In
logic,
quantification is a construct that specifies the quantity of specimens in the
domain of discourse that satisfy an open formula. For example, in arithmetic, it allows the expression of the statement that every natural number has a successor. A language element which generates a quantification (such as "every") is called a
quantifier. The resulting expression is a quantified expression, it is said to be
quantified over the predicate (such as "the natural number
x has a successor") whose
free variable is bound by the quantifier. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The
semantics of the language specifies how the constructor is interpreted. Two fundamental kinds of quantification in
predicate logic are
universal quantification and
existential quantification. The traditional symbol for the universal quantifier "all" is "∀", a rotated letter "
A", and for the existential quantifier "exists" is "∃", a rotated letter "
E". These quantifiers have been generalized beginning with the work of
Mostowski and
Lindström.