Predicate logic · Definition

Negation of quantifiers

logicpredicates

Before you start you need to know

Negating a quantifier swaps \(\forall\) and \(\exists\) and negates the property \(P(x)\).

Definition

For a predicate \(P(x)\) over \(D\), negating a quantifier swaps the quantifier and negates the predicate.

\[\begin{gathered}\neg\forall x P(x)\equiv\exists x\neg P(x)\\\neg\exists x P(x)\equiv\forall x\neg P(x)\end{gathered}\]

Negating a quantifier changes both the quantifier and the property. The negation of "for all" becomes "there exists at least one that does not", and that of "there exists" becomes "for all, not".

Thus, \(\neg\forall x\,P(x)\) is equivalent to \(\exists x\,\neg P(x)\), while \(\neg\exists x\,P(x)\) is equivalent to \(\forall x\,\neg P(x)\).

The difference between "not all" and "none" is crucial: the first asserts that a counterexample exists; the second asserts that all fail the property.

Think about it

What does "not all students passed" express?

That there exists at least one student who did not pass.

Exercises 10

  1. The negation of \(\forall x\,P(x)\) is...
  2. The negation of \(\exists x\,P(x)\) is...
  3. "Not all students passed" is equivalent to...
  4. What operation appears when negating a quantifier?
  5. Which equivalence is correct?
  6. The negation of \(\forall x\,P(x)\) is equivalent to...
  7. The negation of \(\exists x\,P(x)\) is equivalent to...
  8. To negate 'every number is positive' it is enough to state...
  9. Select the correct equivalence.
  10. Confusing \(\neg\forall x\,P\) with \(\forall x\,\neg P\) turns 'there is a counterexample' into...

References

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