Quantifiers: For All and There Exists

Two symbols express nearly everything we want to say about "all" and "some."

∀ means "for all" — every single one, without exception. ∃ means "there exists" — at least one.

These are the hinges on which predicate logic turns. "All swans are white" and "Some swan is black" have completely different logical structures. Get the quantifier wrong, and you've said something entirely different from what you meant.

The difference between claiming "all" and claiming "some" is the difference between an absolute law and a single counterexample.

The Universal Quantifier ∀

∀x P(x) means: for every object x in the domain, P(x) is true.

"All dogs bark": ∀x(Dog(x) → Barks(x)) For every x, if x is a dog, then x barks.

The universal quantifier makes a strong claim. One exception destroys it. If even one dog doesn't bark, ∀x(Dog(x) → Barks(x)) is false.

Think of ∀ as a conjunction spread over the entire domain: P(a) ∧ P(b) ∧ P(c) ∧ ... for every object in the domain.

The Existential Quantifier ∃

∃x P(x) means: there is at least one object x in the domain for which P(x) is true.

"Some dog is brown": ∃x(Dog(x) ∧ Brown(x)) There exists an x such that x is a dog and x is brown.

The existential quantifier makes a weak claim. One example establishes it. Finding even one brown dog makes ∃x(Dog(x) ∧ Brown(x)) true.

Think of ∃ as a disjunction spread over the domain: P(a) ∨ P(b) ∨ P(c) ∨ ... for every object in the domain.

The ∀-→ Pattern

Universal statements use implication: ∀x(Cat(x) → Mammal(x))

Why not conjunction? ∀x(Cat(x) ∧ Mammal(x)) would mean: everything is a cat AND everything is a mammal. That's absurdly false.

The implication says: IF something is a cat, THEN it's a mammal. Non-cats make the implication vacuously true without making the whole statement false.

The ∃-∧ Pattern

Existential statements use conjunction: ∃x(Bird(x) ∧ CanSwim(x))

Why not implication? ∃x(Bird(x) → CanSwim(x)) would be satisfied by any non-bird. "There exists something such that if it's a bird, it can swim" — any rock satisfies this (false → anything is true).

The conjunction says: there's something that IS a bird AND CAN swim. Both conditions must hold.

Quantifier Negation

Negating ∀: ¬∀x P(x) ≡ ∃x ¬P(x) "Not all" = "Some not"

If it's false that all swans are white, then some swan isn't white.

Negating ∃: ¬∃x P(x) ≡ ∀x ¬P(x) "None" = "All not"

If no swan is black, then all swans are not black.

To negate: flip the quantifier, negate the predicate.

Quantifier Scope

The order of quantifiers matters.

∀x ∃y Loves(x, y): For every x, there exists a y such that x loves y. Everyone loves someone (possibly different people).

∃y ∀x Loves(x, y): There exists a y such that for every x, x loves y. There's someone whom everyone loves.

The second is much stronger — it claims a single universal beloved exists.

Same-type quantifiers commute: ∀x ∀y is the same as ∀y ∀x. ∃x ∃y is the same as ∃y ∃x.

Different-type quantifiers don't commute: ∀x ∃y ≢ ∃y ∀x in general.

Restricted Quantifiers

Sometimes we restrict the domain directly:

∀x ∈ S, P(x) means: for all x in set S, P(x). ∃x ∈ S, P(x) means: there exists an x in set S such that P(x).

"All prime numbers greater than 2 are odd": ∀x((Prime(x) ∧ x > 2) → Odd(x))

Or with restricted quantifier: ∀x ∈ Primes, (x > 2 → Odd(x))

Uniqueness: ∃!

Sometimes we want to say exactly one thing exists.

∃!x P(x) means: there exists a unique x such that P(x).

Formally: ∃x(P(x) ∧ ∀y(P(y) → y = x))

There exists an x with property P, and any y with property P must be that same x.

"Every nonzero number has a unique reciprocal": ∀x(x ≠ 0 → ∃!y(x · y = 1))

Vacuous Truth

∀x(Unicorn(x) → Purple(x)) is true.

Why? There are no unicorns. The implication is never tested. When the antecedent is always false, the universal statement is vacuously true.

"All my Olympic gold medals are in the drawer" — true, if I have no Olympic gold medals.

This can seem counterintuitive, but it's essential for the logic to work consistently.

Proving Quantified Statements

To prove ∀x P(x): Take an arbitrary x from the domain and prove P(x). Since x was arbitrary, it holds for all.

To prove ∃x P(x): Exhibit a specific example that satisfies P.

To disprove ∀x P(x): Find one counterexample where P(x) fails.

To disprove ∃x P(x): Show P(x) fails for every x (prove ∀x ¬P(x)).

Common Mistakes

Switching quantifier order: "Everyone has a mother" ≠ "Someone is everyone's mother" ∀x ∃y Mother(y, x) ≠ ∃y ∀x Mother(y, x)

Wrong connective: ∀x(P(x) ∧ Q(x)) makes claims about everything. ∀x(P(x) → Q(x)) only constrains things with property P.

Negation errors: The negation of "All A are B" is "Some A is not B." NOT "No A is B" — that's stronger than needed.

The Power of Quantifiers

With just ∀ and ∃, we can express:

• All: ∀x P(x)
• Some: ∃x P(x)
• None: ¬∃x P(x) ≡ ∀x ¬P(x)
• Not all: ¬∀x P(x) ≡ ∃x ¬P(x)
• Exactly one: ∃!x P(x)
• At least n: chains of ∃ with distinctness conditions
• At most n: chains of ∀ with identity implications

Two symbols. Infinite expressive power.

Part 9 of the Logic series.

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