Scope (logic)

Quantifiers

The scope of a quantifier is the part of a logical expression over which the quantifier exerts control. It is the shortest full sentence written right after the quantifier, often in parentheses; some authors describe this as including the variable written right after the universal or existential quantifier. In the formula ∀xP, for example, P (or xP) is the scope of the quantifier ∀x (or ∀).

This gives rise to the following definitions:

• An occurrence of a quantifier ${\displaystyle \mathrm{\forall }}$ or ${\displaystyle \mathrm{\exists }}$, immediately followed by an occurrence of the variable ${\displaystyle \xi }$, as in ${\displaystyle \mathrm{\forall }\xi }$ or ${\displaystyle \mathrm{\exists }\xi }$, is said to be ${\displaystyle \xi }$-binding.
• An occurrence of a variable ${\displaystyle \xi }$ in a formula ${\displaystyle \varphi }$ is free in ${\displaystyle \varphi }$ if, and only if, it is not in the scope of any ${\displaystyle \xi }$-binding quantifier in ${\displaystyle \varphi }$; otherwise it is bound in ${\displaystyle \varphi }$.
• A closed formula is one in which no variable occurs free; a formula which is not closed is open.
• An occurrence of a quantifier ${\displaystyle \mathrm{\forall }\xi }$ or ${\displaystyle \mathrm{\exists }\xi }$ is vacuous if, and only if, its scope is ${\displaystyle \mathrm{\forall }\xi \psi }$ or ${\displaystyle \mathrm{\exists }\xi \psi }$, and the variable ${\displaystyle \xi }$ does not occur free in ${\displaystyle \psi }$.
• A variable ${\displaystyle \zeta }$ is free for a variable ${\displaystyle \xi }$ if, and only if, no free occurrences of ${\displaystyle \xi }$ lie within the scope of a quantification on ${\displaystyle \zeta }$.
• A quantifier whose scope contains another quantifier is said to have wider scope than the second, which, in turn, is said to have narrower scope than the first.