In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, determining the range in the formula to which the quantifier or connective is applied. The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope.

Connectives

The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. The connective with the largest scope in a formula is called its dominant connective, main connective, main operator, major connective, or principal connective; a connective within the scope of another connective is said to be subordinate to it.

For instance, in the formula ( ( ( P → Q ) ∨ ¬ Q ) ↔ ( ¬ ¬ P ∧ Q ) ) {\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))}, the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧; the first ¬ is also subordinate to the ∨, but not to the →; the second ¬ is subordinate to the ∧, but not to the ∨ or the →; and the third ¬ is subordinate to the second ¬, as well as to the ∧, but not to the ∨ or the →. If an order of precedence is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form ( P → Q ) ∨ ¬ Q ↔ ¬ ¬ P ∧ Q {\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q}, which some may find easier to read.

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 \forall } or ∃ {\displaystyle \exists }, immediately followed by an occurrence of the variable ξ {\displaystyle \xi }, as in ∀ ξ {\displaystyle \forall \xi } or ∃ ξ {\displaystyle \exists \xi }, is said to be ξ {\displaystyle \xi }-binding.
  • An occurrence of a variable ξ {\displaystyle \xi } in a formula ϕ {\displaystyle \phi } is free in ϕ {\displaystyle \phi } if, and only if, it is not in the scope of any ξ {\displaystyle \xi }-binding quantifier in ϕ {\displaystyle \phi }; otherwise it is bound in ϕ {\displaystyle \phi }.
  • A closed formula is one in which no variable occurs free; a formula which is not closed is open.
  • An occurrence of a quantifier ∀ ξ {\displaystyle \forall \xi } or ∃ ξ {\displaystyle \exists \xi } is vacuous if, and only if, its scope is ∀ ξ ψ {\displaystyle \forall \xi \psi } or ∃ ξ ψ {\displaystyle \exists \xi \psi }, and the variable ξ {\displaystyle \xi } does not occur free in ψ {\displaystyle \psi }.
  • A variable ζ {\displaystyle \zeta } is free for a variable ξ {\displaystyle \xi } if, and only if, no free occurrences of ξ {\displaystyle \xi } lie within the scope of a quantification on ζ {\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.

See also

Notes