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The universal quantifier ∀ (for all) and existential quantifier ∃ (there exists) express generality and existence. ∀x P(x) means 'for every object x, P holds'; ∃x P(x) means 'there is at least one object x for which P holds'. Quantifiers bind variables and determine scope.
Translate between English phrases ('all dogs bark', 'some cats are black') and formal quantified formulas. Practice recognizing quantifier scope and how scope affects meaning.
You already know what a predicate is: a property P(x) that is either true or false for each object x in some domain. Predicates by themselves make claims about specific objects — P(alice) says Alice has property P. Quantifiers lift this to claims about *all* or *some* objects in the domain at once, without naming any of them.
The universal quantifier ∀ is a logical "for all." The formula ∀x P(x) means: pick any object x from the domain — P holds. It is equivalent to the conjunction of P over every element, but without having to enumerate them. If your domain is the integers, ∀x (x + 0 = x) says that adding zero is a right identity for every integer. The connection to sets you already know: ∀x P(x) is true in a domain D precisely when the extension of P — the set {x ∈ D : P(x)} — equals the entire domain D.
The existential quantifier ∃ is the logical "there exists." The formula ∃x P(x) means: at least one object in the domain satisfies P. It is the disjunction of P over every element. If your domain is the integers, ∃x (x² = 2) is false over the integers (no integer squares to 2) but true over the reals. Notice that the *same sentence* changes truth value when the domain changes — quantifiers always range over a specific domain of discourse, and specifying that domain is part of giving a formula a meaning.
Scope and variable binding are the subtlest aspects. In ∀x (P(x) → ∃y Q(x, y)), the variable x is bound by the universal quantifier and y is bound by the existential. A bound variable is just a placeholder: ∀x P(x) and ∀z P(z) say exactly the same thing. A free variable — one not bound by any quantifier — makes a formula act like a predicate: it is true or false depending on what value you assign to the free variable. The formula P(x) with free x is open; ∀x P(x) closes it. Quantifier order matters crucially for nested quantifiers: ∀x ∃y (y > x) says "for every number there is a larger one" (true of the integers), while ∃y ∀x (y > x) says "there is a number larger than all numbers" (false). The swap changes the claim entirely.