Formal Logic Flashcards
Everything caused something.
∀x∃yCxy
Everything is caused by something.
∀y∃xCxy
Something caused everything.
∃x∀yCxy
Something is caused by everything.
∃y∀xCxy
Everything caused everything.
∀x∀yCxy
Something caused something.
∃x∃yCxy
Everything caused itself.
∀xCxx
Something caused itself.
∃xCxx
- Mary is taller than John.
- John is taller than Lucy.
- Whoever is taller than someone who is taller than a third person, is taller than this third person.
C. Mary is taller than Lucy.
- Tmj
- Tjl
- ∀x∀y∀z((Txy ∧ Tyz) → Txz)
- Tml
What is logically equivalent to ∃xFx ?
¬∀x¬Fx
What is logically equivalent to ∀xFx
¬∃x¬Fx
What does ¬∀x¬Fx mean?
‘It is not the case that everything is not-F’
What does ¬∃x¬Fx mean?
Nothing is not-F
Polar bears and grizzly bears are dangerous
Anything which is either a polar bear or a grizzly bear is
dangerous:
∀x((Gx ∨ Px) → Dx)
There is at least one unicorn
∃xUx
There is at most one unicorn
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)
There is exactly one unicorn.
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))
There are at least two unicorns.
There are x and y that are unicorns and they are different.
∃x∃y((Ux ∧ Uy) ∧ x ̸= y)
There are at most two unicorns.
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))
There are exactly two unicorns.
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)))
The present King of France is bald.
There is exactly one present King of France and he is bald.
∃x((Fx ∧ ∀y(Fy → x = y)) ∧ Bx)
The present King of France is not bald.
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)
Universal Instantiation
∀xF x → F m
Existential Generalisation
Fm → ∃xF x
