Consider the following statement (using dot notation for multiplication):
All of these variations also apply to universal quantification. Other variations for the universal quantifier are
Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways:
One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used.
Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as,
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 inversed is different:
The maximum depth of nesting of quantifiers in a formula is called its "quantifier rank".
This notation is known as restricted or relativized or bounded quantification. Equivalently one can write,
The existential proposition can be expressed with bounded quantification as
Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:
A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification
None of the quantifiers previously discussed apply to a quantification such asOutline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic