Universal quantification

Universal quantification asserts that a predicate holds for every object in a specified domain. In formal logic, it is commonly written with the symbol ∀.

Predicate logic: A formal logic that represents objects, properties, relations, and quantification. Universal quantification is one of predicate logic’s two main quantifiers.
First-order logic: A formal system that quantifies over individual objects and expresses relations among them. Universal quantification is a basic operator in first-order logic.
Existential quantifier: The logical symbol ∃, which asserts that at least one object satisfies a formula. Replacing ∀ with ∃ changes an all-objects claim into an existence claim.
Universal instantiation: An inference rule that derives a particular instance from a universally quantified formula. It lets a proof apply an all-objects premise to a chosen object.
Quantifier order: The order in which multiple quantifiers occur in a formula, affecting their dependencies and meaning. Swapping universal and existential quantifiers can change a true statement into a false one.
Predicate: An expression that assigns a truth value to each object or tuple of objects in its domain. Universal quantification tests whether the predicate holds throughout its domain.
Existential quantification: The operation of asserting that a predicate holds for at least one object in a domain, written ∃. It expresses existence, unlike universal quantification’s claim about every object.
Unique existence: A claim that exactly one object in a domain satisfies a specified condition. It makes a stronger claim than universal quantification over a predicate’s instances.
Mathematical induction: A proof method establishing a property for all natural numbers from a base case and an inductive step. Its conclusion is universal, though the proof proceeds through a base case and successor step.
Prenex normal form: A logical formula with all quantifiers moved to the front, followed by a quantifier-free matrix. It makes the sequence of universal and existential quantifiers explicit.

Linked from 15 pages