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Game of Logic: Chapter IV -- Hit or Miss -- Questions 01-05

These are worked-out logic exercises from *The Game of Logic* by Lewis Carroll.

In chapters 1-3, Carroll walks you through using diagrams to draw logical conclusions from statements. *The Game of Logic* is sort of a primer that prepares you for the more comprehensive *Symbolic Logic*. *Symbolic Logic* was intended as the first of a three-part graded series, but the other two parts were not completed before Carroll's death.

The 'game boards' are layed out with *x* and *y* in somewhat confusing locations for anyone who's had the x and y axes drilled into them through years of math classes. Originaly, users would place red or gray counters on the boards to represent that something exists (red) or does not exist (gray) in that category. Here, I'm using 1 (exists) and 0 (does not exist) for that purpose.

When you know that something must exist in at least one of two adjacent categories, but you don't know which one, you can place a counter on the line between them; "sitting on the fence", as Carroll liked to call it.

1. Pain is wearisome; no pain is eagerly wished for.

There should only be onec kind of thing here; one kind of thing to fit in our box.

Things:

pain

The small box can handle two attributes; the big box, three.

Attributes:

wearisome
eagerly wished for

So I guess we can use the small box then,

Let:

Universe = Pain
x = wearisome
y = eagerly wished for

"Pain is wearisome" = All pain is wearisome = some pain is wearisome & no pain is not wearisome

some x exists
no x' exists

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

"No pain is eagerly wished for"

no y exists

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

Therefore:

Universe = Pain
x = wearisome
y = eagerly wished for

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

All x are y'
No y exists
No x' exists
All y' are x

Translations:

All wearisome pain is not eagerly wished for pain
No eagerly wished for pain exists
No un-wearisome pain exists
All not eagerly wished for pain is wearisome pain

All pain is wearisome and not eagerly wished for pain.

2. No bald person needs a hairbrush; No lizards have hair

Things:

person
lizard

Hmm... this isn't good. There must be only one kind of thing in the box. So we have to figure out some way to make one a version of the other. Let's say that Carroll is getting a whimsical here, and lizards are a kind of person.

Universe:

People

Attributes:

bald
needs hairbrush
is lizard

Let:

x = bald
y = needs a hairbrush
m = lizard

"No bald person needs a hairbrush"

no xy exists

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

"No lizards have hair" = no person who is a lizard is not bald

no m is x'

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

"No lizards have hair" implies that some lizards do not have hair

some m are x

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

But xy do not exist, therefore, some xy' must exist.

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

x = bald
y = needs a hairbrush
m = lizard

Conclusion:

All m are y' = All lizards are people who don't need a hairbrush.
No xym exist = No bald lizards who need hairbrushes exist

3. All thoughtless people do mischief; No thoughtful person forgets a promise

Things:

persons

Attributes:

thoughtful
forgets promises
does mischief

That's three attributes, so we need the big table.

Let:

Universe = person
x = thoughtful
y = forgets promises
m = does mischief

"All thoughtless people do mischief", so all x' are m. That also implies that no x' are m'.

x'm exists
x'm' does not exist

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

"No thoughtful person forgets a promise", so no xy exists.

no xy exist

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

"All thoughtless people do mischief; No thoughtful person forgets a promise." The existence of thoughtless people and thoughtful persons is implied.

x exists
x' exists

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

Universe = person
x = thoughtful
y = forgets promises
m = does mischief

Conclusions:

all x' are m = All thoughtless people do mischief (we already knew that)
all x are y' = All thoughtful people do not forget promises
no m are xy = No people who do mischief are thoughtful people who forget promises
some m are x' = Some people who do mischief are thoughtless
some y' are x = All people who keep promises are thoughtful

4. I do not like John; Some of my friends like John.

Things:

People

Attributes:

my fiends
like John
me

Let:

Universe = People
x = my friends
y = like John
m = me

First, I think we can conclude that no friends of me are people who are also me. Likewise, all my friends are not me.

xm does not exist
all x are m'

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

But we can also assume that at least one person who is me exists.

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

"I do not like John", so *my* does not exist.

my does not exist

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

Thus mx'y' must exist.

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

"Some of my friends like john", so xy does exist, and there's only one way that could be

xy exists

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

Conclusions:

all m are x'y' = All people who are me are not my friends and do not like John.
all xy are m' = All people who like John and are my friends are not me.
some y are x = Some people who like John are my friends.

5. No potatoes are pine-apples; All pine-apples are nice

Things:

potatoes
pineapple

Hmm...

Let:

Universe = plants

Attributes:

x = potato
y = pineapple
m = nice

Let's tackle "All pine-apples are nice" because it breaks down to:

some pine-apples are nice = ym exists
no pineapples are not nice = ym' does not exist

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

"No potatoes are pine-apples", so xy does not exist.

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

Thus...

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

"No potatoes are pine-apples", implies that potatoes do exist.

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

x = potato
y = pine-apple
m = nice

Conclusions:

all y are mx' = All Pine-apples are nice non-potatoes
all x are y' = All potatoes are not pine-apple (we knew that)
some m are x' = Some nice plants are not potatoes
some y' are x = Some non-pine-apples are potatoes

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✍️ Last Updated: 2022-02-27

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