stylistic:tone, structural:summarization
Notation

The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is " ∀ ", a rotated letter " A ", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is " ∃ ", a rotated letter " E ", which stands for "there exists" or "exists".

An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P (x) the predicate " x likes to dance", and Q (x) the predicate " x likes to go to the beach". Then the above sentence can be written in formal notation as ∀ x ∈ X, (P (x) ∨ Q (x)) {\displaystyle \forall {x}{\in }X,(P(x)\lor Q(x))}, which is read, "for every x that is a member of X, P applies to x or Q applies to x ".

Some other quantified expressions are constructed as follows,

for a formula P. These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance", respectively.Variant notations include, for set X and set members x:

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.

Informally or in natural language, the "∀ x " or "∃ x " might appear after or in the middle of P (x). Formally, however, the phrase that introduces the dummy variable is placed in front.

Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as,

Keywords for uniqueness quantification include:

Further, x may be replaced by a pronoun. For example,