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 and  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 $$(\left( \left( P \rightarrow Q \right) \lor \lnot Q \right) \leftrightarrow \left( \lnot \lnot P \land Q \right))$$, 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 $$\left ( P \rightarrow Q \right) \lor \lnot Q \leftrightarrow \lnot \lnot P \land Q $$, 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 $$\forall$$ or $$\exists$$, immediately followed by an occurrence of the variable $$\xi$$, as in $$\forall \xi$$ or $$\exists \xi$$, is said to be $$\xi$$-binding.
 * An occurrence of a variable $$\xi$$ in a formula $$\phi$$ is free in $$\phi$$ if, and only if, it is not in the scope of any $$\xi$$-binding quantifier in $$\phi$$; otherwise it is bound in $$\phi$$.
 * A closed formula is one in which no variable occurs free; a formula which is not closed is open.
 * An occurrence of a quantifier $$\forall \xi$$ or $$\exists \xi$$ is vacuous if, and only if, its scope is $$\forall \xi \psi$$ or $$\exists \xi \psi$$, 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 $$\zeta$$.