Allquantifikation

Allquantifikation is the universal quantification in formal logic and semantics. It refers to an operator that asserts a predicate holds for every element of the domain of discourse.

In first-order logic the standard notation is ∀x φ(x). The formula ∀x φ(x) is true exactly when φ(d) is true for every element d in the domain. The variable x is bound by the quantifier, and the scope of the quantification is the subformula that follows it. The domain is the set of objects over which the quantifier ranges, and explicit domain restrictions are often written by attaching predicates, for example ∀x (Dog(x) → Likes(x, bone)).

Examples illustrate its meaning. For the real numbers, ∀x x^2 ≥ 0 is true. A statement such as ∀x (P(x) → Q(x)) asserts that every x with property P also has property Q. Universal quantification is often contrasted with existential quantification, ∃x φ(x), which claims that there exists at least one element satisfying φ. They are dual with respect to negation: ¬∃x φ(x) is equivalent to ∀x ¬φ(x), and ¬∀x φ(x) is equivalent to ∃x ¬φ(x). Vacuous truth can arise: if the domain is empty, ∀x φ(x) is true regardless of φ.

Applications of allquantifikation appear across mathematics, computer science, and linguistics. In mathematics, axioms are commonly stated using universal quantification, e.g., ∀x, y, z, a property holds for all objects in a structure. In natural language semantics, universal quantification interacts with negation, conditionals, and plurality readings, influencing scope and interpretation.