Formal Logic

Question 1

Everything caused something.

Answer 1

∀x∃yCxy

Question 2

Everything is caused by something.

Answer 2

∀y∃xCxy

Question 3

Something caused everything.

Answer 3

∃x∀yCxy

Question 4

Something is caused by everything.

Answer 4

∃y∀xCxy

Question 5

Everything caused everything.

Answer 5

∀x∀yCxy

Question 6

Something caused something.

Answer 6

∃x∃yCxy

Question 7

Everything caused itself.

Answer 7

∀xCxx

Question 8

Something caused itself.

Answer 8

∃xCxx

Question 9

1. Mary is taller than John.
2. John is taller than Lucy.
3. Whoever is taller than someone who is taller than a third person, is taller than this third person.
   C. Mary is taller than Lucy.

Answer 9

1. Tmj
2. Tjl
3. ∀x∀y∀z((Txy ∧ Tyz) → Txz)
4. Tml

Question 10

What is logically equivalent to ∃xFx ?

Answer 10

¬∀x¬Fx

Question 11

What is logically equivalent to ∀xFx

Answer 11

¬∃x¬Fx

Question 12

What does ¬∀x¬Fx mean?

Answer 12

‘It is not the case that everything is not-F’

Question 13

What does ¬∃x¬Fx mean?

Answer 13

Nothing is not-F

Question 14

Polar bears and grizzly bears are dangerous

Answer 14

Anything which is either a polar bear or a grizzly bear is
dangerous:
∀x((Gx ∨ Px) → Dx)

Question 15

There is at least one unicorn

Answer 15

∃xUx

Question 16

There is at most one unicorn

Answer 16

If you come across a unicorn and then you come across a unicorn again, they are identical.
If x is a unicorn and y is a unicorn, they must be identical.
∀x∀y((Ux ∧ Uy) → x = y)

Question 17

There is exactly one unicorn.

Answer 17

If you encounter this one unicorn, and you come across another one, that latter unicorn must be identical to the first.
There is a unicorn and anything that is a unicorn is identical to it.
∃x(Ux ∧ ∀y(Uy → x = y))

Question 18

There are at least two unicorns.

Answer 18

There are x and y that are unicorns and they are different.

∃x∃y((Ux ∧ Uy) ∧ x ̸= y)

Question 19

There are at most two unicorns.

Answer 19

If x, y and z are unicorns, then one of them must be identical to another one of them.
∀x∀y∀z(((Ux ∧ Uy) ∧ Uz) → (x = y ∨ x = z))

Question 20

There are exactly two unicorns.

Answer 20

There are different x and y that are unicorns, for any z that is a unicorn, it is identical to one of them.
∃x∃y(((Ux ∧ Uy) ∧ x ̸= y) ∧ ∀z(Uz → (z = x ∨ z = y)))

Question 21

The present King of France is bald.

Answer 21

There is exactly one present King of France and he is bald.

∃x((Fx ∧ ∀y(Fy → x = y)) ∧ Bx)

Question 22

The present King of France is not bald.

Answer 22

It is not the case that there is exactly one present King of France and he is bald.
¬∃x((Fx ∧ ∀y(Fy → x = y)) ∧ Bx)

There is exactly one present King of France and he is not bald.
∃x((Fx ∧ ∀y(Fy → x = y)) ∧ ¬Bx)

Question 23

Universal Instantiation

Answer 23

∀xF x → F m

Question 24

Existential Generalisation

Answer 24

Fm → ∃xF x