Definition
The universal quantifier is the logical operator that asserts a predicate holds for every object in a given domain. Written with the symbol for "for all" — a turned letter A — and followed by a variable and a formula, an expression like "for all x, F of x" claims that whatever value the variable x takes from the domain of discourse, the predicate F is true of it. It is the formal counterpart of the everyday quantifiers "all," "every," and "any."
Universal quantification is one of two basic quantifiers in classical first-order logic, the other being the existential quantifier "there exists." Together they extend propositional logic with the ability to talk about the internal structure of statements — not just whole sentences combined by connectives, but predicates ranging over objects. This jump is what makes predicate logic powerful enough to formalize mathematics and most of science.
Why it matters
How it works
A universally quantified statement is true if, and only if, the inner formula holds for every assignment of values to the bound variable from the domain. To evaluate "for all x, F of x," a logician fixes a domain (say, the natural numbers, or all humans, or all swans) and checks whether F is satisfied no matter which member of that domain x names. A single counterexample — one object in the domain for which F fails — is enough to falsify the whole claim. This sensitivity to counterexamples is what makes universal quantification powerful: a strong claim can be defeated by a single observation.
The universal quantifier has a tight duality with the existential one. Saying "for all x, F of x" is logically equivalent to "there is no x such that not F of x"; symmetrically, "there exists x with F of x" is equivalent to "not for all x, not F of x." This is the quantifier version of De Morgan's laws, and it lets logicians move freely between universal and existential formulations as a matter of convenience. Two further inference rules govern its use in formal proofs: universal instantiation, which lets you drop the quantifier and substitute any specific name for the bound variable, and universal generalization, which lets you introduce the quantifier when a property has been shown to hold for an arbitrary element of the domain.