cuantificator

A cuantificator, or quantifier, is a logical operator used to express the quantity of elements in a domain that satisfy a given predicate. In formal logic, the standard quantifiers are the universal quantifier (for all) and the existential quantifier (there exists). The universal quantifier states that the predicate holds for every element of the domain, while the existential quantifier states that there exists at least one element for which the predicate holds.

Semantics: In a structure M with domain D, the formula ∀x φ(x) is true in M if φ(d) is true in M for every d in D; ∃x φ(x) is true if there exists d in D with φ(d) true. Quantifiers are interpreted with respect to variable assignments, and the truth of quantified formulas is defined recursively from the atomic formulas upward.

Variants: The unique quantifier (∃!x) expresses that there exists exactly one x such that φ(x). Generalized quantifiers extend the basic ones to express properties such as “most,” “finitely many,” or “infinitely many” elements, broadening their applicability beyond the universal and existential cases.

Applications and related ideas: Quantifiers are foundational in mathematics and logic, underpinning definitions, theorems, and proofs. In computer science, they appear in formal specifications, query languages, and program verification under logic and model checking. In linguistics, quantifiers are analyzed for their role in natural language semantics, including issues of scope and interaction with negation.

See also: Quantifier elimination, model theory, first-order logic, generalized quantifiers, syntax and semantics of natural language quantification.