Quantifier (logic)
Range of quantification

Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, say, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as " n " for natural numbers, and " x " for real numbers, although relying exclusively on naming conventions cannot work in general, since ranges of variables can change in the course of a mathematical argument.

A universally quantified formula over an empty range (like ∀ x ∈ ∅ x ≠ x {\displaystyle \forall x\!\in \!\varnothing \;x\neq x}) is always vacuously true. Conversely, an existentially quantified formula over an empty range (like ∃ x ∈ ∅ x = x {\displaystyle \exists x\!\in \!\varnothing \;x=x}) is always false.

A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification

means

In some mathematical theories, a single domain of discourse fixed in advance is assumed. For example, in Zermelo–Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above, to express

in Zermelo–Fraenkel set theory, one would write

where N is the set of all natural numbers.
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