Inlogic, thescope of aquantifier orconnective is the shortest formula in which it occurs,^[1] determining the range in theformula to which the quantifier or connective is applied.^[2][3][4] The notions of afree variable and bound variable are defined in terms of whether that formula iswithin the scope of a quantifier,^[2][5] and the notions of adominant connective andsubordinate connective are defined in terms of whether a connective includes anotherwithin its scope.^[6][7]

The scope of a logical connective occurring within a formula is the smallestwell-formed formula that contains the connective in question.^[2][6][8] The connective with the largest scope in a formula is called itsdominant connective,^[9][10]main connective,^[6][8][7]main operator,^[2]major connective,^[4] orprincipal connective;^[4] a connective within the scope of another connective is said to besubordinate to it.^[6]

For instance, in the formula${\displaystyle (\left(\left(P\to Q\right)\vee \mathrm{\neg }Q\right)\leftrightarrow \left(\mathrm{\neg }\mathrm{\neg }P\wedge Q\right))}$, 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 →.^[6] If anorder 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${\displaystyle \left(P\to Q\right)\vee \mathrm{\neg }Q\leftrightarrow \mathrm{\neg }\mathrm{\neg }P\wedge Q}$, which some may find easier to read.^[6]

The scope of a quantifier is the part of a logical expression over which the quantifier exerts control.^[3] It is the shortest full sentence^[5] written right after the quantifier,^[3][5] often in parentheses;^[3] some authors^[11] describe this as including the variable written right after the universal or existential quantifier. In the formula∀xP, for example,P^[5] (orxP)^[11] is the scope of the quantifier∀x^[5] (or∀).^[11]

This gives rise to the following definitions:^[a]

• 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.^[1][5]
• An occurrence of a variable${\displaystyle \xi }$ in a formula${\displaystyle \varphi }$ isfree in${\displaystyle \varphi }$ if, and only if, it is not in the scope of any${\displaystyle \xi }$-binding quantifier in${\displaystyle \varphi }$; otherwise it isbound in${\displaystyle \varphi }$.^[1][5]
• Aclosed formula is one in which no variable occurs free; a formula which is not closed isopen.^[12][1]
• An occurrence of a quantifier${\displaystyle \mathrm{\forall }\xi }$ or${\displaystyle \mathrm{\exists }\xi }$ isvacuous 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${\displaystyle \psi }$.^[1]
• A variable${\displaystyle \zeta }$ isfree for a variable${\displaystyle \xi }$ if, and only if, no free occurrences of${\displaystyle \xi }$ lie within the scope of a quantification on${\displaystyle \zeta }$.^[12]
• A quantifier whose scope contains another quantifier is said to havewider scope than the second, which, in turn, is said to havenarrower scope than the first.^[13]

• Modal scope fallacy
• Prenex form
• Glossary of logic
• Free variables and bound variables
• Open formula

• Quantifier (logic)
• Predicate logic
• Mathematical terminology

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