kvantifikatoren

Kvantifikatoren, or quantifiers, are operators in logic that specify how many individuals in a given domain satisfy a predicate. The most common are the universal quantifier ∀ and the existential quantifier ∃, often read as “for all” and “there exists” respectively. They are used to form statements such as ∀x P(x) and ∃x P(x).

Semantics: In a structure with a domain D and interpretations of predicates, ∀x P(x) is true iff P(d) holds for every d in D; ∃x P(x) is true iff there exists some d in D with P(d). A quantifier binds the variable that follows it; its scope ends at the end of the subformula or formula it governs. A variable is bound if it lies within the scope; otherwise it is free.

Nested quantifiers illustrate how order matters. For example, ∀x ∃y R(x,y) says that for every x there is a y such that R(x,y) holds, whereas ∃y ∀x R(x,y) says there exists a single y that works for all x. In general, these two statements are not equivalent.

Beyond ∀ and ∃, there are unique quantifiers (∃!x P(x) means exactly one x) and generalized or counting quantifiers (e.g., ∃≥k x P(x)). Such quantifiers extend predicate logic and are used in mathematics, computer science, and linguistics. Quantifiers also relate to techniques like quantifier elimination and Skolemization, which are used to transform quantified formulas into equivalent or simpler forms.