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 $

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Also see

• Results about scope in the context of Logic can be found here.

Sources

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• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 5$: Using Brackets
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): scope