In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, ^[1] determining the range in the formula to which the quantifier or connective is applied. ^{[2] [3] [4]} The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, ^{[2] [5]} and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. ^{[6] [7]}

Connectives

Logical connectives
AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. ^{[2] [6] [8]} The connective with the largest scope in a formula is called its dominant connective, ^{[9] [10]} main connective, ^{[6] [8] [7]} main operator, ^[2] major connective, ^[4] or principal connective; ^[4] a connective within the scope of another connective is said to be subordinate to it. ^[6]

For instance, in the formula ${\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))}$, 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 →. ^[6] 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 ${\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q}$, which some may find easier to read. ^[6]

Quantifiers

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] (or xP) ^[11] is the scope of the quantifier ∀x ^[5] (or ∀). ^[11]

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

• An occurrence of a quantifier ${\displaystyle \forall }$ or ${\displaystyle \exists }$, immediately followed by an occurrence of the variable ${\displaystyle \xi }$, as in ${\displaystyle \forall \xi }$ or ${\displaystyle \exists \xi }$, is said to be ${\displaystyle \xi }$-binding. ^{[1] [5]}
• An occurrence of a variable ${\displaystyle \xi }$ in a formula ${\displaystyle \phi }$ is free in ${\displaystyle \phi }$ if, and only if, it is not in the scope of any ${\displaystyle \xi }$-binding quantifier in ${\displaystyle \phi }$; otherwise it is bound in ${\displaystyle \phi }$. ^{[1] [5]}
• A closed formula is one in which no variable occurs free; a formula which is not closed is open. ^{[12] [1]}
• An occurrence of a quantifier ${\displaystyle \forall \xi }$ or ${\displaystyle \exists \xi }$ is vacuous if, and only if, its scope is ${\displaystyle \forall \xi \psi }$ or ${\displaystyle \exists \xi \psi }$, and the variable ${\displaystyle \xi }$ does not occur free in ${\displaystyle \psi }$. ^[1]
• 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 ${\displaystyle \zeta }$. ^[12]

See also

• Modal scope fallacy
• Prenex form
• Glossary of logic

Notes

References