Complete Works of Lewis Carroll
Page 121
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———-
———-
| | |
12. No x are y. i.e. | 0 | |
| | |
———-
———-
| | |
13. Some x are y, and some are y'. i.e. | 1 | 1 |
| | |
———-
———-
| | |
14. All x are y'. i.e. | 0 | 1 |
| | |
———-
—-
| |
15. No y are x'. i.e. |—-|
| 0 |
—-
—-
| 1 |
16. All y are x. i.e. |—-|
| 0 |
—-
—-
| 0 |
17. No y exist. i.e. |—-|
| 0 |
—-
—-
| |
18. Some y are x'. i.e. |—-|
| 1 |
—-
—-
| |
15. Some y exist. i.e. |-1-|
| |
—-
3. Half of Smaller Diagram.
Symbols interpreted.
__________
1. No x are y'.
2. No x exist.
3. Some x exist.
4. All x are y'.
5. Some x are y. i.e. Some good riddles are hard.
6. All x are y. i.e. All good riddles are hard.
7. No x exist. i.e. No riddles are good.
8. No x are y. i.e. No good riddles are hard.
9. Some x are y'. i.e. Some lobsters are unselfish.
10. No x are y. i.e. No lobsters are selfish.
11. All x are y'. i.e. All lobsters are unselfish.
12. Some x are y, and some are y'. i.e. Some lobsters are selfish, and some are unselfish.
13. All y' are x'. i.e. All invalids are unhappy.
14. Some y' exist. i.e. Some people are unhealthy.
15. Some y' are x, and some are x'. i.e. Some invalids are happy, and some are unhappy.
16. No y' exist. i.e. Nobody is unhealthy.
4. Smaller Diagram.
Propositions represented.
__________
———- ———-
| 1 | | | | |
1. |—-|—-| 2. |—-|—-|
| 0 | | | 1 | |
———- ———-
———- ———-
| | | | | 1 |
3. |—-|—-| 4. |—-|—-|
| | 0 | | | |
———- ———-
———- ———-
| | 1 | | | |
5. |—-|—-| 6. |—-|—-|
| | | | 0 | |
———- ———-
———- ———-
| | | | | |
7. |—-|—-| 8. |—-|—-|
| | 1 | | 0 | 1 |
———- ———-
———- ———-
| | | | | |
9. |—-|-1-| 10. |—-|—-|
| | | | 0 | 0 |
———- ———-
———- ———-
| 1 | | | 1 | 0 |
11. |—-|—-| 12. |—-|—-|
| 1 | | | | 1 |
———- ———-
———-
| | |
13. No x' are y. i.e. |—-|—-|
| 0 | |
———-
———-
| | 0 |
14. All y' are x'. i.e. |—-|—-|
| | 1 |
———-
———-
| | |
15. Some y' exist. i.e. |—-|-1-|
| | |
———-
———-
| 1 | 0 |
16. All y are x, and all x are y. i.e. |—-|—-|
| 0 | |
———-
———-
| | |
17. No x' exist. i.e. |—-|—-|
| 0 | 0 |
———-
———-
| 0 | 1 |
18. All x are y'. i.e. |—-|—-|
| | |
———-
———-
| 0 | |
19. No x are y. i.e. |—-|—-|
| | |
———-
———-
| | |
20. Some x' are y, and some are y'. i.e. |—-|—-|
| 1 | 1 |
———-
———-
| 0 | 1 |
21. No y exist, and some x exist. i.e. |—-|—-|
| 0 | |
———-
———-
| | 1 |
22. All x' are y, and all y' are x. i.e. |—-|—-|
| 1 | 0 |
———-
———-
| 1 | |
17. Some x are y, and some x' are y'. i.e. |—-|—-|
| | 1 |
———-
5. Smaller Diagram.
Symbols interpreted.
__________
1. Some y are not-x, or, Some not-x are y.
2. No not-x are not-y, or, No not-y are not-x.
3. No not-y are x.
4. No not-x exist. i.e. No Things are not-x.
5. No y exist. i.e. No houses are two-storied.
6. Some x' exist. i.e. Some houses are not built of brick.
7. No x are y'. Or, no y' are x. i.e. No houses, built of brick, are other than two-storied. Or, no houses, that are not two-storied, are built of brick.
8. All x' are y'. i.e. All houses, that are not built of brick, are not two-storied.
9. Some x are y, and some are y'. i.e. Some fat boys are active, and some are not.
10. All y' are x'. i.e. All lazy boys are thin.
11. All x are y', and all y' are x. i.e. All fat boys are lazy, and all lazy ones are fat.
12. All y are x, and all x' are y. i.e. All active boys are fat, and all thin ones are lazy.
13. No x exist, and no y' exist. i.e. No cats have green eyes, and none have bad tempers.
14. Some x are y', and some x' are y. Or some y are x', and some y' are x. i.e. Some green-eyed cats are bad-tempered, and some, that have not green eyes, are good-tempered. Or, some good-tempered cats have not green eyes, and some bad-tempered ones have green eyes.
15. Some x are y, and no x' are y'. Or, some y are x, and no y' are x'. i.e. Some green-eyed cats are good-tempered, and none, that are not green-eyed, are bad-tempered. Or, some good-tempered cats have green eyes, and none, that are bad-tempered, have not green eyes.
16. All x are y', and all x' are y. Or, all y are x', and all y' are x. i.e. All green-eyed cats are bad-tempered and all, that have not green eyes, are good-tempered. Or, all good-tempered ones have eyes that are not green, and all bad-tempered ones have green eyes.
6. Larger Diagram.
Propositions represented.
__________
———————- ———————-
| | | | | |
| —-|—- | | —-|—- |
| | 0 | 0 | | | | | | |
1. |—-|—-|—-|—-| 2. |-1-|—-|—-|—-|
| | | | | | | | | |
| —-|—- | | —-|—- |
| | | | | |
———————- ———————-
———————- ———————-
| | | | | 0 |
| —-|—- | | —-|—- |
| | 0 | 0 | | | | | | |
3. |—-|—-|—-|—-| 4. |—-|—-|—-|—-|
| | - | | | | | | |
| —-|—- | | —-|—- |
| | | | | 0 |
———————- ———————-
———————- ———————-
| 0 | | | | |
| —-|—- | | —-|—- |
| | 0 | 0 | | | | 0 | 1 | |
5. |—-|—-|—-|—-| 6. |—-|—-
|—-|—-|
| | 1 | | | | | 0 | | |
| —-|—- | | —-|—- |
| 0 | | | | |
———————- ———————-
———————- ———————-
| | | | | 0 |
| —-|—- | | —-|—- |
| | 0 | 0 | | | | | | |
7. |—-|—-|—-|—-| 8. |—-|—-|—-|—-|
| | 0 | 1 | | | | 0 | 0 | |
| —-|—- | | —-|—- |
| | | | | 0 |
———————- ———————-
———————-
| | |
| —-|—- |
| | 0 | 0 | |
9. No x are m. i.e. |—-|—-|—-|—-|
| | 0 | | |
| —-|—- |
| | |
———————-
———————-
| | |
| —-|—- |
| | | | |
10. Some m' are y. i.e. |-1-|—-|—-|—-|
| | | | |
| —-|—- |
| | |
———————-
———————-
| | |
| —-|—- |
| | | 0 | |
11. All y' are m'. i.e. |—-|—-|—-|-1-|
| | | 0 | |
| —-|—- |
| | |
———————-
———————-
| | |
| —-|—- |
| | 0 | 0 | |
12. All m are x'. i.e. |—-|—-|—-|—-|
| | 1 | |
| —-|—- |
| | |
———————-
———————-
| 0 | |
| —-|—- |
| | 0 | 0 | |
13. No x are m; i.e. |—-|—-|—-|—-|
All y are m. | | 1 | | |
| —-|—- |
| 0 | |
———————-
———————-
| 0 | 0 |
| —-|—- |
| | | | |
14. All m' are y; i.e. |—-|—-|—-|—-|
No x are m'. | | | | |
| —-|—- |
| 1 | 0 |
———————-
———————-
| 0 | 0 |
| —-|—- |
| | 1 | 0 | |
15. All x are m; i.e. |—-|—-|—-|—-|
No m are y'. | | | 0 | |
| —-|—- |
| | |
———————-
———————-
| 0 | 0 |
| —-|—- |
| | | | |
16. All m' are y'; i.e. |—-|—-|—-|—-|
No x are m'. | | | | |
| —-|—- |
| 0 | 1 |
———————-
———————-
| 0 | 0 |
| —-|—- |
| | 1 | 0 | |
17. All x are m; i.e. |—-|—-|—-|—-|
All m are y. | | | 0 | |
| —-|—- |
[See remarks on No. 7, p. 60.] | | |
———————-
———————-
| 0 | |
| —-|—- |
| | | | |
18. No x' are m; i.e. |—-|—-|—-|—-|
No m' are y. | | 0 | 0 | |
| —-|—- |
| 0 | |
———————-
———————-
| | |
| —-|—- |
| | 1 | 0 | |
19. All m are x; i.e. |—-|—-|—-|—-|
All m are y. | | 0 | 0 | |
| —-|—- |
| | |
———————-
20. We had better take "persons" as Universe. We may choose "myself" as 'Middle Term', in which case the Premisses will take the form
I am a-person-who-sent-him-to-bring-a-kitten;
I am a-person-to-whom-he-brought-a-kettle-by-mistake.
Or we may choose "he" as 'Middle Term', in which case the Premisses will take the form
He is a-person-whom-I-sent-to-bring-me-a-kitten;
He is a-person-who-brought-me-a-kettle-by-mistake.
The latter form seems best, as the interest of the anecdote clearly depends on HIS stupidity—not on what happened to ME. Let us then make m = "he"; x = "persons whom I sent, &c."; and y = "persons who brought, &c."
Hence, All m are x;
All m are y. and the required Diagram is
———————- | | | | —-|—- | | | 1 | 0 | | |—-|—-|—-|—-| | | 0 | 0 | | | —-|—- | | | | ———————-
7. Both Diagrams employed.
———-
| 0 | |
1. |—-|—-| i.e. All y are x'.
| 1 | |
———-
———-
| | 1 |
2. |—-|—-| i.e. Some x are y'; or, Some y' are x.
| | |
———-
———-
| | |
3. |—-|—-| i.e. Some y are x'; or, Some x' are y.
| 1 | |
———-
———-
| | |
4. |—-|—-| i.e. No x' are y'; or, No y' are x'.
| | 0 |
———-
———-
| 0 | |
5. |—-|—-| i.e. All y are x'. i.e. All black rabbits
| 1 | | are young.
———-
———-
| | |
6. |—-|—-| i.e. Some y are x'. i.e. Some black
| 1 | | rabbits are young.
———-
———-
| 1 | 0 |
7. |—-|—-| i.e. All x are y. i.e. All well-fed birds
| | | are happy.
———-
———-
| | | i.e. Some x' are y'. i.e. Some birds,
8. |—-|—-| that are not well-fed, are unhappy;
| | 1 | or, Some unhappy birds are not
———- well-fed.
———-
| 1 | 0 |
9. |—-|—-| i.e. All x are y. i.e. John has got a
| | | tooth-ache.
———-
———-
| | |
10. |—-|—-| i.e. No x' are y. i.e. No one, but John,
| 0 | | has got a tooth-ache.
———-
———-
| 1 | |
11. |—-|—-| i.e. Some x are y. i.e. Some one, who
| | | has taken a walk, feels better.
———-
———-
| 1 | | i.e. Some x are y. i.e. Some one,
12. |—-|—-| whom I sent to bring me a kitten,
| | | brought me a kettle by mistake.
———-
———————-
| | 0 |
| —-|—- |
| | 0 | 0 | |
13. |-1-|—-|—-|—-| ———-
| | | | | | | 0 |
| —-|—- | |—-|—-|
| | 0 | | | |
———————- ———-
Let "books" be Universe; m="exciting",
x="that suit feverish patients"; y="that make
one drowsy".
No m are x; &there4 No y' are x.
All m' are y.
i.e. No books suit feverish patients, except such as make
one drowsy.
———————-
| | |
| —-|—- |
| | 1 | 0 | |
14. |—-|—-|—-|—-| ———-
| | | 0 | | | 1 | |
| —-|—- | |—-|—-|
| | | | | |
———————- ———-
Let "perso
ns" be Universe; m="that deserve the fair";
x="that get their deserts"; y="brave".
Some m are x; &there4 Some y are x.
No y' are m.
i.e. Some brave persons get their deserts.
———————-
| 0 | |
| —-|—- |
| | 0 | 0 | |
15. |—-|—-|—-|—-| ———-
| | | | | | 0 | |
| —-|—- | |—-|—-|
| 0 | | | | |
———————- ———-
Let "persons" be Universe; m="patient";
x="children"; y="that can sit still".
No x are m; &there4 No x are y.
No m' are y.
i.e. No children can sit still.
———————-
| 0 | 0 |
| —-|—- |
| | 0 | 1 | |
16. |—-|—-|—-|—-| ———-
| | 0 | | | | 0 | 1 |
| —-|—- | |—-|—-|
| | | | | |
———————- ———-
Let "things" be Universe; m="fat"; x="pigs";
y="skeletons".
All x are m; &there4 All x are y'.
No y are m.
i.e. All pigs are not-skeletons.
———————-
| | |
| —-|—- |
| | 0 | 0 | |
17. |—-|—-|—-|—-| ———-
| | 1 | 0 | | | | |
| —-|—- | |—-|—-|
| | | | 1 | |
———————- ———-
Let "creatures" be Universe; m="monkeys";
x="soldiers"; y="mischievous".
No m are x; &there4 Some y are x'.
All m are y.
i.e. Some mischievous creatures are not soldiers.
———————-
| 0 | |
| —-|—- |
| | 0 | 0 | |
18. |—-|—-|—-|—-| ———-
| | | | | | 0 | |
| —-|—- | |—-|—-|
| 0 | | | | |
———————- ———-
Let "persons" be Universe; m="just";
x="my cousins"; y="judges".
No x are m; &there4 No x are y.
No y are m'.
i.e. None of my cousins are judges.
———————-
| | |
| —-|—- |
| | 1 | 0 | |
19. |—-|—-|—-|—-| ———-
| | | | | | 1 | |
| —-|—- | |—-|—-|
| | | | | |
———————- ———-
Let "periods" be Universe; m="days";
x="rainy"; y="tiresome".
Some m are x; &there4 Some x are y.
All xm are y.
i.e. Some rainy periods are tiresome.
N.B. These are not legitimate Premisses, since the Conclusion is really part of the second Premiss, so that the first Premiss is superfluous. This may be shown, in letters, thus:—
"All xm are y" contains "Some xm are y", which contains "Some x are y". Or, in words, "All rainy days are tiresome" contains "Some rainy days are tiresome", which contains "Some rainy periods are tiresome".
Moreover, the first Premiss, besides being superfluous, is actually contained in the second; since it is equivalent to "Some rainy days exist", which, as we know, is implied in the Proposition "All rainy days are tiresome".