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”.
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