PHIL110
Introduction to logic and reasoning
Week 12: Multiple quantifiers
Instructor: Dr. Bruno Guindon
Multiple quantifiers
• So far, we’ve only been dealing with quantified sentences that contain one
quantifier:
• ∃x(Px ∧ Qx)
• ∀x [(Px ∧ Qx) → (Tx ∨ Oxa)]
• Introducing translations and proofs that use multiple quantifiers.
Multiple quantifiers
• We’ve seen things like [∀x(Px → Qx) ∧ ∃y(Ty ∨ Oya)], but this isn’t a quantified
sentence—it’s a conjunction.
• So, it isn’t a quantified sentence that uses more than one quantifier.
• This week, we are considering sentences of the form:
1. ∀x∀y[(Px ∧ Qy) → Oxy]
2. ∃x∃y[(Px ∧ Qy) ∧ Oxy]
3. ∀x[Px → ∀y(Qy → Oxy)]
4. ∀x[Px → ∃y(Qy ∧ Oxy)]
Translations
• We start with multiple uses of a single quantifier; e.g.,
• ∀x∀y[(Px ∧ Qy) → Oxy]; ∃x∃y [(Px ∧ Qy) ∧ Oxy]
• We’ll then consider quantified sentences with mixed quantifiers; e.g.,
• ∀x[(Px → ∃y(Qy ∧ Oxy)]; ∃x∀yLxy
Four Aristotelian forms
1. All Ps are Qs.
2. Some Ps are Qs.
3. No Ps are Qs.
4. Some Ps are not Qs.
1. ∀x(Px → Qx)
2. ∃x(Px ∧ Qx)
3. ∀x(Px → ¬Qx)
4. ∃x(Px ∧ ¬Qx)
• Many quantified sentences in English can be seen as belonging to one of
these Aristotelian forms.
Multiple uses of ∃
• Consider the following symbolization key:
UD: Everything
Axy: x is afraid of y
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
1. Some dog is bigger than some cat.
Form: Some Ps are Qs.
i.
∃x(… ∧ …)
How to express ‘x is a dog’?
ii. ∃x(Dx ∧ …)
How to express ‘x is bigger than some cat’?
iii. ∃y(Cy ∧ Bxy)
Put those things together:
iv. ∃x(Dx ∧ ∃y(Cy ∧ Bxy))
Prenex: ∃x∃y((Dx ∧ Cy) ∧ Bxy)
Multiple uses of ∃
UD: Everything
Axy: x is afraid of y
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
•
Some small cat is faster than some large dog.
Form: Some Ps are Qs.
i.
∃x(… ∧ …)
How to express ‘x is a small cat’?
ii. ∃x((Cx ∧ Sx) ∧ …)
How to express ‘x is faster than some large dog’?
iii. ∃y((Ly ∧ Dy) ∧ Fxy)
Put those things together:
iv. ∃x((Cx ∧ Sx) ∧ ∃y((Ly ∧ Dy) ∧ Fxy))
Prenex: ∃x∃y((Cx ∧ Sx) ∧ ((Ly ∧ Dy) ∧ Fxy))
Multiple uses of ∃
UD: Everything
•
Axy: x is afraid of y •
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
Some large dog isn’t afraid of any small cat.
Form: Some Ps are not Qs.
i.
∃x(… ∧ ¬…)
How to express ‘x is a large dog’?
ii. ∃x((Lx ∧ Dx) ∧ …)
How to express ‘x is not afraid of any small cat’?
iii. ¬∃y((Sy ∧ Cy) ∧ Axy)
Put those things together:
iv. ∃x((Lx ∧ Dx) ∧ ¬∃y((Sy ∧ Cy) ∧ Axy))
Prenex: (Requires the use of mixed quantifiers)
Multiple uses of ∀
UD: Everything
Axy: x is afraid of y
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
•
Every small dog is afraid of every big cat.
Form: All Ps are Qs.
i.
∀x(… → …)
How to express ‘x is small dog’?
ii. ∀x((Sx ∧ Dx) → …)
How to express ‘x is afraid of every big cat’?
iii. ∀y((By ∧ Cy) → Axy)
Put those things together:
iv. ∀x((Sx ∧ Dx) → ∀y((By ∧ Cy) → Axy))
Prenex: ∀x∀y[((Sx ∧ Dx) ∧ (By ∧ Cy)) → Axy]
Multiple uses of ∀
UD: Everything
Axy: x is afraid of y
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
•
No large cat is bigger than every small dog.
Form: No Ps are Qs.
i.
∀x(… → ¬…)
Express ‘x is a large cat’:
ii. ∀x((Lx ∧ Cx) → ¬…)
Express ‘x is bigger than every small dog’:
iii. ∀y((Sy ∧ Dy) → Bxy)
Put those things together:
iv. ∀x((Lx ∧ Cx) → ¬∀y((Sy ∧ Dy) → Bxy))
Prenex: Requires mixed quantifiers
Multiple uses of ∀
UD: Everything
Axy: x is afraid of y
Bxy: x is bigger than y
Cx: x is a cat
Dx: x is a dog
Fxy: x is faster than y
Lx: x is large
Sx: x is small
•
All dogs are afraid of each other.
Form: All Ps are Qs.
i.
∀x(Dx → ∀y(Dy → Axy)), or
ii. ∀x∀y[(Dx ∧ Dy) → Axy]
Incorrect: Both i. and ii. state that every dog is afraid of
every dog, including themselves.
Correct translations:
iv. ∀x(Dx → ∀y[(Dy ∧ y ≠ x) → Axy], or
v. ∀x∀y[((Dx ∧ Dy) ∧ y ≠ x) → Axy]
Mixed quantifiers
• Translating sentences with mixed quantifiers can be a bit tricker, but if we use our step-by•
•
•
•
•
•
•
step method of translation, we should have no issue.
Some dog is bigger than every small cat.
Form: Some Ps are Qs.
∃x(Dx ∧ ….)
Express ‘x is bigger than every small cat’:
∀y((Sy ∧ Cy) → Bxy)
Taken together: ∃x[Dx ∧ ∀y((Sy ∧ Cy) → Bxy)], or
∃x∀y(Dx ∧ [(Sy ∧ Cy) → Bxy])
Mixed quantifiers
• Every small dog is afraid of some cat.
• Form: All Ps are Qs.
• ∀x((Sx ∧ Dx) → …)
• Express ‘x is afraid of some cat’: ∃y(Cy ∧ Axy)
• Taken together: ∀x[(Sx ∧ Dx) → ∃y(Cy ∧ Axy)], or
• ∀x∃y[(Sx ∧ Dx) → (Cy ∧ Axy)]
Mixed quantifiers
• No large cat is afraid of some dog.
• Form: No Ps are Qs.
• ∀x((Lx ∧ Cx) → ¬…)
• How to translate “x is afraid of some dog?
• ∃y(Dy ∧ Axy)
• Put those things together: ∀x((Lx ∧ Cx) → ¬∃y(Dy ∧ Axy))
Caution
• When dealing with multiple instances of a single quantifier, the orders
of variables and quantifiers don’t matter.
• The following expressions all mean that something is bigger than
something:
• ∃x∃yBxy;
∃x∃yByx;
∃y∃xBxy;
∃y∃xByx
• Similarly, the following expressions all mean that everything is afraid of
everything:
• ∀x∀yAxy;
∀x∀yAyx;
∀y∀xAxy;
∀y∀xAyx
Caution
• When dealing with mixed variables the orders of variables and quantifiers
matter a great deal.
• Let the UD consist of people. Now, consider the following sentences:
1. ∀x∃yAxy 1. Everyone is afraid of someone.
2. ∃y∀xAxy
2. There is someone of whom everyone is afraid.
3. ∀x∃yAyx
3. Everyone has someone who’s afraid of them.
4. ∃y∀xAyx
4. Someone is afraid of everyone.
Proofs using multiple quantifiers
• Proofs with sentences using multiple quantifiers are really no different than
proofs using sentences with just one quantifier.
• If we use our rules correctly, and work backwards, we should have no issue.
• Show that the following sentences are logically equivalent:
• ∀x[Cx → ∀y(Dy → Fxy)]; ∀x∀y[(Cx ∧ Dy) → Fxy]
More examples
• Someone is someone’s father. Anyone who’s the father of anyone is not their
son. So, someone is not someone’s son.