An open sentence becomes a statement in a second way: instead of fixing the variable, say how many values satisfy it. "For every" claims the sentence never fails; "there exists" claims it succeeds at least once. When two quantifiers appear together their order decides the meaning, and swapping them usually changes the claim entirely.

Compare "every lock has a key that opens it" with "there is a key that opens every lock". The first is unremarkable; the second describes a master key. The words are the same and only the order differs.

Write the quantifier before the sentence it governs. Reading a nested pair from left to right matters: each quantifier is chosen knowing only the variables named before it.

• $\forall x\, P(x)$ says $P$ holds for every $x$. One failure is enough to refute it.
• $\exists x\, P(x)$ says holds for at least one . One success is enough to establish it.

Denying that something always happens is exactly claiming a case where it does not. Reading the denial in plain words in one direction, and the witness back into a denial in the other, gives both halves.

Start from the denial and see what it forces.

That is what the denial says, read in plain words.

Failing to hold everywhere means failing somewhere, and that somewhere is a single value.

A value where $P$ fails is precisely a value where $\neg P$ holds.

One counterexample is enough to bring down a universal claim, which gives the other direction.

:::note{title="One failure refutes "for every""} A universal claim covers every case, so a single case where it fails is enough to bring it down. No further examples are needed. :::

$\forall n\, (n^2 > n)$ is false, since $n = 1$ gives $1 > 1$.

One value where the sentence fails settles it, however many values satisfy it.

:::note{title="One success establishes "there exists""} An existential claim asks for at least one case. Exhibiting a single value that works proves it completely. :::

$\exists n\, (n^2 = 9)$ is true, since $n = 3$ works.

Naming one value that satisfies the sentence is a full proof of an existential claim.

Denying that every value is positive claims one value that is not.

$\forall x\, \exists y\, (y > x)$ is true, while $\exists y\, \forall x\, (y > x)$ is false.

Every number has a larger one, but no single number exceeds them all.

Writing a range on a quantifier is shorthand, and the two kinds expand differently.

• One counterexample refutes a universal; one witness establishes an existential.
• The order of unlike quantifiers is part of the claim and must not be changed.
• A bound expands with an implication under a universal and with an and under an existential.
• Like quantifiers may be swapped freely.

Great work! You now know how to:

• Refute a universal with one counterexample and establish an existential with one witness
• Expand a bounded quantifier correctly
• Say why the order of unlike quantifiers changes the claim