Predicate logic · Definition

Universal quantifier

logicpredicates

Before you start you need to know

The universal quantifier \(\forall\) expresses that a property holds for all elements of a domain.

Definition

Given a predicate \(P(x)\) over a domain \(D\), the universal quantifier asserts that \(P(x)\) is true for every \(x\in D\).

\[\forall x\,P(x)\]

The universal quantifier, also called "for all", asserts that a property holds for every element of the domain of interpretation. It is written \(\forall x\,P(x)\) and read "for all \(x\), \(P(x)\)".

To refute a universal statement it is enough to find a counterexample: an element of the domain for which P(x) is false.

The domain is not a decorative detail. Changing it can change the truth value of the universal statement.

Think about it

What is enough to refute \(\forall x\,P(x)\)?

A single element of the domain that makes \(P(x)\) false.

Exercises 10

  1. The symbol of the universal quantifier is...
  2. ∀x P(x) means that P(x) holds...
  3. To refute a universal statement it is enough to find...
  4. Which statement is universal?
  5. If P(x) is true for every x in the domain, ∀x P(x) is...
  6. The symbol ∀ is read...
  7. ∀x P(x) is true when...
  8. To refute ∀x P(x) it is enough to find...
  9. Which statement is equivalent to \(\forall x\,P(x)\)?
  10. In an empty domain, ∀x P(x) is considered...

References

  • Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.