i.e. Some lessons need attention.
23. All clever people are popular;
All obliging people are popular.
Univ. is "people"; m = popular; x = clever; y = obliging.
x_{1}m'_{0} + y_{1}m'_{0} > nothing.
[Fallacy of Like Eliminands not asserted to exist.]
24. Thoughtless people do mischief;
No thoughtful person forgets a promise.
Univ. is "persons"; m = thoughtful; x = mischievous; y = forgetful of
promises.
m'_{1}x'_{0} + my_{0} > x'y_{0}
i.e. No one, who forgets a promise, fails to do mischief.
_Solutions for Sec. 6._ SL6-B
1. xm_{1} + my'_{0} > xy_{1} [Fig. II.] Concl. right.
2. x_{1}m'_{0} + ym'_{0} Fallacy of Like Eliminands not asserted to exist.
3. xm'_{1} + y'_{1}m'_{0} > xy_{1} [Fig. II.] Concl. right.
pg149
4. x_{1}m'_{0} + ym_{0} > x_{1}y_{0} [Fig. I (a).] Concl. right.
5. m'x'_{1} + m'y_{0} > x'y'_{1} [Fig. II.] "
6. x'm_{0} + y_{1}m_{0} Fallacy of Like Eliminands not asserted to exist.
7. m'x'_{1} + y'_{1}m_{0} Fallacy of Unlike Eliminands with an
Entity-Premiss.
8. m'x'_{0} + y'_{1}m_{0} > y'_{1}x'_{0} [Fig. I (a).] Concl. right.
9. mx'_{1} + my_{0} > x'y'_{1} [Fig. II.] "
10. m'_{1}x_{0} + m'_{1}y'_{0} > x'y_{1} [Fig. III.] "
11. x_{1}m_{0} + ym_{1} > x'y_{1} [Fig. II.] "
12. xm_{0} + m'y'_{0} > xy'_{0} [Fig. I.] "
13. xm_{0} + y'_{1}m'_{0} > y'_{1}x_{0} [Fig. I (a).] "
14. m'_{1}x_{0} + m'_{1}y'_{0} > x'y_{1} [Fig. III.] "
15. mx'_{1} + y_{1}m_{0} > x'y'_{1} [Fig. II.] "
16. x'm_{0} + y'_{1}m_{0} Fallacy of Like Eliminands not asserted to exist.
17. m'x_{0} + m'_{1}y_{0} > x'y'_{1} [Fig. III.] Concl. right.
18. x'm_{0} + my_{1} > xy_{1} [Fig. II.] "
19. mx'_{1} + m_{1}y'_{0} > x'y_{1} [ " ] "
20. x'm'_{0} + m'y'_{1} > xy'_{1} [ " ] "
21. mx_{0} + m_{1}y_{0} > x'y'_{1} [Fig. III.] "
22. x'_{1}m'_{0} + ym'_{1} > xy_{1} [Fig. II.] Concl. wrong:
the right one is "Some x are y."
23. m_{1}x'_{0} + m'y'_{0} > x'y'_{0} [Fig. I.] Concl. right.
24. x_{1}m_{0} + m'_{1}y'_{0} > x_{1}y'_{0} [Fig. I (a).] "
25. xm'_{0} + m_{1}y'_{0} > xy'_{0} [Fig. I.] "
26. m_{1}x_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).] "
27. x_{1}m'_{0} + my'_{0} > x_{1}y'_{0} [ " ] "
28. x_{1}m'_{0} + y'm'_{0} Fallacy of Like Eliminands not asserted to
exist.
29. x'm_{0} + m'y'_{0} > x'y'_{0} [Fig. I.] Concl. right.
30. x_{1}m'_{0} + m_{1}y_{0} > x_{1}y_{0} [Fig. I (a).] "
31. x'_{1}m_{0} + y'm'_{0} > x'_{1}y'_{0} [ " ] "
32. xm_{0} + y'm'_{0} > xy'_{0} [Fig. I.] "
33. m_{1}x_{0} + y'_{1}m'_{0} > y'_{1}x_{0} [Fig. I (a).] "
34. x_{1}m_{0} + ym'_{1} Fallacy of Unlike Eliminands with an
Entity-Premiss.
35. xm_{1} + m_{1}y'_{0} > xy_{1} [Fig. II.] Concl. right.
36. m_{1}x_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).] "
37. mx'_{0} + m_{1}y_{0} > xy'_{1} [Fig. III.] "
38. xm_{0} + my'_{0} Fallacy of Like Eliminands not asserted to exist.
39. mx_{0} + my'_{1} > x'y'_{1} [Fig. II.] Concl. right.
40. mx'_{0} + ym_{1} > xy_{1} [Fig. II.] "
pg150
_Solutions for Sec. 7._ SL7-B
1. No doctors are enthusiastic;
You are enthusiastic.
You are not a doctor.
Univ. "persons"; m = enthusiastic; x = doctors; y = you.
xm_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
Conclusion right.
2. Dictionaries are useful;
Useful books are valuable.
Dictionaries are valuable.
Univ. "books"; m = useful; x = dictionaries; y = valuable.
x_{1}m'_{0} + m_{1}y'_{0} > x_{1}y'_{0} [Fig. I (a).
Conclusion 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.
mx'_{0} + m'y_{0} > x'y_{0} [Fig. I.
Conclusion 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.
xm'_{1} + y_{1}m'_{0} > xy'_{1} [Fig. II.
Conclusion wrong: right one is "Some epicures 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.
m_{1}x'_{0} + m'y_{0} > x'y_{0} [Fig. I.
Conclusion right.
6. Some healthy people are fat;
No unhealthy people are strong.
Some fat people are not strong.
Univ. "people"; m = healthy; x = fat; y = strong.
mx_{1} + m'y_{0}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
7. I saw it in a newspaper;
All newspapers tell lies.
It was a lie.
Univ. "publications"; m = newspapers; x = publications in which I saw
it; y = telling lies.
x_{1}m'_{0} + m_{1}y'_{0} > x_{1}y'_{0} [Fig. I (a).
Conclusion wrong: right one is "The publication, in which I saw it,
tells lies."
pg151
8. Some cravats are not artistic;
I admire anything artistic.
There are some cravats that I do not admire.
Univ. "things"; m = artistic; x = cravats; y = things that I admire.
xm_{1} + m_{1}y_{0}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
9. His songs never last an hour.
A song, that lasts an hour, is tedious.
His songs are never tedious.
Univ. "songs"; m = lasting an hour; x = his; y = tedious.
x_{1}m_{0} + m_{1}y'_{0} > x'y_{1} [Fig. III.
Conclusion wrong: right one is "Some tedious songs are not his."
10. Some candles give very little light;
Candles are meant to give light.
Some things, that are meant to give light, give very little.
Univ. "things"; m = candles; x = giving &c.; y = meant &c.
mx_{1} + m_{1}y'_{0} > xy_{1} [Fig. II.
Conclusion right.
11. All, who are anxious to learn, work hard.
Some of these boys work hard.
Some of these boys are anxious to learn.
Univ. "persons"; m = hard-working; x = anxious to learn; y = these boys.
x_{1}m'_{0} + ym_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
12. All lions are fierce;
Some lions do not drink coffee.
Some creatures that drink coffee are not fierce.
Univ. "creatures"; m = lions; x = fierce; y = creatures that drink
coffee.
m_{1}x'_{0} + my'_{1} > xy'_{1} [Fig. II.
Conclusion wrong: right one is "Some fierce creatures do not drink
coffee."
13. No misers are generous;
Some old men are ungenerous.
Some old men are misers.
Univ. "persons"; m = generous; x = misers; y = old men.
xm_{0} + ym'_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
14. No fossil can be crossed in love;
An oyster may be crossed in love.
Oysters are not fossils.
Univ. "things"; m = things that can be crossed in love; x = fossils;
y = oysters.
xm_{0} + y_{1}m'_{0} > y_{1}x_{0} [Fig. I (a).
Conclusion right.
pg152
15. All uneducated people are shallow;
Students are all educated.
No students are shallow.
Univ. "people"; m = educated; x = shallow; y = students.
m'_{1}x'_{0} + y_{1}m'_{0} > xy'_{1} [Fig. III.
Conclusion wrong: right one is "Some shallow people are not students."
16. All young lambs jump;
No young animals are healthy, unless they jump.
All young lambs are healthy.
Univ. "young animals"; m = young animals that jump; x = lambs;
y = healthy.
x_{1}m'_{0} + m'y_{0}
No Conclusion. [Fallacy of Like Eliminands not asserted to exist.]
17. Ill-managed business is unprofitable;
Railways are never ill-managed.
All railways are profitable.
Univ. "business"; m = ill-managed; x = profitable; y = railways.
m_{1}x_{0} + y_{1}m_{0} > x'y'_{1} [Fig. III.
Conclusion wrong: right one is "Some business, other than railways, is
profitable."
18. No Professors are ignorant;
All ignorant people are vain.
No Professors are vain.
Univ. "people"; m = ignorant; x = Professors; y = vain.
xm_{0} + m_{1}y'_{0} > x'y_{1} [Fig. III.
Conclusion wrong: right one is "Some vain persons are not Professors."
19. A prudent man shuns hyaenas.
No banker is imprudent.
No banker fails to shun hyaenas.
Univ. "men"; m = prudent; x = shunning hyaenas; y = bankers.
m_{1}x'_{0} + ym'_{0} > x'y_{0} [Fig. I.
Conclusion right.
20. All wasps are unfriendly;
No puppies are unfriendly.
No puppies are wasps.
Univ. "creatures"; m = friendly; x = wasps; y = puppies.
x_{1}m_{0} + ym'_{0} > x_{1}y_{0} [Fig. I (a).
Conclusion incomplete: complete one is "Wasps are not puppies".
21. No Jews are honest;
Some Gentiles are rich.
Some rich people are dishonest.
Univ. "persons"; m = Jews; x = honest; y = rich.
mx_{0} + m'y_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
pg153
22. No idlers win fame;
Some painters are not idle.
Some painters win fame.
Univ. "persons"; m = idlers; x = persons who win fame; y = painters.
mx_{0} + ym'_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
23. No monkeys are soldiers;
All monkeys are mischievous.
Some mischievous creatures are not soldiers.
Univ. "creatures"; m = monkeys; x = soldiers; y = mischievous.
mx_{0} + m_{1}y'_{0} > x'y_{1} [Fig. III.
Conclusion right.
24. All these bonbons are chocolate-creams;
All these bonbons are delicious.
Chocolate-creams are delicious.
Univ. "food"; m = these bonbons; x = chocolate-creams; y = delicious.
m_{1}x'_{0} + m_{1}y'_{0} > xy_{1} [Fig. III.
Conclusion wrong, being in excess of the right one, which is "Some
chocolate-creams are delicious."
25. No muffins are wholesome;
All buns are unwholesome.
Buns are not muffins.
Univ. "food"; m = wholesome; x = muffins; y = buns.
xm_{0} + y_{1}m_{0}
No Conclusion. [Fallacy of Like Eliminands not asserted to exist.]
26. Some unauthorised reports are false;
All authorised reports are trustworthy.
Some false reports are not trustworthy.
Univ. "reports"; m = authorised; x = true; y = trustworthy.
m'x'_{1} + m_{1}y'_{0}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
27. Some pillows are soft;
No pokers are soft.
Some pokers are not pillows.
Univ. "things"; m = soft; x = pillows; y = pokers.
xm_{1} + ym_{0} > xy'_{1} [Fig. II.
Conclusion wrong: right one is "Some pillows are not pokers."
28. Improbable stories are not easily believed;
None of his stories are probable.
None of his stories are easily believed.
Univ. "stories"; m = probable; x = easily believed; y = his.
m'_{1}x_{0} + ym_{0} > xy_{0} [Fig. I.
Conclusion right.
pg154
29. No thieves are honest;
Some dishonest people are found out.
Some thieves are found out.
Univ. "people"; m = honest; x = thieves; y = found out.
xm_{0} + m'y_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
30. No muffins are wholesome;
All puffy food is unwholesome.
All muffins are puffy.
Univ. is "food"; m = wholesome; x = muffins; y = puffy.
xm_{0} + y_{1}m_{0}
No Conclusion. [Fallacy of Like Eliminands not asserted to exist.]
31. No birds, except peacocks, are proud of their tails;
Some birds, that are proud of their tails, cannot sing.
Some peacocks cannot sing.
Univ. "birds"; m = proud of their tails; x = peacocks; y = birds that
cannot sing.
x'm_{0} + my'_{1} > xy'_{1} [Fig. II.
Conclusion right.
32. Warmth relieves pain;
Nothing, that does not relieve pain, is useful in toothache.
Warmth is useful in toothache.
Univ. "applications"; m = relieving pain; x = warmth; y = useful in
toothache.
x_{1}m'_{0} + m'y_{0}
No Conclusion. [Fallacy of Like Eliminands not asserted to exist.]
33. No bankrupts are rich;
Some merchants are not bankrupts.
Some merchants are rich.
Univ. "persons"; m = bankrupts; x = rich; y = merchants.
mx_{0} + ym'_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
34. Bores are dreaded;
No bore is ever begged to prolong his visit.
No one, who is dreaded, is ever begged to prolong his visit.
Univ. "persons"; m = bores; x = dreaded; y = begged to prolong their
visits.
m_{1}x'_{0} + my_{0} > xy'_{1} [Fig. III.
Conclusion wrong: the right one is "Some dreaded persons are not begged
to prolong their visits."
35. All wise men walk on their feet;
All unwise men walk on their hands.
No man walks on both.
Univ. "men"; m = wise; x = walking on their feet; y = walking on their
hands.
m_{1}x'_{0} + m'_{1}y'_{0} > x'y'_{0} [Fig. I.
Conclusion wrong: right one is "No man walks on neither."
pg155
36. No wheelbarrows are comfortable;
No uncomfortable vehicles are popular.
No wheelbarrows are popular.
Univ. "vehicles"; m = comfortable; x = wheelbarrows; y = popular.
xm_{0} + m'x_{0} > xy_{0} [Fig. I.
Conclusion right.
37. No frogs are poetical;
Some ducks are unpoetical.
Some ducks are not frogs.
Univ. "creatures"; m = poetical; x = frogs; y = ducks.
xm_{0} + ym'_{1}
No Conclusion. [Fallacy of Unlike Eliminands with an Entity-Premiss.]
38. No emperors are dentists;
All dentists are dreaded by children.
No emperors are dreaded by children.
Univ. "persons"; m = dentists; x = emperors; y = dreaded by children.
xm_{0} + m_{1}y'_{0} > x'y_{1} [Fig. III.
Conclusion wrong: right one is "Some persons, dreaded by children, are
not emperors."
39. Sugar is sweet;
Salt is not sweet.
Salt is not sugar.
Univ. "things"; m = sweet; x = sugar; y = salt.
x_{1}m'_{0} + y_{1}m_{0} > (x_{1}y_{0} + y_{1}x_{0}) [Fig. I (b).
Conclusion incomplete: omitted portion is "Sugar is not salt."
40. Every eagle can fly;
Some pigs cannot fly.
Some pigs are not eagles.
Univ. "creatures"; m = creatures that can fly; x = eagles; y = pigs.
x_{1}m'_{0} + ym'_{1} > x'y_{1} [Fig. II.
Conclusion right.
_Solutions for Sec. 8._ SL8
1 2 3
1. cd_{0} + a_{1}d'_{0} + b_{1}c'_{0};
1 2 3
cd + ad' + bc' > ab_{0} + a_{1} + b_{1}
-- = =
i.e. > a_{1}b_{0} + b_{1}a_{0}
1 2 3
2. d_{1}b'_{0} + ac'_{0} + bc_{0};
1 3 2
db' + bc + ac' > da_{0} + d_{1} i.e. > d_{1}a_{0}
- =- =
1 2 3
3. ba_{0} + cd'_{0} + d_{1}b'_{0};
1 3 2
ba + db' + cd' > ac_{0}
- -= =
1 2 3
4. bc_{0} + a_{1}b'_{0} + c'd_{0};
1 2 3
bc + ab' + c'd > ad_{0} + a_{1} i.e. > a_{1}d_{0}
-- = =
pg156
1 2 3 1 2 3
5. b'_{1}a_{0} + bc_{0} + a'd_{0}; b'a + bc + a'd > cd_{0}
- - = =
1 2 3
6. a_{1}b_{0} + b'c_{0} + d_{1}a'_{0};
1 2 3
ab + b'c + da' > cd_{0} + d_{1} i.e. > d_{1}c_{0}
-- = =
1 2 3 1 2 3
7. db'_{0} + b_{1}a'_{0} + cd'_{0}; db' + ba' + cd' > a'c_{0}
-- = =
1 2 3
8. b'd_{0} + a'b_{0} + c_{1}d'_{0};
1 2 3
b'd + a'b + cd' > a'c_{0} + c_{1} i.e. > c_{1}a'_{0}
- - = =
1 2 3
9. b'_{1}a'_{0} + ad_{0} + b_{1}c'_{0};
1 2 3
b'a' + ad + bc' > dc'_{0}
- - = =
1 2 3
10. cd_{0} + b_{1}c'_{0} + ad'_{0};
1 2 3
cd + bc' + ad' > ba_{0} + b_{1} i.e. > b_{1}a_{0}
-- = =
1 2 3
11. bc_{0} + d_{1}a'_{0} + c'_{1}a_{0};
1 3 2
bc + c'a + da' > bd_{0} + d_{1} i.e. > d_{1}b_{0}
- = - =
1 2 3
12. cb'_{0} + c'_{1}d_{0} + b_{1}a'_{0};
1 2 3
cb' + c'd + ba' > da'_{0}
-- = =
1 2 3 4
13. d_{1}e'_{0} + c_{1}a'_{0} + bd'_{0} + e_{1}a_{0};
1 3 4 2
de' + bd' + ea + ca' > bc_{0} + c_{1} i.e. > c_{1}b_{0}
-- = =- =
1 2 3 4
14. c_{1}b'_{0} + a_{1}e'_{0} + d_{1}b_{0} + a'_{1}c'_{0};
1 3 4 2
cb' + db + a'c' + ae' > de'_{0} + d_{1} i.e. > d_{1}e'_{0}
-- = - = =
1 2 3 4
15. b'd_{0} + e_{1}c'_{0} + b_{1}a'_{0} + d'_{1}c_{0};
1 3 4 2
b'd + ba' + d'c + ec' > a'e_{0} + e_{1} i.e. > e_{1}a'_{0}
- - = = - =
1 2 3 4
16. a'e_{0} + d_{1}c_{0} + a_{1}b'_{0} + e'_{1}d'_{0};
1 3 4 2
a'e + ab' + e'd' + dc > b'c_{0}
- - = = - =
1 2 3 4
17. d_{1}c'_{0} + a_{1}e'_{0} + bd'_{0} + c_{1}e_{0};
1 3 4 2
dc' + bd' + ce + ae' > ba_{0} + a_{1} i.e. > a_{1}b_{0}
-- = =- =
1 2 3 4
18. a_{1}b'_{0} + d_{1}e'_{0} + a'_{1}c_{0} + be_{0};
1 3 4 2
ab' + a'c + be + de' > cd_{0} + d_{1} i.e. > d_{1}c_{0}
-- = =- =
1 2 3 4 5
19. bc_{0} + e_{1}h'_{0} + a_{1}b'_{0} + dh_{0} + e'_{1}c'_{0};
1 3 5 2 4
bc + ab' + e'c' + eh' + dh > ad_{0} + a_{1}
-- = - = =- =
i.e. > a_{1}d_{0}
1 2 3 4 5
20. dh'_{0} + ce_{0} + h_{1}b'_{0} + ad'_{0} + be'_{0};
1 3 4 5 2
dh' + hb' + ad' + be' + ce > ac_{0}
-- =- = =- =
1 2 3 4 5
21. b_{1}a'_{0} + dh_{0} + ce_{0} + ah'_{0} + c'_{1}b'_{0};
1 4 2 5 3
ba' + ah' + dh + c'b' + ce > de_{0}
-- =- = - = =
1 2 3 4 5
22. e_{1}d_{0} + b'h'_{0} + c'_{1}d'_{0} + a_{1}e'_{0} + ch_{0};
1 3 4 5 2
ed + c'd' + ae' + ch + b'h' > ab'_{0} + a_{1}
-- - = = =- =
i.e. > a_{1}b_{0}
pg157
1 2 3 4 5
23. b'_{1}a_{0} + de'_{0} + h_{1}b_{0} + ce_{0} + d'_{1}a'_{0};
1 3 5 2 4
b'a + hb + d'a' + de' + ce
- - = - = =- =
> hc_{0} + h_{1} i.e. > h_{1}c_{0}
1 2 3 4 5 6
24. h'_{1}k_{0} + b'a_{0} + c_{1}d'_{0} + e_{1}h_{0} + dk'_{0} + bc'_{0};
1 4 5 3 6 2
h'k + eh + dk' + cd' + bc' + b'a > ea_{0} + e_{1}
- - = -= -= -= =
i.e. > e_{1}a_{0}
1 2 3 4
25. a_{1}d'_{0} + k_{1}b'_{0} + e_{1}h'_{0} + a'b_{0}
5 6
+ d_{1}c'_{0} + h_{1}k'_{0};
1 4 2 5 6 3
ad' + a'b + kb' + dc' + hk' + eh' > c'e_{0} + e_{1}
-- = - -= = -= =
i.e. > e_{1}c'_{0}
1 2 3 4 5
26. a'_{1}h'_{0} + d'k'_{0} + e_{1}b_{0} + hk_{0} + a_{1}c'_{0}
6
+ b'd_{0};
1 4 2 5 6 3
a'h' + hk + d'k' + ac' + b'd + eb > c'e_{0} + e_{1}
- - =- - = = - = =
i.e. > e_{1}c'_{0}
1 2 3 4 5
27. e_{1}d_{0} + hb_{0} + a'_{1}k'_{0} + ce'_{0} + b'_{1}d'_{0}
6
+ ac'_{0};
1 4 5 2 6 3
ed + ce' + b'd' + hb + ac' + a'k' > hk'_{0}
-- -= - = = -= =
1 2 3 4 5
28. a'k_{0} + e_{1}b'_{0} + hk'_{0} + d'c_{0} + ab_{0}
6
+ c'_{1}h'_{0};
1 3 5 2 6 4
a'k + hk' + ab + eb' + c'h' + d'c > ed'_{0} + e_{1}
- - -= =- = - = =
i.e. > e_{1}d'_{0}
1 2 3 4 5
29. ek_{0} + b'm_{0} + ac'_{0} + h'_{1}e'_{0} + d_{1}k'_{0}
6 7 8
+ cb_{0} + d'_{1}l'_{0} + hm'_{0};
1 4 5 7 8 2 6 3
ek + h'e' + dk' + d'l' + hm' + b'm + cb + ac'
-- - = -= = =- - = -= =
> l'a_{0}
1 2 3 4 5
30. n_{1}m'_{0} + a'_{1}e'_{0} + c'l_{0} + k_{1}r_{0} + ah'_{0}
6 7 8 9 10
+ dl'_{0} + cn'_{0} + e_{1}b'_{0} + m_{1}r'_{0} + h_{1}d'_{0};
1 7 3 6 9 4 10 5 2 8
nm' + cn' + c'l + dl' + mr' + kr + hd' + ah' + a'e' + eb'
-- -= = - -= =- = -= -= = - =
> kb'_{0} + k_{1} i.e. > k_{1}b'_{0}
_Solutions for Sec. 9._ SL9
1.
1 2 3
b_{1}d_{0} + ac_{0} + d'_{1}c'_{0};
1 3 2
bd + d'c' + ac > ba_{0} + b_{1}, i.e. > b_{1}a_{0}
- = - =
i.e. Babies cannot manage crocodiles.
2.
1 2 3
a_{1}b'_{0} + d_{1}c'_{0} + bc_{0};
1 3 2
ab' + bc + dc' > ad_{0} + d_{1}, i.e. > d_{1}a_{0}
- =- =
i.e. _Your_ presents to me are not made of tin.
pg158
3.
1 2 3
da_{0} + c_{1}b'_{0} + a'b_{0};
1 3 2
da + a'b + cb' > dc_{0} + c_{1}, i.e. > c_{1}d_{0}
- = - =
i.e. All my potatoes in this dish are old ones.
4.
1 2 3
ba_{0} + b'd_{0} + c_{1}a'_{0};
1 2 3
ba + b'd + ca' > dc_{0} + c_{1}, i.e. > c_{1}d_{0}
-- = =
i.e. My servants never say "shpoonj."
5.
1 2 3
ad_{0} + cd'_{0} + b_{1}a'_{0};
1 2 3
ad + cd' + ba' > cb_{0} + b_{1}, i.e. > b_{1}c_{0}
-- = =
i.e. My poultry are not officers.
6.
1 2 3 1 2 3
c_{1}a'_{0} + c'b_{0} + da_{0}; ca' + c'b + da > bd_{0}
-- = =
i.e. None of your sons are fit to serve on a jury.
7.
1 2 3 1 3 2
cb_{0} + da_{0} + b'_{1}a'_{0}; cb + b'a' + da > cd_{0}
- = - =
i.e. No pencils of mine are sugarplums.
8.
1 2 3
cb'_{0} + d_{1}a'_{0} + ba_{0};
1 3 2
cb' + ba + da' > cd_{0} + d_{1}, i.e. > d_{1}c_{0}
- =- =
i.e. Jenkins is inexperienced.
9.
1 2 3 1 2 3
cd_{0} + d'a_{0} + c'b_{0}; cd + d'a + c'b > ab_{0}
-- = =
i.e. No comet has a curly tail.
10.
1 2 3 1 3 2
d'c_{0} + ba_{0} + a'_{1}d_{0}; d'c + a'd + ba > cb_{0}
- - = =
i.e. No hedgehog takes in the _Times_.
11.
1 2 3
b_{1}a'_{0} + c_{1}b'_{0} + ad_{0};
1 2 3
ba' + cb' + ad > cd_{0} + c_{1}, i.e. > c_{1}d_{0}
-- = =
i.e. This dish is unwholesome.
12.
1 2 3
b_{1}c'_{0} + d'a_{0} + a'c_{0};
1 3 2
bc' + a'c + d'a > bd'_{0} + b_{1}, i.e. > b_{1}d'_{0}
- - = =
i.e. My gardener is very old.
13.
1 2 3
a_{1}d'_{0} + bc_{0} + c'_{1}d_{0};
1 3 2
ad' + c'd + bc > ab_{0} + a_{1}, i.e. > a_{1}b_{0}
- - = =
i.e. All humming-birds are small.
pg159
14.
1 2 3 1 3 2
c'b_{0} + a_{1}d'_{0} + ca'_{0}; c'b + ca' + ad' > bd'_{0}
- =- =
i.e. No one with a hooked nose ever fails to make money.
15.
1 2 3 1 2 3
b_{1}a'_{0} + b'_{1}d_{0} + ca_{0}; ba' + b'd + ca > dc_{0}
-- = =
i.e. No gray ducks in this village wear lace collars.
16.
1 2 3 1 2 3
d_{1}b'_{0} + cd'_{0} + ba_{0}; db' + cd' + ba > ca_{0}
-- = =
i.e. No jug in this cupboard will hold water.
17.
1 2 3
b'_{1}d_{0} + c_{1}d'_{0} + ab_{0};
1 2 3
b'd + cd' + ab > ca_{0} + c_{1}, i.e. > c_{1}a_{0}
- - = =
i.e. These apples were grown in the sun.
18.
1 2 3
d'_{1}b'_{0} + c_{1}b_{0} + c'a_{0};
1 2 3
d'b' + cb + c'a > d'a_{0} + d'_{1}, i.e. > d'_{1}a_{0}
- -= =
i.e. Puppies, that will not lie still, never care to do worsted-work.
19.
1 2 3 1 3 2
bd'_{0} + a_{1}c'_{0} + a'd_{0}; bd' + a'd + ac' > bc'_{0}
- - = =
i.e. No name in this list is unmelodious.
20.
1 2 3 1 3 2
a_{1}b'_{0} + dc_{0} + a'_{1}d'_{0}; ab' + a'd' + dc > b'c_{0}
- = - =
i.e. No M.P. should ride in a donkey-race, unless he has perfect
self-command.
21.
1 2 3 1 3 2
bd_{0} + c'a_{0} + b'c_{0}; bd + b'c + c'a > da_{0}
- = - =
i.e. No goods in this shop, that are still on sale, may be carried away.
22.
1 2 3 1 3 2
a'b_{0} + cd_{0} + d'a_{0}; a'b + d'a + cd > bc_{0}
- - = =
i.e. No acrobatic feat, which involves turning a quadruple somersault,
is ever attempted in a circus.
23.
1 2 3
dc'_{0} + a_{1}b'_{0} + bc_{0};
1 3 2
dc' + bc + ab' > da_{0} + a_{1}, i.e. > a_{1}d_{0}
- -= =
i.e. Guinea-pigs never really appreciate Beethoven.
pg160
24.
1 2 3 1 3 2
a_{1}d'_{0} + b'_{1}c_{0} + ba'_{0}; ad' + ba' + b'c > d'c_{0}
- -= =
i.e. No scentless flowers please me.
25.
1 2 3
c_{1}d'_{0} + ba'_{0} + d_{1}a_{0};
1 3 2
cd' + da + ba' > cb_{0} + c_{1}, i.e. > c_{1}b_{0}
- =- =
i.e. Showy talkers are not really well-informed.
26.
1 2 3 4
ea_{0} + b_{1}d'_{0} + a'_{1}c_{0} + e'b'_{0};
1 3 4 2
ea + a'c + e'b' + bd' > cd'_{0}
-- = = - =
i.e. None but red-haired boys learn Greek in this school.
27.
1 2 3 4
b_{1}d_{0} + ac'_{0} + e_{1}d'_{0} + c_{1}b'_{0};
1 3 4 2
bd + ed' + cb' + ac' > ea_{0} + e_{1}, i.e. > e_{1}a_{0}
-- = -= =
i.e. Wedding-cake always disagrees with me.
28.
1 2 3 4
ad_{0} + e'_{1}b'_{0} + c_{1}d'_{0} + e_{1}a'_{0};
1 3 4 2
ad + cd' + ea' + e'b' > cb'_{0} + c_{1}, i.e. > c_{1}b'_{0}
-- = -= =
i.e. Discussions, that go on while Tomkins is in the chair, endanger the
peacefulness of our Debating-Club.
29.
1 2 3 4
d_{1}a_{0} + e'c_{0} + b_{1}a'_{0} + d'e_{0};
1 3 4 2
da + ba' + d'e + e'c > bc_{0} + b_{1}, i.e. > b_{1}c_{0}
-- = = - =
i.e. All gluttons in my family are unhealthy.
30.
1 2 3 4
d_{1}e_{0} + c'a_{0} + b_{1}e'_{0} + c_{1}d'_{0};
1 3 4 2
de + be' + cd' + c'a > ba_{0} + b_{1}, i.e. > b_{1}a_{0}
-- = -= =
i.e. An egg of the Great Auk is not to be had for a song.
31.
1 2 3 4
d'b_{0} + a_{1}c'_{0} + c_{1}e'_{0} + a'd_{0};
1 4 2 3
d'b + a'd + ac' + ce' > be'_{0}
- - = =- =
i.e. No books sold here have gilt edges unless they are priced at 5s.
and upwards.
32.
1 2 3 4
a'_{1}c'_{0} + d_{1}b_{0} + a_{1}e'_{0} + c_{1}b'_{0};
1 3 4 2
a'c' + ae' + cb' + db > e'd_{0} + d_{1}, i.e. > d_{1}e'_{0}
- - = =- =
i.e. When you cut your finger, you will find Tincture of Calendula
useful.
33.
1 2 3 4
d'b_{0} + a_{1}e'_{0} + ec_{0} + d_{1}a'_{0};
1 4 2 3
d'b + da' + ae' + ec > bc_{0}
- =- =- =
i.e. _I_ have never come across a mermaid at sea.
pg161
34.
1 2 3 4
c'_{1}b_{0} + a_{1}e'_{0} + d_{1}b'_{0} + a'_{1}c_{0};
1 3 4 2
c'b + db' + a'c + ae' > de'_{0} + d_{1}, i.e. > d_{1}e'_{0}
- - = - = =
i.e. All the romances in this library are well-written.
35.
1 2 3 4
e'd_{0} + c'a_{0} + eb_{0} + d'c_{0};
1 3 4 2
e'd + eb + d'c + c'a > ba_{0}
- - = = - =
i.e. No bird in this aviary lives on mince-pies.
36.
1 2 3 4
d'_{1}c'_{0} + e_{1}a'_{0} + c_{1}b_{0} + e'd_{0};
1 3 4 2
d'c' + cb + e'd + ea' > ba'_{0}
- - = - = =
i.e. No plum-pudding, that has not been boiled in a cloth, can be
distinguished from soup.
37.
1 2 3 4 5
ce'_{0} + b'a'_{0} + h_{1}d'_{0} + ae_{0} + bd_{0};
1 4 2 5 3
ce' + ae + b'a' + bd + hd' > ch_{0} + h_{1}, i.e. > h_{1}c_{0}
- -= - = =- =
i.e. All _your_ poems are uninteresting.
38.
1 2 3 4 5
b'_{1}a'_{0} + db_{0} + he'_{0} + ec_{0} + a_{1}h'_{0};
1 2 5 3 4
b'a' + db + ah' + he' + ec > dc_{0}
- - = =- =- =
i.e. None of my peaches have been grown in a hothouse.
39.
1 2 3 4 5
c_{1}d_{0} + h_{1}e'_{0} + c'_{1}a'_{0} + h'b_{0} + e_{1}d'_{0};
1 3 5 2 4
cd + c'a' + ed' + he' + h'b > a'b_{0}
-- = -= -= =
i.e. No pawnbroker is dishonest.
40.
1 2 3 4 5
bd'_{0} + c'h_{0} + e_{1}b'_{0} + da_{0} + e'c_{0};
1 3 4 5 2
bd' + eb' + da + e'c + c'h > ah_{0}
-- -= = = - =
i.e. No kitten with green eyes will play with a gorilla.
41.
1 2 3 4 5
c_{1}a'_{0} + h'b_{0} + ae_{0} + d_{1}c'_{0} + h_{1}e'_{0};
1 3 4 5 2
ca' + ae + dc' + he' + h'b > db_{0} + d_{1}, i.e. > d_{1}b_{0}
-- =- = -= =
i.e. All _my_ friends in this College dine at the lower table.
42.
1 2 3 4 5
ca_{0} + h_{1}d'_{0} + c'_{1}e'_{0} + b'a'_{0} + d_{1}e_{0};
1 3 4 5 2
ca + c'e' + b'a' + de + hd' > b'h_{0} + h_{1},
-- = - = -= =
i.e. > h_{1}b'_{0}
i.e. My writing-desk is full of live scorpions.
43.
1 2 3 4 5
b'_{1}e_{0} + ah_{0} + dc_{0} + e'_{1}a'_{0} + bc'_{0}
1 4 2 5 3
b'e + e'a' + ah + bc' + dc > hd_{0}
- - = - = =- =
i.e. No Mandarin ever reads Hogg's poems.
pg162
44.
1 2 3 4 5
e_{1}b'_{0} + a'd_{0} + c_{1}h'_{0} + e'a_{0} + d'h_{0};
1 4 2 5 3
eb' + e'a + a'd + d'h + ch' > b'c_{0} + c_{1},
- = - = - = - =
i.e. > c_{1}b'_{0}
i.e. Shakespeare was clever.
45.
1 2 3 4 5
e'_{1}c'_{0} + hb'_{0} + d_{1}a_{0} + e_{1}a'_{0} + c_{1}b_{0};
1 4 3 5 2
e'c' + ea' + da + cb + hb' > dh_{0} + d_{1}, i.e. > d_{1}h_{0}
- - =- = =- =
i.e. Rainbows are not worth writing odes to.
46.
1 2 3 4 5
c'_{1}h'_{0} + e_{1}a_{0} + bd_{0} + a'_{1}h_{0} + d'c_{0};
1 4 2 5 3
c'h' + a'h + ea + d'c + bd > eb_{0} + e_{1}, i.e. > e_{1}b_{0}
- - - = = - = =
i.e. These Sorites-examples are difficult.
47.
1 2 3 4 5 6
a'_{1}e'_{0} + bk_{0} + c'a_{0} + eh'_{0} + d_{1}b'_{0} + k'h_{0};
1 3 4 6 2 5
a'e' + c'a + eh' + k'h + bk + db' > c'd_{0} + d_{1},
- - = =- - = -= =
i.e. > d_{1}c'_{0}
i.e. All my dreams come true.
48.
1 2 3 4 5 6
a'h_{0} + c'k_{0} + a_{1}d'_{0} + e_{1}h'_{0} + b_{1}k'_{0} + c_{1}e'_{0};
1 3 4 6 2 5
a'h + ad' + eh' + ce' + c'k + bk' > d'b_{0} + b_{1},
- - = -= -= = - =
i.e. > b_{1}d'_{0}
i.e. All the English pictures here are painted in oils.
49.
1 2 3 4 5 6
k'_{1}e_{0} + c_{1}h_{0} + b_{1}a'_{0} + kd_{0} + h'a_{0} + b'_{1}e'_{0};
1 4 6 3 5 2
k'e + kd + b'e' + ba' + h'a + ch > dc_{0} + c_{1},
- - = - = =- - = =
i.e. > c_{1}d_{0}
i.e. Donkeys are not easy to swallow.
50.
1 2 3 4 5 6
ab'_{0} + h'd_{0} + e_{1}c_{0} + b_{1}d'_{0} + a'k_{0} + c'_{1}h_{0};
1 4 2 5 6 3
ab' + bd' + h'd + a'k + c'h + ec > ke_{0} + e_{1},
-- =- - = = - = =
i.e. > e_{1}k_{0}
i.e. Opium-eaters never wear white kid gloves.
51.
1 2 3 4 5 6
bc_{0} + k_{1}a'_{0} + eh_{0} + d_{1}b'_{0} + h'c'_{0} + k'_{1}e'_{0};
1 4 5 3 6 2
bc + db' + h'c' + eh + k'e' + ka' > da'_{0} + d_{1},
-- = - = -= - = =
i.e. > d_{1}a'_{0}
i.e. A good husband always comes home for his tea.
52.
1 2 3 4 5 6
a'_{1}k'_{0} + ch_{0} + h'k_{0} + b_{1}d'_{0} + ea_{0} + d_{1}c'_{0}
1 3 2 6 4 5
a'k' + h'k + ch + dc' + bd' + ea > be_{0} + b_{1},
- - - = -= -= = =
i.e. > b_{1}e_{0}
i.e. Bathing-machines are never made of mother-of-pearl.
pg163
53.
1 2 3 4 5
da'_{0} + k_{1}b'_{0} + c_{1}h_{0} + d'_{1}k'_{0} + e_{1}c'_{0}
6
+ a_{1}h'_{0};
1 4 2 6 3 5
da' + d'k' + kb' + ah' + ch + ec'
-- = - = =- -= =
> b'e_{0} + e_{1}, i.e. > e_{1}b'_{0}
i.e. Rainy days are always cloudy.
54.
1 2 3 4 5 6
kb'_{0} + a'_{1}c'_{0} + d'b_{0} + k'_{1}h'_{0} + ea_{0} + d_{1}c_{0};
1 3 4 6 2 5
kb' + d'b + k'h' + dc + a'c' + ea
-- - = = =- - = =
> h'e_{0}
i.e. No heavy fish is unkind to children.
55.
1 2 3 4 5 6
k'_{1}b'_{0} + eh'_{0} + c'd_{0} + hb_{0} + ac_{0} + kd'_{0};
1 4 2 6 3 5
k'b' + hb + eh' + kd' + c'd + ac > ea_{0}
- - -= = =- - = =
i.e. No engine-driver lives on barley-sugar.
56.
1 2 3 4 5
h_{1}b'_{0} + c_{1}d'_{0} + k'a_{0} + e_{1}h'_{0} + b_{1}a'_{0}
6
+ k_{1}c'_{0};
1 4 5 3 6 2
hb' + eh' + ba' + k'a + kc' + cd'
-- = =- - = =- =
> ed'_{0} + e_{1}, i.e. > e_{1}d'_{0}
i.e. All the animals in the yard gnaw bones.
57.
1 2 3 4 5 6
h'_{1}d'_{0} + e_{1}c'_{0} + k'a_{0} + cb_{0} + d_{1}l'_{0} + e'h_{0}
7
+ kl_{0};
1 5 7 3 6 2 4
h'd' + dl' + kl + k'a + e'h + ec' + cb > ab_{0}
- - =- -= = - = =- =
i.e. No badger can guess a conundrum.
58.
1 2 3 4 5 6
b'h_{0} + d'_{1}l'_{0} + ca_{0} + d_{1}k'_{0} + h'_{1}e'_{0} + mc'_{0}
7 8
+ a'b_{0} + ek_{0};
1 5 7 3 6 8 4 2
b'h + h'e' + a'b + ca + mc' + ek + dk' + d'l' > ml'_{0}
- - = - - = -= = =- -= =
i.e. No cheque of yours, received by me, is payable to order.
59.
1 2 3 4 5 6
c_{1}l'_{0} + h'e_{0} + kd_{0} + mc'_{0} + b'_{1}e'_{0} + n_{1}a'_{0}
7 8 9
+ l_{1}d'_{0} + m'b_{0} + ah_{0};
1 4 7 3 8 5 2 9 6
cl' + mc' + ld' + kd + m'b + b'e' + h'e + ah + na'
-- -= =- = = - = - - = -= =
> kn_{0}
i.e. I cannot read any of Brown's letters.
60.
1 2 3 4 5 6
e_{1}c'_{0} + l_{1}n'_{0} + d_{1}a'_{0} + m'b_{0} + ck'_{0} + e'r_{0}
7 8 9 10
+ h_{1}n_{0} + b'k_{0} + r'_{1}d'_{0} + m_{1}l'_{0};
1 5 6 8 4 9 3 10 2 7
ec' + ck' + e'r + b'k + m'b + r'd' + da' + ml' + ln' + hn
-- =- = - - = - = = - = =- =- =
> a'h_{0} + h_{1}, i.e. > h_{1}a'_{0}
i.e. I always avoid a kangaroo.
pg164
NOTES.
(A) [See p. 80].
One of the favourite objections, brought against the Science of Logic by
its detractors, is that a Syllogism has no real validity as an argument,
since it involves the Fallacy of _Petitio Principii_ (i.e. "Begging the
Question", the essence of which is that the whole Conclusion is involved
in _one_ of the Premisses).
This formidable objection is refuted, with beautiful clearness and
simplicity, by these three Diagrams, which show us that, in each of the
three Figures, the Conclusion is really involved in the _two_ Premisses
taken together, each contributing its share.
Thus, in Fig. I., the Premiss xm_{0} empties the _Inner_ Cell of the
N.W. Quarter, while the Premiss ym_{0} empties its _Outer_ Cell. Hence
it needs the _two_ Premisses to empty the _whole_ of the N.W. Quarter,
and thus to prove the Conclusion xy_{0}.
Again, in Fig. II., the Premiss xm_{0} empties the Inner Cell of the
N.W. Quarter. The Premiss ym_{1} merely tells us that the Inner Portion
of the W. Half is _occupied_, so that we may place a 'I' in it,
_somewhere_; but, if this were the _whole_ of our information, we should
not know in _which_ Cell to place it, so that it would have to 'sit on
the fence': it is only when we learn, from the other Premiss, that the
_upper_ of these two Cells is _empty_, that we feel authorised to place
the 'I' in the _lower_ Cell, and thus to prove the Conclusion x'y_{1}.
Lastly, in Fig. III., the information, that m _exists_, merely
authorises us to place a 'I' _somewhere_ in the Inner Square----but it
has large choice of fences to sit upon! It needs the Premiss xm_{0} to
drive it out of the N. Half of that Square; and it needs the Premiss
ym_{0} to drive it out of the W. Half. Hence it needs the _two_
Premisses to drive it into the Inner Portion of the S.E. Quarter, and
thus to prove the Conclusion x'y'_{1}.
pg165
APPENDIX,
ADDRESSED TO TEACHERS.
Sec. 1.
_Introductory._
There are several matters, too hard to discuss with _Learners_, which
nevertheless need to be explained to any _Teachers_, into whose hands
this book may fall, in order that they may thoroughly understand what my
Symbolic Method _is_, and in what respects it differs from the many
other Methods already published.
These matters are as follows:--
The "Existential Import" of Propositions.
The use of "is-not" (or "are-not") as a Copula.
The theory "two Negative Premisses prove nothing."
Euler's Method of Diagrams.
Venn's Method of Diagrams.
My Method of Diagrams.
The Solution of a Syllogism by various Methods.
My Method of treating Syllogisms and Sorites.
Some account of Parts II, III.
Sec. 2.
_The "Existential Import" of Propositions._
The writers, and editors, of the Logical text-books which run in the
ordinary grooves----to whom I shall hereafter refer by the (I hope
inoffensive) title "The Logicians"----take, on this subject, what seems
to me to be a more humble position than is at all necessary. They speak
of the Copula of a Proposition "with bated breath", almost as if it were
a living, conscious Entity, capable of declaring for itself what it
chose to mean, and that we, poor human creatures, had nothing to do but
to ascertain _what_ was its sovereign will and pleasure, and submit to
it.
pg166
In opposition to this view, I maintain that any writer of a book is
fully authorised in attaching any meaning he likes to any word or phrase
he intends to use. If I find an author saying, at the beginning of his
book, "Let it be understood that by the word '_black_' I shall always
mean '_white_', and that by the word '_white_' I shall always mean
'_black_'," I meekly accept his ruling, however injudicious I may think
it.
And so, with regard to the question whether a Proposition is or is not
to be understood as asserting the existence of its Subject, I maintain
that every writer may adopt his own rule, provided of course that it is
consistent with itself and with the accepted facts of Logic.
Let us consider certain views that may _logically_ be held, and thus
settle which of them may _conveniently_ be held; after which I shall
hold myself free to declare which of them _I_ intend to hold.
The _kinds_ of Propositions, to be considered, are those that begin with
"some", with "no", and with "all". These are usually called Propositions
"in _I_", "in _E_", and "in _A_".
First, then, a Proposition in _I_ may be understood as asserting, or
else as _not_ asserting, the existence of its Subject. (By "existence" I
mean of course whatever kind of existence suits its nature. The two
Propositions, "_dreams_ exist" and "_drums_ exist", denote two totally
different kinds of "existence". A _dream_ is an aggregate of ideas, and
exists only in the _mind of a dreamer_: whereas a _drum_ is an aggregate
of wood and parchment, and exists in _the hands of a drummer_.)
First, let us suppose that _I_ "asserts" (i.e. "asserts the existence of
its Subject").
Here, of course, we must regard a Proposition in _A_ as making the
_same_ assertion, since it necessarily _contains_ a Proposition in _I_.
We now have _I_ and _A_ "asserting". Does this leave us free to make
what supposition we choose as to _E_? My answer is "No. We are tied down
to the supposition that _E_ does _not_ assert." This can be proved as
follows:--
If possible, let _E_ "assert". Then (taking x, y, and z to represent
Attributes) we see that, if the Proposition "No xy are z" be true, some
things exist with the Attributes x and y: i.e. "Some x are y."
pg167
Also we know that, if the Proposition "Some xy are z" be true, the same
result follows.
But these two Propositions are Contradictories, so that one or other of
them _must_ be true. Hence this result is _always_ true: i.e. the
Proposition "Some x are y" is _always_ true!
_Quod est absurdum._ (See Note (A), p. 195).
We see, then, that the supposition "_I_ asserts" necessarily leads to
"_A_ asserts, but _E_ does not". And this is the _first_ of the various
views that may conceivably be held.
Next, let us suppose that _I_ does _not_ "assert." And, along with this,
let us take the supposition that _E_ _does_ "assert."
Hence the Proposition "No x are y" means "Some x exist, and none of them
are y": i.e. "_all_ of them are _not_-y," which is a Proposition in _A_.
We also know, of course, that the Proposition "All x are not-y" proves
"No x are y." Now two Propositions, each of which proves the other, are
_equivalent_. Hence every Proposition in _A_ is equivalent to one in
_E_, and therefore "_asserts_".
Hence our _second_ conceivable view is "_E_ and _A_ assert, but _I_ does
not."
This view does not seen to involve any necessary contradiction with
itself or with the accepted facts of Logic. But, when we come to _test_
it, as applied to the actual _facts_ of life, we shall find I think,
that it fits in with them so badly that its adoption would be, to say
the least of it, singularly inconvenient for ordinary folk.
Let me record a little dialogue I have just held with my friend Jones,
who is trying to form a new Club, to be regulated on strictly _Logical_
principles.
_Author._ "Well, Jones! Have you got your new Club started yet?"
_Jones_ (_rubbing his hands_). "You'll be glad to hear that some of the
Members (mind, I only say '_some_') are millionaires! Rolling in gold,
my boy!"
_Author._ "That sounds well. And how many Members have entered?"
_Jones_ (_staring_). "None at all. We haven't got it started yet. What
makes you think we have?"
_Author._ "Why, I thought you said that some of the Members----"
pg168
_Jones_ (_contemptuously_). "You don't seem to be aware that we're
working on strictly _Logical_ principles. A _Particular_ Proposition
does _not_ assert the existence of its Subject. I merely meant to say
that we've made a Rule not to admit _any_ Members till we have at least
_three_ Candidates whose incomes are over ten thousand a year!"
_Author._ "Oh, _that's_ what you meant, is it? Let's hear some more of
your Rules."
_Jones._ "Another is, that no one, who has been convicted seven times of
forgery, is admissible."
_Author._ "And here, again, I suppose you don't mean to assert there
_are_ any such convicts in existence?"
_Jones._ "Why, that's exactly what I _do_ mean to assert! Don't you know
that a Universal Negative _asserts_ the existence of its Subject? _Of
course_ we didn't make that Rule till we had satisfied ourselves that
there are several such convicts now living."
The Reader can now decide for himself how far this _second_ conceivable
view would fit in with the facts of life. He will, I think, agree with
me that Jones' view, of the 'Existential Import' of Propositions, would
lead to some inconvenience.
Thirdly, let us suppose that neither _I_ nor _E_ "asserts".
Now the supposition that the two Propositions, "Some x are y" and "No x
are not-y", do _not_ "assert", necessarily involves the supposition that
"All x are y" does _not_ "assert", since it would be absurd to suppose
that they assert, when combined, more than they do when taken
separately.
Hence the _third_ (and last) of the conceivable views is that neither
_I_, nor _E_, nor _A_, "asserts".
The advocates of this third view would interpret the Proposition "Some x
are y" to mean "If there _were_ any x in existence, some of them _would_
be y"; and so with _E_ and _A_.
It admits of proof that this view, as regards _A_, conflicts with the
accepted facts of Logic.
Let us take the Syllogism _Darapti_, which is universally accepted as
valid. Its form is
"All m are x;
All m are y.
.'. Some y are x".
pg169
This they would interpret as follows:--
"If there were any m in existence, all of them would be x;
If there were any m in existence, all of them would be y.
.'. If there were any y in existence, some of them
would be x".
That this Conclusion does _not_ follow has been so briefly and clearly
explained by Mr. Keynes (in his "Formal Logic", dated 1894, pp. 356,
357), that I prefer to quote his words:--
"_Let no proposition imply the existence either of its subject or of its
predicate._
"Take, as an example, a syllogism in _Darapti_:--
'_All M is P_,
_All M is S_,
_.'. Some S is P_.'
"Taking S, M, P, as the minor, middle, and major terms respectively, the
conclusion will imply that, if there is an S, there is some P. Will the
premisses also imply this? If so, then the syllogism is valid; but not
otherwise.
"The conclusion implies that if S exists P exists; but, consistently
with the premisses, S may be existent while M and P are both
non-existent. An implication is, therefore, contained in the conclusion,
which is not justified by the premisses."
This seems to _me_ entirely clear and convincing. Still, "to make
sicker", I may as well throw the above (_soi-disant_) Syllogism into a
concrete form, which will be within the grasp of even a _non_-logical
Reader.
Let us suppose that a Boys' School has been set up, with the following
system of Rules:--
"All boys in the First (the highest) Class are to do French, Greek, and
Latin. All in the Second Class are to do Greek only. All in the Third
Class are to do Latin only."
Suppose also that there _are_ boys in the Third Class, and in the
Second; but that no boy has yet risen into the First.
It is evident that there are no boys in the School doing French: still
we know, by the Rules, what would happen if there _were_ any.
pg170
We are authorised, then, by the _Data_, to assert the following two
Propositions:--
"If there were any boys doing French, all of them would
be doing Greek;
If there were any boys doing French, all of them would
be doing Latin."
And the Conclusion, according to "The Logicians" would be
"If there were any boys doing Latin, some of them would
be doing Greek."
Here, then, we have two _true_ Premisses and a _false_ Conclusion (since
we know that there _are_ boys doing Latin, and that _none_ of them are
doing Greek). Hence the argument is _invalid_.
Similarly it may be shown that this "non-existential" interpretation
destroys the validity of _Disamis_, _Datisi_, _Felapton_, and
_Fresison_.
Some of "The Logicians" will, no doubt, be ready to reply "But we are
not _Aldrichians_! Why should _we_ be responsible for the validity of
the Syllogisms of so antiquated an author as Aldrich?"
Very good. Then, for the _special_ benefit of these "friends" of mine
(with what ominous emphasis that name is sometimes used! "I must have a
private interview with _you_, my young _friend_," says the bland Dr.
Birch, "in my library, at 9 a.m. tomorrow. And you will please to be
_punctual_!"), for their _special_ benefit, I say, I will produce
_another_ charge against this "non-existential" interpretation.
It actually invalidates the ordinary Process of "Conversion", as applied
to Proposition in '_I_'.
_Every_ logician, Aldrichian or otherwise, accepts it as an established
fact that "Some x are y" may be legitimately converted into "Some y are
x."
But is it equally clear that the Proposition "If there _were_ any x,
some of them _would_ be y" may be legitimately converted into "If there
_were_ any y, some of them would be x"? I trow not.
The example I have already used----of a Boys' School with a non-existent
First Class----will serve admirably to illustrate this new flaw in the
theory of "The Logicians."
pg171
Let us suppose that there is yet _another_ Rule in this School, viz. "In
each Class, at the end of the Term, the head boy and the second boy
shall receive prizes."
This Rule entirely authorises us to assert (in the sense in which "The
Logicians" would use the words) "Some boys in the First Class will
receive prizes", for this simply means (according to them) "If there
_were_ any boys in the First Class, some of them _would_ receive
prizes."
Now the Converse of this Proposition is, of course, "Some boys, who will
receive prizes, are in the First Class", which means (according to "The
Logicians") "If there _were_ any boys about to receive prizes, some of
them _would_ be in the First Class" (which Class we know to be _empty_).
Of this Pair of Converse Propositions, the first is undoubtedly _true_:
the second, _as_ undoubtedly, _false_.
It is always sad to see a batsman knock down his own wicket: one pities
him, as a man and a brother, but, as a _cricketer_, one can but
pronounce him "Out!"
We see, then, that, among all the conceivable views we have here
considered, there are only _two_ which can _logically_ be held, viz.
_I_ and _A_ "assert", but _E_ does not.
_E_ and _A_ "assert", but _I_ does not.
The _second_ of these I have shown to involve great practical
inconvenience.
The _first_ is the one adopted in this book. (See p. 19.)
Some further remarks on this subject will be found in Note (B), at p.
196.
Sec. 3.
_The use of "is-not" (or "are-not") as a Copula._
Is it better to say "John _is-not_ in-the-house" or "John _is_
not-in-the-house"? "Some of my acquaintances _are-not_
men-I-should-like-to-be-seen-with" or "Some of my acquaintances _are_
men-I-should-_not_-like-to-be-seen-with"? That is the sort of question
we have now to discuss.
pg172
This is no question of Logical Right and Wrong: it is merely a matter of
_taste_, since the two forms mean exactly the same thing. And here,
again, "The Logicians" seem to me to take much too humble a position.
When they are putting the final touches to the grouping of their
Proposition, just before the curtain goes up, and when the
Copula----always a rather fussy 'heavy father', asks them "Am _I_ to
have the 'not', or will you tack it on to the Predicate?" they are much
too ready to answer, like the subtle cab-driver, "Leave it to _you_,
Sir!" The result seems to be, that the grasping Copula constantly gets a
"not" that had better have been merged in the Predicate, and that
Propositions are differentiated which had better have been recognised as
precisely similar. Surely it is simpler to treat "Some men are Jews" and
"Some men are Gentiles" as being both of them, _affirmative_
Propositions, instead of translating the latter into "Some men are-not
Jews", and regarding it as a _negative_ Propositions?
The fact is, "The Logicians" have somehow acquired a perfectly _morbid_
dread of negative Attributes, which makes them shut their eyes, like
frightened children, when they come across such terrible Propositions as
"All not-x are y"; and thus they exclude from their system many very
useful forms of Syllogisms.
Under the influence of this unreasoning terror, they plead that, in
Dichotomy by Contradiction, the _negative_ part is too large to deal
with, so that it is better to regard each Thing as either included in,
or excluded from, the _positive_ part. I see no force in this plea: and
the facts often go the other way. As a personal question, dear Reader,
if _you_ were to group your acquaintances into the two Classes, men that
you _would_ like to be seen with, and men that you would _not_ like to
be seen with, do you think the latter group would be so _very_ much the
larger of the two?
For the purposes of Symbolic Logic, it is so _much_ the most convenient
plan to regard the two sub-divisions, produced by Dichotomy, on the
_same_ footing, and to say, of any Thing, either that it "is" in the
one, or that it "is" in the other, that I do not think any Reader of
this book is likely to demur to my adopting that course.
pg173
Sec. 4.
_The theory that "two Negative Premisses prove nothing"._
This I consider to be _another_ craze of "The Logicians", fully as
morbid as their dread of a negative Attribute.
It is, perhaps, best refuted by the method of _Instantia Contraria_.
Take the following Pairs of Premisses:--
"None of my boys are conceited;
None of my girls are greedy".
"None of my boys are clever;
None but a clever boy could solve this problem".
"None of my boys are learned;
Some of my boys are not choristers".
(This last Proposition is, in _my_ system, an _affirmative_ one, since I
should read it "are not-choristers"; but, in dealing with "The
Logicians," I may fairly treat it as a _negative_ one, since _they_
would read it "are-not choristers".)
If you, dear Reader, declare, after full consideration of these Pairs of
Premisses, that you cannot deduce a Conclusion from _any_ of
them----why, all I can say is that, like the Duke in Patience, you "will
have to be contented with our heart-felt sympathy"! [See Note (C), p.
196.]
Sec. 5.
_Euler's Method of Diagrams._
Diagrams seem to have been used, at first, to represent _Propositions_
only. In Euler's well-known Circles, each was supposed to contain a
class, and the Diagram consisted of two circles, which exhibited the
relations, as to inclusion and exclusion, existing between the two
Classes.
_____
_/ ___ \_
/ / y \ \
| \___/ |
\_ x _/
\_____/
Thus, the Diagram, here given, exhibits the two Classes, whose
respective Attributes are x and y, as so related to each other that the
following Propositions are all simultaneously true:--"All x are y", "No
x are not-y", "Some x are y", "Some y are not-x", "Some not-y are
not-x", and, of course, the Converses of the last four.
pg174
_____
_/ ___ \_
/ / y \ \
| \___/ |
\_ x _/
\_____/
Similarly, with this Diagram, the following Propositions are true:--"All
y are x", "No y are not-x", "Some y are x", "Some x are not-y", "Some
not-x are not-y", and, of course, the Converses of the last four.
_____ _____
_/ \_ _/ \_
/ \ / \
| x | | y |
\_ _/ \_ _/
\_____/ \_____/
Similarly, with this Diagram, the following are true:--"All x are
not-y", "All y are not-x", "No x are y", "Some x are not-y", "Some y are
not-x", "Some not-x are not-y", and the Converses of the last four.
_____ _____
_/ \_/ \_
/ / \ \
| x | | y |
\_ \_/ _/
\_____/ \_____/
Similarly, with this Diagram, the following are true:--"Some x are y",
"Some x are not-y", "Some not-x are y", "Some not-x are not-y", and of
course, their four Converses.
Note that _all_ Euler's Diagrams assert "Some not-x are not-y."
Apparently it never occured to him that it might _sometimes_ fail to be
true!
Now, to represent "All x are y", the _first_ of these Diagrams would
suffice. Similarly, to represent "No x are y", the _third_ would
suffice. But to represent any _Particular_ Proposition, at least _three_
Diagrams would be needed (in order to include all the possible cases),
and, for "Some not-x are not-y", all the _four_.
Sec. 6.
_Venn's Method of Diagrams._
Let us represent "not-x" by "x'".
Mr. Venn's Method of Diagrams is a great advance on the above Method.
He uses the last of the above Diagrams to represent _any_ desired
relation between x and y, by simply shading a Compartment known to be
_empty_, and placing a + in one known to be _occupied_.
Thus, he would represent the three Propositions "Some x are y", "No x
are y", and "All x are y", as follows:--
_____ _____
_/ \_/ \_
/ / \ \
| | + | |
\_ \_/ _/
\_____/ \_____/
_____ _____
_/ \_/ \_
/ /#\ \
| |###| |
\_ \#/ _/
\_____/ \_____/
_____ _____
_/#####\_/ \_
/#######/ \ \
|#######| + | |
\#######\_/ _/
\#####/ \_____/
pg175
It will be seen that, of the _four_ Classes, whose peculiar Sets of
Attributes are xy, xy', x'y, and x'y', only _three_ are here provided
with closed Compartments, while the _fourth_ is allowed the rest of the
Infinite Plane to range about in!
This arrangement would involve us in very serious trouble, if we ever
attempted to represent "No x' are y'." Mr. Venn _once_ (at p. 281)
encounters this awful task; but evades it, in a quite masterly fashion,
by the simple foot-note "We have not troubled to shade the outside of
this diagram"!
To represent _two_ Propositions (containing a common Term) _together_, a
_three_-letter Diagram is needed. This is the one used by Mr. Venn.
_____
_/ \_
_/___ x ___\_
_/| \_/ |\_
/ \_ / \ _/ \
| \|___|/ |
\_ m \_/ y _/
\_____/ \_____/
Here, again, we have only _seven_ closed Compartments, to accommodate
the _eight_ Classes whose peculiar Sets of Attributes are xym, xym', &c.
"With four terms in request," Mr. Venn says, "the most simple and
symmetrical diagram seems to me that produced by making four ellipses
intersect one another in the desired manner". This, however, provides
only _fifteen_ closed compartments.
b ____ ____ c
/ \ / \
a ___/___ \/ ___\___ d
/ \ \ /\ / / \
/ \ \/ \/ / \
\ \ /\ /\ / /
\ \/ \/ \/ /
\ /\ /\ /\ /
\ \ \/ \/ / /
\ \/\ /\/ /
\ /\_\/_/\ /
\__\______/__/
For _five_ letters, "the simplest diagram I can suggest," Mr. Venn says,
"is one like this (the small ellipse in the centre is to be regarded as
a portion of the _outside_ of c; i.e. its four component portions are
inside b and d but are no part of c). It must be admitted that such a
diagram is not quite so simple to draw as one might wish it to be; but
then consider what the alternative is of one undertakes to deal with
five terms and all their combinations--nothing short of the disagreeable
task of writing out, or in some way putting before us, all the 32
combinations involved."
b c d
______ ____ ______
______/_ \/ \/ _\______
a / / \ /\ /\ / \ \ e
/ / \ / \ / \ / \ \
/ \ \/ \/ \/ / \
/ \ /\ /\ /\ / \
/ \ / \ / \ / \ / \
| \/ \/ __ \/ \/ |
| /\ /\/ \/\ /\ |
| / \ / /\ /\ \ / \ |
| | \/ | \/ | \/ | |
| | /\ | /\ | /\ | |
| \ / \ \/ \/ / \ / |
\ \/ \/\__/\/ \/ /
\ /\ /\ /\ /\ /
\ / \ / \ / \ / \ /
\ / \/ \/ \/ \ /
\ \ /\ /\ /\ / /
\ \ / \ / \ / \ / /
\ \/ \/ \/ \/ /
\ /\____/\____/\____/\ /
\ / \ /
\| |/
\____________________/
pg176
This Diagram gives us 31 closed compartments.
For _six_ letters, Mr. Venn suggests that we might use _two_ Diagrams,
like the above, one for the f-part, and the other for the not-f-part, of
all the other combinations. "This", he says, "would give the desired 64
subdivisions." This, however, would only give 62 closed Compartments,
and _one_ infinite area, which the two Classes, a'b'c'd'e'f and
a'b'c'd'e'f', would have to share between them.
Beyond _six_ letters Mr. Venn does not go.
Sec. 7.
_My Method of Diagrams._
My Method of Diagrams _resembles_ Mr. Venn's, in having separate
Compartments assigned to the various Classes, and in marking these
Compartments as _occupied_ or as _empty_; but it _differs_ from his
Method, in assigning a _closed_ area to the _Universe of Discourse_, so
that the Class which, under Mr. Venn's liberal sway, has been ranging at
will through Infinite Space, is suddenly dismayed to find itself
"cabin'd, cribb'd, confined", in a limited Cell like any other Class!
Also I use _rectilinear_, instead of _curvilinear_, Figures; and I mark
an _occupied_ Cell with a 'I' (meaning that there is at least _one_
Thing in it), and an _empty_ Cell with a 'O' (meaning that there is _no_
Thing in it).
For _two_ letters, I use this Diagram, in which the North Half is
assigned to 'x', the South to 'not-x' (or 'x''), the West to y, and the
East to y'. Thus the N.W. Cell contains the xy-Class, the N.E. Cell the
xy'-Class, and so on.
.-------.
| | |
|---|---|
| | |
.-------.
For _three_ letters, I subdivide these four Cells, by drawing an _Inner_
Square, which I assign to m, the _Outer_ Border being assigned to m'. I
thus get _eight_ Cells that are needed to accommodate the eight Classes,
whose peculiar Sets of Attributes are xym, xym', &c.
.---------------.
| | |
| .---|---. |
| | | | |
|---|---|---|---|
| | | | |
| .---|---. |
| | |
.---------------.
This last Diagram is the most complex that I use in the _Elementary_
Part of my 'Symbolic Logic.' But I may as well take this opportunity of
describing the more complex ones which will appear in Part II.
pg177
For _four_ letters (which I call a, b, c, d) I use this Diagram;
assigning the North Half to a (and of course the _rest_ of the Diagram
to a'), the West Half to b, the Horizontal Oblong to c, and the Upright
Oblong to d. We have now got 16 Cells.
.---------------------.
| | |
| .---|---. |
| | | | |
| .---|---|---|---. |
| | | | | | |
|--|---|---|---|---|--|
| | | | | | |
| .---|---|---|---. |
| | | | |
| .---|---. |
| | |
.---------------------.
For _five_ letters (adding e) I subdivide the 16 Cells of the previous
Diagram by _oblique_ partitions, assigning all the _upper_ portions to
e, and all the _lower_ portions to e'. Here, I admit, we lose the
advantage of having the e-Class all _together_, "in a ring-fence", like
the other 4 Classes. Still, it is very easy to find; and the operation,
of erasing it, is nearly as easy as that of erasing any other Class. We
have now got 32 Cells.
.---------------------.
| / | / |
| / .---|---. / |
| / | / | / | / |
| .---|---|---|---. |
| | / | / | / | / | |
|--|---|---|---|---|--|
| | / | / | / | / | |
| .---|---|---|---. |
| / | / | / | / |
| / .---|---. / |
| / | / |
.---------------------.
For _six_ letters (adding h, as I avoid _tailed_ letters) I substitute
upright crosses for the oblique partitions, assigning the 4 portions,
into which each of the 16 Cells is thus divided, to the four Classes eh,
eh', e'h, e'h'. We have now got 64 Cells.
#=============================#
H | H | H
H | #=====H=====# | H
H | H | H | H | H
H-----|--H--|--H--|--H--|-----H
H | H | H | H | H
H #=====H=====H=====H=====# H
H H | H | H | H | H H
H H--|--H--|--H--|--H--|--H H
H H | H | H | H | H H
H==H=====H=====H=====H=====H--H
H H | H | H | H | H H
H H--|--H--|--H--|--H--|--H H
H H | H | H | H | H H
H #=====H=====H=====H=====# H
H | H | H | H | H
H-----|--H--|--H--|--H--|-----H
H | H | H | H | H
H | #=====H=====# | H
H | H | H
#=============================#
pg178
For _seven_ letters (adding k) I add, to each upright cross, a little
inner square. All these 16 little squares are assigned to the k-Class,
and all outside them to the k'-Class; so that 8 little Cells (into which
each of the 16 Cells is divided) are respectively assigned to the 8
Classes ehk, ehk', &c. We have now got 128 Cells.
#=====================================================#
H | H | H
H | #===========H===========# | H
H | H | H | H | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | | | H | | | H | | | H | | | H
H-----|--|--|--H--|--|--|--H--|--|--|--H--|--|--|-----H
H | | | H | | | H | | | H | | | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | H | H | H | H
H #===========H===========H===========H===========# H
H H | H | H | H | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | | | H | | | H | | | H | | | H H
H H--|--|--|--H--|--|--|--H--|--|--|--H--|--|--|--H H
H H | | | H | | | H | | | H | | | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | H | H | H | H H
H==H===========H===========H===========H===========H==H
H H | H | H | H | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | | | H | | | H | | | H | | | H H
H H--|--|--|--H--|--|--|--H--|--|--|--H--|--|--|--H H
H H | | | H | | | H | | | H | | | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | H | H | H | H H
H #===========H===========H===========H===========# H
H | H | H | H | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | | | H | | | H | | | H | | | H
H-----|--|--|--H--|--|--|--H--|--|--|--H--|--|--|-----H
H | | | H | | | H | | | H | | | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | H | H | H | H
H | #===========H===========# | H
H | H | H
.=====================================================#
For _eight_ letters (adding l) I place, in each of the 16 Cells, a
_lattice_, which is a reduced copy of the whole Diagram; and, just as
the 16 large Cells of the whole Diagram are assigned to the 16 Classes
abcd, abcd', &c., so the 16 little Cells of each lattice are assigned to
the 16 Classes ehkl, ehkl', &c. Thus, the lattice in the N.W. corner
serves to accommodate the 16 Classes abc'd'ehkl, abc'd'eh'kl', &c. This
Octoliteral Diagram (see next page) contains 256 Cells.
For _nine_ letters, I place 2 Octoliteral Diagrams side by side,
assigning one of them to m, and the other to m'. We have now got 512
Cells.
pg179
#=====================================================================#
H | H | H
H | #===============H===============# | H
H | H | H | H | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | | | H | | | H | | | H | | | H
H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H
H | | | | | H | | | | | H | | | | | H | | | | | H
H----|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|----H
H | | | | | H | | | | | H | | | | | H | | | | | H
H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H
H | | | H | | | H | | | H | | | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | H | H | H | H
H #===============H===============H===============H===============# H
H H | H | H | H | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | | | H | | | H | | | H | | | H H
H H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H H
H H | | | | | H | | | | | H | | | | | H | | | | | H H
H H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H H
H H | | | | | H | | | | | H | | | | | H | | | | | H H
H H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H H
H H | | | H | | | H | | | H | | | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | H | H | H | H H
H==H===============H===============H===============H===============H==H
H H | H | H | H | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | | | H | | | H | | | H | | | H H
H H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H H
H H | | | | | H | | | | | H | | | | | H | | | | | H H
H H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H H
H H | | | | | H | | | | | H | | | | | H | | | | | H H
H H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H H
H H | | | H | | | H | | | H | | | H H
H H .--|--. H .--|--. H .--|--. H .--|--. H H
H H | H | H | H | H H
H #===============H===============H===============H===============# H
H | H | H | H | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | | | H | | | H | | | H | | | H
H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H
H | | | | | H | | | | | H | | | | | H | | | | | H
H----|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|----H
H | | | | | H | | | | | H | | | | | H | | | | | H
H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H .--|--|--|--. H
H | | | H | | | H | | | H | | | H
H .--|--. H .--|--. H .--|--. H .--|--. H
H | H | H | H | H
H | #===============H===============# | H
H | H | H
#=====================================================================#
Finally, for _ten_ letters, I arrange 4 Octoliteral Diagrams, like the
above, in a square, assigning them to the 4 Classes mn, mn', m'n, m'n'.
We have now got 1024 Cells.
Sec. 8.
_Solution of a Syllogism by various Methods._
The best way, I think, to exhibit the differences between these various
Methods of solving Syllogisms, will be to take a concrete example, and
solve it by each Method in turn. Let us take, as our example, No. 29
(see p. 102).
"No philosophers are conceited;
Some conceited persons are not gamblers.
.'. Some persons, who are not gamblers, are not philosophers."
pg180
(1) _Solution by ordinary Method._
These Premisses, as they stand, will give no Conclusion, as they are
both negative.
If by 'Permutation' or 'Obversion', we write the Minor Premiss thus,
'Some conceited persons are not-gamblers,'
we can get a Conclusion in _Fresison_, viz.
"No philosophers are conceited;
Some conceited persons are not-gamblers.
.'. Some not-gamblers are not philosophers"
This can be proved by reduction to _Ferio_, thus:--
"No conceited persons are philosophers;
Some not-gamblers are conceited.
.'. Some not-gamblers are not philosophers".
The validity of _Ferio_ follows directly from the Axiom '_De Omni et
Nullo_'.
(2) _Symbolic Representation._
Before proceeding to discuss other Methods of Solution, it is necessary
to translate our Syllogism into an _abstract_ form.
Let us take "persons" as our 'Universe of Discourse'; and let
x = "philosophers", m = "conceited", and y = "gamblers."
Then the Syllogism may be written thus:--
"No x are m;
Some m are y'.
.'. Some y' are x'."
(3) _Solution by Euler's Method of Diagrams._
The Major Premiss requires only _one_ Diagram, viz.
1
_____ _____
_/ \_ _/ \_
/ \ / \
| x | | m |
\_ _/ \_ _/
\_____/ \_____/
pg181
The Minor requires _three_, viz.
2
_____ _____
_/ \_ _/ \_
/ \ / \
| y | | m |
\_ _/ \_ _/
\_____/ \_____/
3
_____ _____
_/ \_/ \_
/ / \ \
| y | | m |
\_ \_/ _/
\_____/ \_____/
4
_____
_/ ___ \_
/ / y \ \
| \___/ |
\_ m _/
\_____/
The combination of Major and Minor, in every possible way requires
_nine_, viz.
Figs. 1 and 2 give
5
_____ _____ _____
_/ \_ _/ \_ _/ \_
/ \ / \ / \
| x | | y | | m |
\_ _/ \_ _/ \_ _/
\_____/ \_____/ \_____/
6
_____ _____ _____
_/ \_/ \_ _/ \_
/ / \ \ / \
| x | | y | | m |
\_ \_/ _/ \_ _/
\_____/ \_____/ \_____/
7
_____ _____
_/ \_ _/ \_
/ \ / \
| xy | | m |
\_ _/ \_ _/
\_____/ \_____/
8
_____ _____
_/ ___ \_ _/ \_
/ / x \ \ / \
| \___/ | | m |
\_ y _/ \_ _/
\_____/ \_____/
9
_____ _____
_/ ___ \_ _/ \_
/ / y \ \ / \
| \___/ | | m |
\_ x _/ \_ _/
\_____/ \_____/
Figs. 1 and 3 give
10
_____ _____ _____
_/ \_ _/ \_/ \_
/ \ / / \ \
| x | | y | | m |
\_ _/ \_ \_/ _/
\_____/ \_____/ \_____/
11
_____ _____ _____
_/ \_/ \_/ \_
/ / \ / \ \
| x | | y | | m |
\_ \_/ \_/ _/
\_____/ \_____/ \_____/
12
_____ _____
_/___ \_/ \_
/ / x \ / \ \
| \___/| | m |
\_ y \_/ _/
\_____/ \_____/
Figs. 1 and 4 give
13
_____ _____
_/ \_ _/ ___ \_
/ \ / / y \ \
| x | | \___/ |
\_ _/ \_ m _/
\_____/ \_____/
From this group (Figs. 5 to 13) we have, by disregarding m, to find the
relation of x and y. On examination we find that Figs. 5, 10, 13 express
the relation of entire mutual exclusion; that Figs. 6, 11 express
partial inclusion and partial exclusion; that Fig. 7 expresses
coincidence; that Figs. 8, 12 express entire inclusion of x in y; and
that Fig. 9 expresses entire inclusion of y in x.
pg182
We thus get five Biliteral Diagrams for x and y, viz.
14
_____ _____
_/ \_ _/ \_
/ \ / \
| x | | y |
\_ _/ \_ _/
\_____/ \_____/
15
_____ _____
_/ \_/ \_
/ / \ \
| x | | y |
\_ \_/ _/
\_____/ \_____/
16
_____
_/ \_
/ \
| xy |
\_ _/
\_____/
17
_____
_/ ___ \_
/ / x \ \
| \___/ |
\_ y _/
\_____/
18
_____
_/ ___ \_
/ / y \ \
| \___/ |
\_ x _/
\_____/
where the only Proposition, represented by them all, is "Some not-y are
not-x," i.e. "Some persons, who are not gamblers, are not
philosophers"----a result which Euler would hardly have regarded as a
_valuable_ one, since he seems to have assumed that a Proposition of
this form is _always_ true!
(4) _Solution by Venn's Method of Diagrams._
The following Solution has been kindly supplied to me Mr. Venn himself.
"The Minor Premiss declares that some of the constituents in my' must be
saved: mark these constituents with a cross.
_____
_/ + \_
_/___ ___\_
_/|##+#\_/ |\_
/ \###/#\ _/ \
| + \|_#_|/ |
\_ m \_/ y _/
\_____/ \_____/
The Major declares that all xm must be destroyed; erase it.
Then, as some my' is to be saved, it must clearly be my'x'. That is,
there must exist my'x'; or eliminating m, y'x'. In common phraseology,
'Some y' are x',' or, 'Some not-gamblers are not-philosophers.'"
pg183
(5) _Solution by my Method of Diagrams._
The first Premiss asserts that no xm exist: so we mark the
xm-Compartment as empty, by placing a 'O' in each of its Cells.
The second asserts that some my' exist: so we mark the my'-Compartment
as occupied, by placing a 'I' in its only available Cell.
.---------------.
| | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| | |(I)| |
| .---|---. |
| | |
.---------------.
The only information, that this gives us as to x and y, is that the
x'y'-Compartment is _occupied_, i.e. that some x'y' exist.
Hence "Some x' are y'": i.e. "Some persons, who are not philosophers,
are not gamblers".
(6) _Solution by my Method of Subscripts._
xm_{0} + my'_{1} > x'y'_{1}
i.e. "Some persons, who are not philosophers, are not gamblers."
Sec. 9.
_My Method of treating Syllogisms and Sorites._
Of all the strange things, that are to be met with in the ordinary
text-books of Formal Logic, perhaps the strangest is the violent
contrast one finds to exist between their ways of dealing with these two
subjects. While they have elaborately discussed no less than _nineteen_
different forms of _Syllogisms_----each with its own special and
exasperating Rules, while the whole constitute an almost useless
machine, for practical purposes, many of the Conclusions being
incomplete, and many quite legitimate forms being ignored----they have
limited _Sorites_ to _two_ forms only, of childish simplicity; and these
they have dignified with special _names_, apparently under the
impression that no other possible forms existed!
As to _Syllogisms_, I find that their nineteen forms, with about a score
of others which they have ignored, can all be arranged under _three_
forms, each with a very simple Rule of its own; and the only question
the Reader has to settle, in working any one of the 101 Examples given
at p. 101 of this book, is "Does it belong to Fig. I., II., or III.?"
pg184
As to _Sorites_, the only two forms, recognised by the text-books, are
the _Aristotelian_, whose Premisses are a series of Propositions in A,
so arranged that the Predicate of each is the Subject of the next, and
the _Goclenian_, whose Premisses are the very same series, written
backwards. Goclenius, it seems, was the first who noticed the startling
fact that it does not affect the force of a Syllogism to invert the
order of its Premisses, and who applied this discovery to a Sorites. If
we assume (as surely we may?) that he is the _same_ man as that
transcendent genius who first noticed that 4 times 5 is the same thing
as 5 times 4, we may apply to him what somebody (Edmund Yates, I think
it was) has said of Tupper, viz., "here is a man who, beyond all others
of his generation, has been favoured with Glimpses of the Obvious!"
These puerile----not to say infantine----forms of a Sorites I have, in
this book, ignored from the very first, and have not only admitted
freely Propositions in _E_, but have purposely stated the Premisses in
random order, leaving to the Reader the useful task of arranging them,
for himself, in an order which can be worked as a series of regular
Syllogisms. In doing this, he can begin with _any one_ of them he likes.
I have tabulated, for curiosity, the various orders in which the
Premisses of the Aristotelian Sorites
1. All a are b;
2. All b are c;
3. All c are d;
4. All d are e;
5. All e are h.
.'. All a are h.
may be syllogistically arranged, and I find there are no less than
_sixteen_ such orders, viz., 12345, 21345, 23145, 23415, 23451, 32145,
32415, 32451, 34215, 34251, 34521, 43215, 43251, 43521, 45321, 54321. Of
these the _first_ and the _last_ have been dignified with names; but the
other _fourteen_----first enumerated by an obscure Writer on Logic,
towards the end of the Nineteenth Century----remain without a name!
pg185
Sec. 10.
_Some account of Parts II, III._
In Part II. will be found some of the matters mentioned in this
Appendix, viz., the "Existential Import" of Propositions, the use of a
_negative_ Copula, and the theory that "two negative Premisses prove
nothing." I shall also extend the range of Syllogisms, by introducing
Propositions containing alternatives (such as "Not-all x are y"),
Propositions containing 3 or more Terms (such as "All ab are c", which,
taken along with "Some bc' are d", would prove "Some d are a'"), &c. I
shall also discuss Sorites containing Entities, and the _very_ puzzling
subjects of Hypotheticals and Dilemmas. I hope, in the course of Part
II., to go over all the ground usually traversed in the text-books used
in our Schools and Universities, and to enable my Readers to solve
Problems of the same kind as, and far harder than, those that are at
present set in their Examinations.
In Part III. I hope to deal with many curious and out-of-the-way
subjects, some of which are not even alluded to in any of the treatises
I have met with. In this Part will be found such matters as the Analysis
of Propositions into their Elements (let the Reader, who has never gone
into this branch of the subject, try to make out for himself what
_additional_ Proposition would be needed to convert "Some a are b" into
"Some a are bc"), the treatment of Numerical and Geometrical Problems,
the construction of Problems, and the solution of Syllogisms and Sorites
containing Propositions more complex than any that I have used in Part
II.
I will conclude with eight Problems, as a taste of what is coming in
Part II. I shall be very glad to receive, from any Reader, who thinks he
has solved any one of them (more especially if he has done so _without_
using any Method of Symbols), what he conceives to be its complete
Conclusion.
It may be well to explain what I mean by the _complete_ Conclusion of a
Syllogism or a Sorites. I distinguish their Terms as being of two
kinds----those which _can_ be eliminated (e.g. the Middle Term of a
Syllogism), which I call the "Eliminands," and those which _cannot_,
which I call the "Retinends"; and I do not call the Conclusion
_complete_, unless it states _all_ the relations among the Retinends
only, which can be deduced from the Premisses.
pg186
1.
All the boys, in a certain School, sit together in one large room every
evening. They are of no less than _five_ nationalities----English,
Scotch, Welsh, Irish, and German. One of the Monitors (who is a great
reader of Wilkie Collins' novels) is very observant, and takes MS. notes
of almost everything that happens, with the view of being a good
sensational witness, in case any conspiracy to commit a murder should be
on foot. The following are some of his notes:--
(1) Whenever some of the English boys are singing "Rule Britannia", and
some not, some of the Monitors are wide-awake;
(2) Whenever some of the Scotch are dancing reels, and some of the Irish
fighting, some of the Welsh are eating toasted cheese;
(3) Whenever all the Germans are playing chess, some of the Eleven are
_not_ oiling their bats;
(4) Whenever some of the Monitors are asleep, and some not, some of the
Irish are fighting;
(5) Whenever some of the Germans are playing chess, and none of the
Scotch are dancing reels, some of the Welsh are _not_ eating toasted
cheese;
(6) Whenever some of the Scotch are _not_ dancing reels, and some of the
Irish _not_ fighting, some of the Germans are playing chess;
(7) Whenever some of the Monitors are awake, and some of the Welsh are
eating toasted cheese, none of the Scotch are dancing reels;
(8) Whenever some of the Germans are _not_ playing chess, and some of
the Welsh are _not_ eating toasted cheese, none of the Irish are
fighting;
pg187
(9) Whenever all the English are singing "Rule Britannia," and some of
the Scotch are _not_ dancing reels, none of the Germans are playing
chess;
(10) Whenever some of the English are singing "Rule Britannia", and some
of the Monitors are asleep, some of the Irish are _not_ fighting;
(11) Whenever some of the Monitors are awake, and some of the Eleven are
_not_ oiling their bats, some of the Scotch are dancing reels;
(12) Whenever some of the English are singing "Rule Britannia", and some
of the Scotch are _not_ dancing reels, * * * *
Here the MS. breaks off suddenly. The Problem is to complete the
sentence, if possible.
[N.B. In solving this Problem, it is necessary to remember that
the Proposition "All x are y" is a _Double_ Proposition, and is
equivalent to "Some x are y, and none are y'." See p. 17.]
2.
(1) A logician, who eats pork-chops for supper, will probably lose
money;
(2) A gambler, whose appetite is not ravenous, will probably lose money;
(3) A man who is depressed, having lost money and being likely to lose
more, always rises at 5 a.m.;
(4) A man, who neither gambles nor eats pork-chops for supper, is sure
to have a ravenous appetite;
(5) A lively man, who goes to bed before 4 a.m., had better take to
cab-driving;
(6) A man with a ravenous appetite, who has not lost money and does not
rise at 5 a.m., always eats pork-chops for supper;
(7) A logician, who is in danger of losing money, had better take to
cab-driving;
(8) An earnest gambler, who is depressed though he has not lost money,
is in no danger of losing any;
(9) A man, who does not gamble, and whose appetite is not ravenous, is
always lively;
pg188
(10) A lively logician, who is really in earnest, is in no danger of
losing money;
(11) A man with a ravenous appetite has no need to take to cab-driving,
if he is really in earnest;
(12) A gambler, who is depressed though in no danger of losing money,
sits up till 4 a.m.
(13) A man, who has lost money and does not eat pork-chops for supper,
had better take to cab-driving, unless he gets up at 5 a.m.
(14) A gambler, who goes to bed before 4 a.m., need not take to
cab-driving, unless he has a ravenous appetite;
(15) A man with a ravenous appetite, who is depressed though in no
danger of losing, is a gambler.
Univ. "men"; a = earnest; b = eating pork-chops for supper;
c = gamblers; d = getting up at 5; e = having lost money; h = having a
ravenous appetite; k = likely to lose money; l = lively; m = logicians;
n = men who had better take to cab-driving; r = sitting up till 4.
[N.B. In this Problem, clauses, beginning with "though", are
intended to be treated as _essential_ parts of the Propositions
in which they occur, just as if they had begun with "and".]
3.
(1) When the day is fine, I tell Froggy "You're quite the dandy, old
chap!";
(2) Whenever I let Froggy forget that L10 he owes me, and he begins to
strut about like a peacock, his mother declares "He shall _not_ go out
a-wooing!";
(3) Now that Froggy's hair is out of curl, he has put away his gorgeous
waistcoat;
(4) Whenever I go out on the roof to enjoy a quiet cigar, I'm sure to
discover that my purse is empty;
(5) When my tailor calls with his little bill, and I remind Froggy of
that L10 he owes me, he does _not_ grin like a hyaena;
pg189
(6) When it is very hot, the thermometer is high;
(7) When the day is fine, and I'm not in the humour for a cigar, and
Froggy is grinning like a hyaena, I never venture to hint that he's quite
the dandy;
(8) When my tailor calls with his little bill and finds me with an empty
purse, I remind Froggy of that L10 he owes me;
(9) My railway-shares are going up like anything!
(10) When my purse is empty, and when, noticing that Froggy has got his
gorgeous waistcoat on, I venture to remind him of that L10 he owes me,
things are apt to get rather warm;
(11) Now that it looks like rain, and Froggy is grinning like a hyaena, I
can do without my cigar;
(12) When the thermometer is high, you need not trouble yourself to take
an umbrella;
(13) When Froggy has his gorgeous waistcoat on, but is _not_ strutting
about like a peacock, I betake myself to a quiet cigar;
(14) When I tell Froggy that he's quite the dandy, he grins like a
hyaena;
(15) When my purse is tolerably full, and Froggy's hair is one mass of
curls, and when he is _not_ strutting about like a peacock, I go out on
the roof;
(16) When my railway-shares are going up, and when it is chilly and
looks like rain, I have a quiet cigar;
(17) When Froggy's mother lets him go a-wooing, he seems nearly mad with
joy, and puts on a waistcoat that is gorgeous beyond words;
(18) When it is going to rain, and I am having a quiet cigar, and Froggy
is _not_ intending to go a-wooing, you had better take an umbrella;
(19) When my railway-shares are going up, and Froggy seems nearly mad
with joy, _that_ is the time my tailor always chooses for calling with
his little bill;
(20) When the day is cool and the thermometer low, and I say nothing to
Froggy about his being quite the dandy, and there's not the ghost of a
grin on his face, I haven't the heart for my cigar!
pg190
4.
(1) Any one, fit to be an M.P., who is not always speaking, is a public
benefactor;
(2) Clear-headed people, who express themselves well, have had a good
education;
(3) A woman, who deserves praise, is one who can keep a secret;
(4) People, who benefit the public, but do not use their influence for
good purpose, are not fit to go into Parliament;
(5) People, who are worth their weight in gold and who deserve praise,
are always unassuming;
(6) Public benefactors, who use their influence for good objects,
deserve praise;
(7) People, who are unpopular and not worth their weight in gold, never
can keep a secret;
(8) People, who can talk for ever and are fit to be Members of
Parliament, deserve praise;
(9) Any one, who can keep a secret and who is unassuming, is a
never-to-be-forgotten public benefactor;
(10) A woman, who benefits the public, is always popular;
(11) People, who are worth their weight in gold, who never leave off
talking, and whom it is impossible to forget, are just the people whose
photographs are in all the shop-windows;
(12) An ill-educated woman, who is not clear-headed, is not fit to go
into Parliament;
(13) Any one, who can keep a secret and is not for ever talking, is sure
to be unpopular;
(14) A clear-headed person, who has influence and uses it for good
objects, is a public benefactor;
(15) A public benefactor, who is unassuming, is not the sort of person
whose photograph is in every shop-window;
(16) People, who can keep a secret and who use their influence for good
purposes, are worth their weight in gold;
(17) A person, who has no power of expression and who cannot influence
others, is certainly not a _woman_;
pg191
(18) People, who are popular and worthy of praise, either are public
benefactors or else are unassuming.
Univ. "persons"; a = able to keep a secret; b = clear-headed;
c = constantly talking; d = deserving praise; e = exhibited in
shop-windows; h = expressing oneself well; k = fit to be an M.P.;
l = influential; m = never-to-be-forgotten; n = popular; r = public
benefactors; s = unassuming; t = using one's influence for good objects;
v = well-educated; w = women; z = worth one's weight in gold.
5.
Six friends, and their six wives, are staying in the same hotel; and
they all walk out daily, in parties of various size and composition. To
ensure variety in these daily walks, they have agree to observe the
following Rules:--
(1) If Acres is with (i.e. is in the same party with) his wife, and
Barry with his, and Eden with Mrs. Hall, Cole must be with Mrs. Dix;
(2) If Acres is with his wife, and Hall with his, and Barry with Mrs.
Cole, Dix must _not_ be with Mrs. Eden;
(3) If Cole and Dix and their wives are all in the same party, and Acres
_not_ with Mrs. Barry, Eden must _not_ be with Mrs. Hall;
(4) If Acres is with his wife, and Dix with his, and Barry _not_ with
Mrs. Cole, Eden must be with Mrs. Hall;
(5) If Eden is with his wife, and Hall with his, and Cole with Mrs. Dix,
Acres must _not_ be with Mrs. Barry;
(6) If Barry and Cole and their wives are all in the same party, and
Eden _not_ with Mrs. Hall, Dix must be with Mrs. Eden.
The Problem is to prove that there must be, every day, at least _one_
married couple who are not in the same party.
pg192
6.
After the six friends, named in Problem 5, had returned from their tour,
three of them, Barry, Cole, and Dix, agreed, with two other friends of
theirs, Lang and Mill, that the five should meet, every day, at a
certain _table d'hote_. Remembering how much amusement they had derived
from their code of rules for walking-parties, they devised the following
rules to be observed whenever beef appeared on the table:--
(1) If Barry takes salt, then either Cole or Lang takes _one_ only of
the two condiments, salt and mustard: if he takes mustard, then either
Dix takes neither condiment, or Mill takes both.
(2) If Cole takes salt, then either Barry takes only _one_ condiment, or
Mill takes neither: if he takes mustard, then either Dix or Lang takes
both.
(3) If Dix takes salt, then either Barry takes neither condiment or Cole
take both: if he takes mustard, then either Lang or Mill takes neither.
(4) If Lang takes salt, then Barry or Dix takes only _one_ condiment: if
he takes mustard, then either Cole or Mill takes neither.
(5) If Mill takes salt, then either Barry or Lang takes both condiments:
if he takes mustard, then either Cole or Dix takes only _one_.
The Problem is to discover whether these rules are _compatible_; and, if
so, what arrangements are possible.
[N.B. In this Problem, it is assumed that the phrase "if Barry
takes salt" allows of _two_ possible cases, viz. (1) "he takes
salt _only_"; (2) "he takes _both_ condiments". And so with all
similar phrases.
It is also assumed that the phrase "either Cole or Lang takes
_one_ only of the two condiments" allows _three_ possible cases,
viz. (1) "Cole takes _one_ only, Lang takes both or neither";
(2) "Cole takes both or neither, Lang takes _one_ only"; (3)
"Cole takes _one_ only, Lang takes _one_ only". And so with all
similar phrases.
It is also assumed that every rule is to be understood as
implying the words "and _vice versa_." Thus the first rule would
imply the addition "and, if either Cole or Lang takes only _one_
condiment, then Barry takes salt."]
pg193
7.
(1) Brothers, who are much admired, are apt to be self-conscious;
(2) When two men of the same height are on opposite sides in Politics,
if one of them has his admirers, so also has the other;
(3) Brothers, who avoid general Society, look well when walking
together;
(4) Whenever you find two men, who differ in Politics and in their views
of Society, and who are not both of them ugly, you may be sure that they
look well when walking together;
(5) Ugly men, who look well when walking together, are not both of them
free from self-consciousness;
(6) Brothers, who differs in Politics, and are not both of them
handsome, never give themselves airs;
(7) John declines to go into Society, but never gives himself airs;
(8) Brothers, who are apt to be self-conscious, though not _both_ of
them handsome, usually dislike Society;
(9) Men of the same height, who do not give themselves airs, are free
from self-consciousness;
(10) Men, who agree on questions of Art, though they differ in Politics,
and who are not both of them ugly, are always admired;
(11) Men, who hold opposite views about Art and are not admired, always
give themselves airs;
(12) Brothers of the same height always differ in Politics;
(13) Two handsome men, who are neither both of them admired nor both of
them self-conscious, are no doubt of different heights;
(14) Brothers, who are self-conscious, and do not both of them like
Society, never look well when walking together.
[N.B. See Note at end of Problem 2.]
pg194
8.
(1) A man can always master his father;
(2) An inferior of a man's uncle owes that man money;
(3) The father of an enemy of a friend of a man owes that man nothing;
(4) A man is always persecuted by his son's creditors;
(5) An inferior of the master of a man's son is senior to that man;
(6) A grandson of a man's junior is not his nephew;
(7) A servant of an inferior of a friend of a man's enemy is never
persecuted by that man;
(8) A friend of a superior of the master of a man's victim is that man's
enemy;
(9) An enemy of a persecutor of a servant of a man's father is that
man's friend.
The Problem is to deduce some fact about great-grandsons.
[N.B. In this Problem, it is assumed that all the men, here
referred to, live in the same town, and that every pair of them
are either "friends" or "enemies," that every pair are related
as "senior and junior", "superior and inferior", and that
certain pairs are related as "creditor and debtor", "father and
son", "master and servant", "persecutor and victim", "uncle and
nephew".]
9.
"Jack Sprat could eat no fat:
His wife could eat no lean:
And so, between them both,
They licked the platter clean."
Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion
to be proved. It is permitted to use, as Premisses, not only all that is
here _asserted_, but also all that we may reasonably understand to be
_implied_.
pg195
NOTES TO APPENDIX.
(A) [See p. 167, line 6.]
It may, perhaps, occur to the Reader, who has studied Formal Logic that
the argument, here applied to the Propositions I and E, will apply
equally well to the Propositions I and A (since, in the ordinary
text-books, the Propositions "All xy are z" and "Some xy are not z" are
regarded as Contradictories). Hence it may appear to him that the
argument might have been put as follows:--
"We now have I and A 'asserting.' Hence, if the Proposition 'All xy are
z' be true, some things exist with the Attributes x and y: i.e. 'Some x
are y.'
"Also we know that, if the Proposition 'Some xy are not-z' be true the
same result follows.
"But these two Propositions are Contradictories, so that one or other of
them _must_ be true. Hence this result is always true: i.e. the
Proposition 'Some x are y' is _always_ true!
"_Quod est absurdum._ Hence I _cannot_ assert."
This matter will be discussed in Part II; but I may as well give here
what seems to me to be an irresistable proof that this view (that _A_
and _I_ are Contradictories), though adopted in the ordinary text-books,
is untenable. The proof is as follows:--
With regard to the relationship existing between the Class 'xy' and the
two Classes 'z' and 'not-z', there are _four_ conceivable states of
things, viz.
(1) Some xy are z, and some are not-z;
(2) " " none "
(3) No xy " some "
(4) " " none "
Of these four, No. (2) is equivalent to "All xy are z", No. (3) is
equivalent to "All xy are not-z", and No. (4) is equivalent to "No xy
exist."
Now it is quite undeniable that, of these _four_ states of things, each
is, _a priori_, _possible_, some _one must_ be true, and the other three
_must_ be false.
Hence the Contradictory to (2) is "Either (1) or (3) or (4) is true."
Now the assertion "Either (1) or (3) is true" is equivalent to "Some xy
are not-z"; and the assertion "(4) is true" is equivalent to "No xy
exist." Hence the Contradictory to "All xy are z" may be expressed as
the Alternative Proposition "Either some xy are not-z, or no xy exist,"
but _not_ as the Categorical Proposition "Some y are not-z."
pg196
(B) [See p. 171, at end of Section 2.]
There are yet _other_ views current among "The Logicians", as to the
"Existential Import" of Propositions, which have not been mentioned in
this Section.
One is, that the Proposition "some x are y" is to be interpreted,
neither as "Some x _exist_ and are y", nor yet as "If there _were_ any x
in existence, some of them _would_ be y", but merely as "Some x _can be_
y; i.e. the Attributes x and y are _compatible_". On _this_ theory,
there would be nothing offensive in my telling my friend Jones "Some of
your brothers are swindlers"; since, if he indignantly retorted "What do
you _mean_ by such insulting language, you scoundrel?", I should calmly
reply "I merely mean that the thing is _conceivable_----that some of
your brothers _might possibly_ be swindlers". But it may well be doubted
whether such an explanation would _entirely_ appease the wrath of Jones!
Another view is, that the Proposition "All x are y" _sometimes_ implies
the actual _existence_ of x, and _sometimes_ does _not_ imply it; and
that we cannot tell, without having it in _concrete_ form, _which_
interpretation we are to give to it. _This_ view is, I think, strongly
supported by common usage; and it will be fully discussed in Part II:
but the difficulties, which it introduces, seem to me too formidable to
be even alluded to in Part I, which I am trying to make, as far as
possible, easily intelligible to mere _beginners_.
(C) [See p. 173, Sec. 4.]
The three Conclusions are
"No conceited child of mine is greedy";
"None of my boys could solve this problem";
"Some unlearned boys are not choristers."
pg197
INDEX.
Sec. 1.
_Tables._
I. Biliteral Diagram. Attributes of Classes, and
Compartments, or Cells, assigned to them 25
II. do. Representation of Uniliteral Propositions of
Existence 34
III. do. Representation of Biliteral Propositions of
Existence and of Relation 35
IV. Triliteral Diagram. Attributes of Classes, and
Compartments, or Cells, assigned to them 42
V. do. Representation of Particular and Universal
Negative Propositions, of Existence and of Relation,
in terms of x and m 46
VI. do. do., in terms of y and m 47
VII. do. Representation of Universal Affirmative
Propositions of Relation, in terms of x and m 48
VIII. do. do. in terms of y and m 49
IX. Method of Subscripts. Formulae and Rules for Syllogisms 78
Sec. 2.
_Words &c. explained._
'Abstract' Proposition 59
'Adjuncts' 1
'Affirmative' Proposition 10
'Attributes' 1
'Biliteral' Diagram 22
" Proposition 27
'Class' 1 1/2
Classes, arbitrary limits of 3 1/2
" , subdivision of 4
pg198
'Classification' 1 1/2
'Codivisional' Classes 3
'Complete' Conclusion of a Sorites 85
'Conclusion' of a Sorites "
" " Syllogism 56
'Concrete' Proposition 59
'Consequent' in a Sorites 85
" " Syllogism 56
'Converse' Propositions 31
'Conversion' of a Proposition "
'Copula' of a Proposition 9
'Definition' 6
'Dichotomy' 3 1/2
'Differentia' 1 1/2
'Division' 3
'Eliminands' of a Sorites 85
" " Syllogism 56
'Entity' 70
'Equivalent' Propositions 17
'Fallacy' 81
'Genus' 1 1/2
'Imaginary' Class "
" Name 4 1/2
'Individual' 2
'Like', and 'Unlike', Signs of Terms 70
'Name' 4
'Negative' Proposition 10
'Normal' form of a Proposition 9
" " of Existence 11
" " of Relation 12
'Nullity' 70
'Partial' Conclusion of a Sorites 85
'Particular' Proposition 9
'Peculiar' Attributes 1 1/2
'Predicate' of a Proposition 9
" of a Proposition of Existence 11
" " Relation 12
'Premisses' of a Sorites 85
" " Syllogism 56
pg199
'Proposition' 8
" 'in _I_', 'in _E_', and 'in _A_' 9
" 'in terms of' certain Letters 27
" of Existence 11
" of Relation 12
'Real' Class 1 1/2
'Retinends' of a Sorites 85
" " Syllogism 56
'Sign of Quantity' in a Proposition 9
'Sitting on the Fence' 26
'Some', technical meaning of 8
'Sorites' 85
'Species' 1 1/2
'Subject' of a Proposition 9
" " of Existence 11
" " of Relation 12
'Subscripts' of Terms 70
'Syllogism' 56
Symbol ".'." "
" "+" and ">" 70
'Terms' of a Proposition 9
'Things' 1
Translation of Proposition from 'concrete' to 'abstract' 59
" " 'abstract' to 'subscript' 72
'Triliteral' Diagram 39
'Underscoring' of letters 91
'Uniliteral' Proposition 27
'Universal' " 10
'Universe of Discourse' (or 'Univ.') 12
'Unreal' Class 1 1/2
'Unreal' Name 4 1/2
pg200
THE END.
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N.B.--This book contains 395 pages--nearly as much as the two
Alice' books put together.
SYLVIE AND BRUNO CONCLUDED. With Forty-six Illustrations by HARRY
FURNISS. (First published in 1893.) Crown 8vo, cloth, gilt edges, price
7_s._ 6_d._ net. Third Thousand.
N.B.--This book contains 411 pages.
ORIGINAL GAMES AND PUZZLES. Crown 8vo, cloth, gilt
edges. [_In preparation._
THREE SUNSETS, and Other Poems. With Twelve Illustrations by E. GERTRUDE
THOMSON. Fcap. 4to, cloth, gilt edges.
N.B.--This will be a reprint, possibly with a few additions, of the
serious portion of "Phantasmagoria, and other Poems," published in 1869.
* * * * *
ADVICE TO WRITERS.
Buy "THE WONDERLAND CASE FOR POSTAGE-STAMPS," invented by LEWIS CARROL,
Oct. 29, 1888, size 4 inches by 3, containing 12 separate pockets for
stamps of different values, 2 Coloured Pictorial Surprises taken
from _Alice in Wonderland_, and 8 or 9 Wise Words about Letter-Writing.
It is published by Messrs. EMBERLIN & SON, 4 Magdalene Street, Oxford.
Price 1_s._
N.B.--If ordered by Post, an additional payment will be required, to
cover cost of postage, as follows:--
One copy, 1 1/2_d._ Two or three do., 2_d._ Four do., 2 1/2_d._ Five to
fourteen do., 3_d._ Each subsequent fourteen or fraction thereof, 1 1/2_d._
* * * * *
* * * * *
Transcriber's Note
This book makes extensive use of page references. To assist the reader,
page markers in the forms "pg-ix", "pg193" & "px-3" have been included
in the right margin at points corresponding closely to the tops of the
original pages. These may be searched for to locate the material
referred to. In the main section, these page markers are always given
with 3 digits including, if necessary, leading zeroes.
This book contains a large number of line drawn illustrations which are
un-credited. As these cannot be rendered here in the original manner
they have been reproduced as well as possible in the manner known as
"ASCII art".
In this ASCII file, the greater-than symbol (>) has bee substituted
for the Pilcrow symbol.
A number of transcription errors were found in the original book. As
these were clearly not part of the Author's intention they have, as far
as possible, been identified and corrected in accordance with the
methods given by the Author. These corrections are listed here below
with their locations and original text. In these notes the word
'natural' identifies a letter symbol occurring without a prime mark.
Page viii: "189" corrected to "194".
Page viii: "188" corrected to "192".
Page xv, '=Real=' and '=Unreal=', or '=Imaginary=', Classes: "1 1/2"
corrected to "2".
Page xix, Propositions beginning with "Some": "18" corrected to "19".
Page xix, Rules: "19" corrected to "20".
Page xix, The West and East Halves ...: "24" corrected to "23".
Page xxi, The Proposition "All x are y" ...: was originally shown as
occurring in page 34.
Page xxii, And of three other similar arrangements: "37" corrected to
"36".
Page xxiii, The Proposition 'No xm exist': "43" corrected to "44".
Page 060: "contruct" corrected to "construct".
Page 094, Paper V: missing word "it" supplied.
Page 111, #28: "No h are k-natural" corrected to "No h are k-prime".
Page 111, #30: "No a are h-natural" corrected to "No a are h-prime".
Page 111, #30: "No c are n-natural" corrected to "No c are n-prime".
Page 123, #57 (d): "mortally offended if I fail to notice them"
corrected to "mortally offended".
Page 128, #34: "Some x-prime are y" corrected to "Some x-natural are y".
Page 146, #30: "x_{1}m-prime_{0}" corrected to "x_{1}m-natural_{0}".
Page 147, #32: "xm-natural_{0}" corrected to "xm-prime_{0}".
Page 156, #18 (4/1): "be" corrected to "be-sub:zero".
Page 156, #22 (5/1): "ch" corrected to "ch-sub:zero".
Page 157, #23 (5/2): second underline corrected from single to double.
Page 157, #23 (2/2): first underline corrected from single to double.
Page 157, #23 (>2): "h_{1}c" corrected to "h_{1}c-sub:zero".
Page 157, #26 (5/1): "a_{1}c-natural_{0}" corrected to
"a_{1}c-prime_{0}".
Page 157, #26 (>2): "e_{1}c-natural_{0}" corrected to
"e_{1}c-prime_{0}".
Page 157, #29 (8/2): first underline corrected from single to double.
Page 161, #36 (4/2): "e'd-prime" corrected to "e'd-natural".
Page 161, #39 (1/2): "c-prime+d" corrected to "c-natural+d".
Page 161, #40: "a" and "b" interchanged.
Page 161, #43 (1/1): "b-natural_{1}e_{0}" corrected to
"b-prime_{1}e_{0}".
Page 162, #52 (3/1): "h'k-prime_{0}" corrected to "h'k-natural_{0}".
Page 163, #55 (3/2): "c'd-prime" corrected to "c'd-natural".
Page 163, #57 (1/2): "h-natural+d'" corrected to "h-prime+d'".
In the original book at the top of page 97, the following text occurred:
[N.B. The numbers at the foot of each page indicate the pages
where the corresponding matter may be found.]
In accordance with the un-paged medium here this has been changed to:
[N.B. Reference tags for Examples, Answers & Solutions will be
found in the right margin.]
The part of the book to which this relates contains, by sections,
"Examples" (Exercises for the student), "Answers" (to the Examples) &
"Solutions" (Worked Answers). All Example sections have corresponding
Answer sections. For Sections 2 & 3, worked Solutions are not supplied;
for Sections 4-7, Solutions are given by 2 different methods. In
association with this, the original text contained editorial notes at
the foot of each page giving the page numbers for the related Sections.
In this version, these notes are replaced with marginal tags such as
EX3, AN4, SL5, & SL6-A and SL7-B which are placed at the top of each
Section to identify the current location. As with page tags, these may
be searched for to locate the material refered to. (A search for "SL4"
should find successively both "SL4-A" & "SL4-B".) These tags are unique
regardless of case.