universal quantifier

Thân thiện

A student writes a universal quantifier on the whiteboard.

Definition

Noun: A logical quantifier that asserts a statement or proposition is true for every member of a specified class or domain. It is a fundamental concept in predicate logic, used to express generalizations.

Usage

The universal quantifier is used in formal logic, mathematics, computer science, and philosophy to make statements about all objects in a given set. It is typically symbolized by the inverted "A" (∀).

Examples

• In the logical statement "∀x (P(x))", the universal quantifier "∀" indicates that the property P holds for all x in the domain.
• The sentence "All humans are mortal" can be formalized using a universal quantifier: ∀x (Human(x) → Mortal(x)).
• In mathematics, the commutative law for addition is expressed with a universal quantifier: ∀a ∀b (a + b = b + a).

Advanced Usage

• Scope: The part of the logical statement that the quantifier governs is called its scope. In "∀x (P(x) ∧ Q(x))", the scope of ∀x is the entire conjunction (P(x) ∧ Q(x)).
• Negation: The negation of a universally quantified statement is an existentially quantified negation. ¬(∀x P(x)) is logically equivalent to ∃x ¬P(x). For example, "It is not true that all birds can fly" is equivalent to "There exists a bird that cannot fly."

Variants and Related Words

• Universal Quantification (n): The act or process of using a universal quantifier to create a generalized statement.
• Existential Quantifier (n): The logical quantifier (symbol ∃) that asserts a proposition is true for at least one member of a class. It is the conceptual counterpart to the universal quantifier.

Synonyms

• For-all quantifier

Related Phrases and Concepts

• Bound Variable: In a quantified statement like ∀x P(x), the variable 'x' is bound by the quantifier.
• Domain of Discourse: The set of objects over which the quantifier ranges. The meaning of "∀x" depends on what this domain is defined to be (e.g., all numbers, all people).

Học thuật

Noun

1. a logical quantifier of a proposition that asserts that the proposition is true for all members of a class of things