text: Definition:Scope (Logic)
  
  Definition
  
  Let $\LL_0$ be the language of propositional logic.
  
  Let $\circ$ be a connective of $\LL_0$.
  
  Let $\mathbf W$ be a well-formed formula of $\LL_0$.
  
  Connective of Propositional Logic
  
  The scope of an occurrence of $\circ$ in $\mathbf W$ is defined as:
  
  the smallest well-formed part of $\mathbf W$ containing this occurrence of $\circ$.
  
  Quantifier of Predicate Logic
  
  Let $\mathbf A$ be a WFF of the language of predicate logic.
  
  Let $Q$ be an occurrence of a quantifier in $\mathbf A$.
  
  Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$.
  
  That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$.
  
  $\mathbf B$ is called the scope of the quantifier $Q$.
  
  Examples
  
  Arbitrary Example $1$
  
  Let $\circ$ be a binary logical connective in a compound statement $p \circ q$.
  
  The scope of $\circ$ is $p$ and $q$.
  
  Arbitrary Example $2$
  
  Consider the statement:
  
  $\paren {p \land \paren {q \lor r} } \implies \paren {s \iff \neg \, t}$
  
  We have:
  
  $(1): \quad$ The scope of $\land$ is $p$ and $\paren {q \lor r}$.
  $(2): \quad$ The scope of $\lor$ is $q$ and $r$.
  $(3): \quad$ The scope of $\implies$ is $\paren {p \land \paren {q \lor r} }$ and $\paren {s \iff \neg \, t}$.
  $(4): \quad$ The scope of $\iff$ is $s$ and $\neg \, t$.
  $(5): \quad$ The scope of $\neg$ is $t$.
  
  Arbitrary Example $3$
  
  Consider the statement:
  
  $\exists x: \paren {x < y} \lor y = 0$
  
  We have:
  
  $(1): \quad$ The scope of $\exists$ is $x$.
  $(2): \quad$ The scope of $\exists x$ is $\exists x: \paren {x < y}$.
  $(3): \quad$ The scope of $=$ is $y$ and $0$.
  $(4): \quad$ The scope of $\lor$ is $\exists x: \paren {x < y}$ and $y = 0$.
  Scopes for Connectives and Quantifiers in the formula: $\exists x: \paren { x < y} \lor y = 0$
  Scope (Logic)/Connective
  Scope (Logic)/Connective Scope (Logic)/Connective Scope 1 Scope 2
  $=$ $y=0$ $y$ $0$
  $\lor$ $\exists x: \paren { x < y } \lor y=0$ $\exists x: \paren { x < y }$ $y=0$
  Quantifier Scope (Logic)/Quantifier Scope
  $\exists x$ $\exists x: \paren{ x < y }$ $x < y$
  
  Also see
  
  • Results about scope in the context of Logic can be found here.