Nested Quantifiers

When we use multiple quantifiers in succession, we talk about nested quantifiers. In such cases, it is very important what order they appear in, because this changes the meaning of the statement.

Order of ∀ and ∃

This means: for every x there exists a y that is greater than it. This is true among real numbers.

This means: there exists a y that is greater than every x. This is false, because there is no largest number.

Examples of the Difference

• “Every person has a parent.” → ∀x ∃y (y is parent of x).
• “There is a person who is everyone's parent.” → ∃y ∀x (y is parent of x).

It is clear that the order gives a completely different meaning to the statement.

Nesting with Multiple Variables

This is a universal statement on two variables: for every x and every y, x + y = y + x holds.

This means: there is such an x that for every y satisfies x·y = y. This is true only for x = 1.

Summary

The order of nested quantifiers is crucial. ∀x ∃y and ∃y ∀x mean completely different statements. Understanding logical precision requires their correct use.

Practice Exercise