Quantifier (logic)
Order of quantifiers (nesting)

The order of quantifiers is critical to meaning, as is illustrated by the following two propositions:

This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is reversed is different:

This is clearly false; it asserts that there is a single natural number s that is the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables.

A less trivial example from mathematical analysis regards the concepts of uniform and pointwise continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called

In the former case, the particular value chosen for δ can be a function of both ε and x, the variables that precede it.In the latter case, δ can be a function only of ε (i.e., it has to be chosen independent of x). For example, f (x) = x satisfies pointwise, but not uniform continuity (its slope is unbound).In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning.

As a general rule, swapping two adjacent universal quantifiers with the same scope (or swapping two adjacent existential quantifiers with the same scope) doesn't change the meaning of the formula (see Example here), but swapping an existential quantifier and an adjacent universal quantifier may change its meaning.

The maximum depth of nesting of quantifiers in a formula is called its " quantifier rank ".
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