Scope (logic)

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.

Logical connectives
NOT ${\displaystyle \neg A,-A,{\overline {A}},\sim A}$ AND ${\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}$ NAND ${\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B,A+B,A\mid B,A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}$ XNOR ${\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}$ └ equivalent ${\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B,A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}$ nonimplication (NIMPLY) ${\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}$ converse ${\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}$ converse nonimplication ${\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}$
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. 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 ${\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))}$, 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 ${\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q}$, which some may find easier to read.

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 ${\displaystyle \xi }$, as in ${\displaystyle \forall \xi }$or ${\displaystyle \exists \xi }$, is said to be ${\displaystyle \xi }$-binding.
• 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 ${\displaystyle \phi }$.
• A closed formula is one in which no variable occurs free; a formula which is not closed is open.
• An occurrence of a quantifier ${\displaystyle \forall \xi }$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 ${\displaystyle \psi }$.
• 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.

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