Predicate logic · Definition

Existential quantifier

logicpredicates

Before you start you need to know

The existential quantifier \(\exists\) lets us assert that some object in a domain satisfies a property.

Definition

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

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

The existential quantifier, also called "there exists", asserts that at least one element of the domain satisfies a property. It is written \(\exists x\,P(x)\) and read "there exists an \(x\) such that \(P(x)\)".

To prove an existential statement it is enough to present a witness: a concrete element of the domain that makes the property true.

If the domain is empty, no witness is available and the existential statement turns out to be false.

Think about it

What is a witness of \(\exists x\,P(x)\)?

A concrete element of the domain for which \(P(x)\) is true.

Exercises 10

  1. The symbol of the existential quantifier is...
  2. ∃x P(x) means that P(x) holds...
  3. To prove an existential statement it is enough to provide...
  4. Which statement is existential?
  5. If there is at least one x in the domain with P(x) true, ∃x P(x) is...
  6. The symbol ∃ is read...
  7. ∃x P(x) is true when...
  8. To prove ∃x P(x), it is usually enough to exhibit...
  9. Select expressions that imply there is at least one object.
  10. Let D be empty. What is the value of ¬∃x P(x) in D?

References

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