**Definition**: Quantifier, Bound Variables, Free Variables

A **quantifier** is a non-empty string over an alphabet used in a logical calculus to make quantitative statements about how many values from the domain of discourse a given variable can take. Depending on this quantity, the string containing the variable with its quantifier can be valued as true or false.

More exactly, let $L$ be a formal language, $U$ the domain of discourse, and $I(U,L)$ the corresponding interpretation. If an interpretable string $s\in L$ contains a variable, a **quantifier attached to** that variable is a symbol expressing how many values in $U$ the variable can take. Depending on this quality, the valuation $[[s]]_I$ can be either true or false.

A variable with a quantifier attached to it is called a **bound variable**, otherwise, it is called a **free variable**.

Unlike different types of quantifiers in natural languages like “many”, “a lot”, “no”, “for some”, “a few”, logical calculi generally use two types of quantifiers:

**existential quantifier**\(\exists\): read “there exists”, symbolized by rotated letter “E”,**universal quantifier**\(\forall\): read “for all” or “for every”.

| | | | | Contributors: *bookofproofs*, *guest*

## 1.**Example**: Examples of Quantifiers in a Logical Calculus

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