Quantification - Logic

Logic

In language and logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that apply to (or 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 is called a quantifier. The resulting expression is a quantified expression; and the expression is said to be quantified over the predicate or function expression whose free variable is bound by the quantifier. Quantification is used in both natural languages and formal languages. Examples of quantifiers in English are for all, for some, many, few, a lot, and no. 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 as an extent of validity.

The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. The traditional symbol for the universal quantifier "all" is "∀", an inverted 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.