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The Game of Logic: Chapter IV -- Hit or Miss -- Questions 06-10

More exercises from Lewis Carroll's *The Game of Logic*.

6. No pins are ambitious; No needles are pins

Universe = sharp things?

Attributes:

• x = pin
• y = needle
• m = ambitious

"No pins are ambitious" = no x is m & some x exist

• no xm exist
• some x exist

      y       y'
  ┌───────┬───────┐
  │       1       │
x │   ┌───┼───┐   │
  │   │ 0 │ 0 │   │
  ├───┼───m───┼───┤
  │   │   │   │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

"No needles are pins" = no y are x

• no yx exists

      y       y'
  ┌───────┬───────┐
  │ 0     1       │
x │   ┌───┼───┐   │
  │   │ 0 │ 0 │   │
  ├───┼───m───┼───┤
  │   │   │   │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

Now we see that the 1 must be in *xy'*

      y       y'
  ┌───────┬───────┐
  │ 0     │     1 │
x │   ┌───┼───┐   │
  │   │ 0 │ 0 │   │
  ├───┼───m───┼───┤
  │   │   │   │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

• x = pin
• y = needle
• m = ambitious

Conclusion:

• all x are m'y' = all pins are unambitious non-needles

But it's also implied that needles do exist, so we can update the diagram:

      y       y'
  ┌───────┬───────┐
  │ 0     │     1 │
x │   ┌───┼───┐   │
  │   │ 0 │ 0 │   │
  ├───┼───m───┼───┤
  │   │   │   │   │
x'│   1───┼───┘   │
  │       │       │
  └───────┴───────m'

Conclusion:

• all x are m'y' = all pins are unambitious non-needles
• all y are x' = all needles are non-pins (not really new information)
• no xym exist = no sharp things are ambitious pin and ambitious needles at the same time

7. All my friends have colds; No one can sing who has a cold

Universe = friends?

Attributes:

• has a cold
• can sing

OK, how about Universe = People

• x = has a cold
• y = can sing
• m = my friend

"All my friends have colds" = all m are x

• mx exists
• mx' does not exist

      y       y'
  ┌───────┬───────┐
  │       │       │
x │   ┌───┼───┐   │
  │   │   1   │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

"No one can sing who has a cold" = xy does not exist

      y       y'
  ┌───────┬───────┐
  │ 0     │       │
x │   ┌───┼───┐   │
  │   │ 0 1   │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

Therefore:

      y       y'
  ┌───────┬───────┐
  │ 0     │       │
x │   ┌───┼───┐   │
  │   │ 0 │ 1 │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

Conclusion:

• all m are xy' = all my fiends are people who have colds and cannot sing

8. All these dishes are well-cooked; Some dishes are unwholesome if well-cooked

Universe = dishes?

Attributes:

• x = well-cooked
• y = wholesome
• m = 'these' dishes

"All these dishes are well-cooked" = all m are x

• mx exists
• mx' does not exist

      y       y'
  ┌───────┬───────┐
  │       │       │
x │   ┌───┼───┐   │
  │   │   1   │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

"Some dishes are unwholesome if well-cooked" = Some not-well-cooked dishes are not wholesome = some x'y' exist

• x'y' exists

      y       y'
  ┌───────┬───────┐
  │       │       │
x │   ┌───┼───┐   │
  │   │   1   │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───1   │
  │       │       │
  └───────┴───────m'

Thus...

      y       y'
  ┌───────┬───────┐
  │       │       │
x │   ┌───┼───┐   │
  │   │   1   │   │
  ├───┼───m───┼───┤
  │   │ 0 │ 0 │   │
x'│   └───┼───┘   │
  │       │     1 │
  └───────┴───────m'

• x = well-cooked
• y = wholesome
• m = 'these' dishes

Conclusion:

• x'y'm' exists = Some dishes are not these dishes and are not well-cooked and unwholesome

9. No medicine is nice; Senna is a medicine

Universe = medicine

Attributes:

• x = nice
• y = Senna

Hmm...we haven't done many small diagrams.

"No medicine is nice" = no x

    y   y'
  ┌───┬───┐
x │ 0 │ 0 │
  ├───┼───┤
x'│   │   │
  └───┴───┘

"Senna is a medicine" = Some medicine called 'Senna' exists = y exists.

    y   y'
  ┌───┬───┐
x │ 0 │ 0 │
  ├─1─┼───┤
x'│   │   │
  └───┴───┘

Thus...

    y   y'
  ┌───┬───┐
x │ 0 │ 0 │
  ├───┼───┤
x'│ 1 │   │
  └───┴───┘

• x = nice
• y = Senna

Conclusion:

• All medicines call 'Senna' are not-nice medicines

10. Some oysters are silent; No silent creatures are amusing

Universe = creatures

Attributes:

• x = oyster
• y = silent
• m = amusing

"Some oysters are silent" = some xy exist

"No silent creatures are amusing" = no ym exist

• xy exists
• ym does not exist

      y       y'
  ┌───────┬───────┐
  │       │       │
x │   1───┼───┐   │
  │   │ 0 │   │   │
  ├───┼───m───┼───┤
  │   │ 0 │   │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

Thus...

      y       y'
  ┌───────┬───────┐
  │ 1     │       │
x │   ┌───┼───┐   │
  │   │ 0 │   │   │
  ├───┼───m───┼───┤
  │   │ 0 │   │   │
x'│   └───┼───┘   │
  │       │       │
  └───────┴───────m'

• x = oyster
• y = silent
• m = amusing

Conclusion:

• xym' exists = some oysters are silent, un-amusing oysters

---'--,-<@    @>-'--,---

✍️ Last Updated: 2022-02-27

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