Scope (logic)

Template:Short description Template:Use mdy dates 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 Template:Glossary link and Template:Glossary link are defined in terms of whether a connective includes another within its scope.

Connectives

Template:Sidebar 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 $(((P\to Q)\vee \mathrm{\neg }Q)\leftrightarrow (\mathrm{\neg }\mathrm{\neg }P\wedge Q))$, 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 $(P\to Q)\vee \mathrm{\neg }Q\leftrightarrow \mathrm{\neg }\mathrm{\neg }P\wedge Q$, 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 Template:Math, for example, Template:Math (or Template:Math) is the scope of the quantifier Template:Math (or Template:Math).

This gives rise to the following definitions:Template:Refn

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

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