Quantification is some jargon from formal logic. There is the universal quantifier written as ∀ in a context such as “∀x(x=x)” which is taken to mean that for all x, x is x. Tarski wrote “x(x=x)” in the same context. The other quantifier is ∃ where “∃x(x>4)” means that for some x, x is greater than 4. Tarski wrote “x(x>4)”.

To quantify over sets means that the variable following the quantifier is to range over sets.

Unicode together with font mongers do not support Tarski’s notation well.

Contrast:
For every x there is a y such that x divides y.
There is a y such that for every x x divides y.

Symbolically:
∀x∃y y mod x = 0.
∃y∀x y mod x = 0.

The first is true and the second is false.

John Maynard Keynes famously said: “In the long run we are all dead.”. Let’s pull that apart:

1. It might mean that for each person, x, there is a future date d, such that x is dead at d.
2. It might mean that there is a future date d, such that for each person, x, x is dead at d.
Under ordinary assumptions 1 is certainly true but 2 is certainly false.

“∀x” is the universal quantifier and “∃y” is the existential quantifier. If φ is some expression, perhaps involving y then.
∃yφ = ¬∀y¬φ
This is often taken as a definition for ∃ while ∀ is primitive.

Universal quantifiers commute with each other and so do existential quantifiers. They do not commute with each other.

The sentence:
“For every x (x divides y) for some y.” can be read as either:
“For every x ((x divides y) for some y).” or
“(For every x (blah blah x blah y)) for some y.”.

The meanings, as above, are different. Professional mathematicians often make this mistake.

∀ε (ε>0 → ∀x∃δ (δ > 0 ∧ ∀z(|x − z| < δ → |f(x) − f(z)| < ε))) ↔︎ The function f is continuous.
∀ε (ε>0 → ∃δ∀x (δ > 0 ∧ ∀z(|x − z| < δ → |f(x) − f(z)| < ε))) ↔︎ The function f is uniformly continuous.
One must be careful with quantifiers.

More logic