Quantifier

A quantifier is a logical operator that states how many objects in a domain satisfy a condition. In predicate logic, the existential and universal quantifiers express “at least one” and “every.”

Universal quantification: A quantifier asserting that a predicate holds for every object in a specified domain. It expresses the “every object” case of quantification.
Predicate logic: A formal system that represents objects, their properties, and relations using variables and quantifiers. Quantifiers extend its expressive power beyond whole-proposition logic.
Bounded quantifier: A quantifier restricted to objects satisfying a specified condition or belonging to a specified range. Restrictions make quantified claims more precise, as in “for every integer greater than zero.”
Generalized quantifier: A quantifier-like expression that specifies a relation between sets, such as “most” or “exactly three.” It extends the basic existential and universal patterns to richer quantity claims.
Existential quantification: A quantifier asserting that a predicate holds for at least one object in a specified domain. It expresses the “at least one object” case.
Predicate: An expression that denotes a property or relation and becomes a proposition when applied to suitable arguments. A quantifier states how many domain objects satisfy a predicate.
Quantifier alternation: The occurrence of different quantifiers in sequence within a logical formula. Alternating quantifiers express dependencies that appear in mathematical definitions and complexity theory.
Counting quantifier: A logical quantifier asserting that exactly, at least, or at most a specified number of objects satisfy a condition. It makes numerical quantity explicit rather than merely existential or universal.
Bound variable: A variable whose occurrences are governed by a quantifier or another binding operator. Quantifiers bind variables to the objects over which they range.
Variable (mathematics): A symbol that represents an unspecified or changing mathematical object. Quantified variables range over possible objects in the domain.

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