# Definition:Scope (Logic)

## Contents

## Definition

The **scope** of a logical connective is defined as the statements that it connects, whether this be simple or compound.

In the case of a unary connective, there will be only one such statement.

### Connective of Propositional Logic

Let $\mathcal L_0$ be the language of propositional logic.

Let $\circ$ be a connective of $\mathcal L_0$.

Let $\mathbf W$ be a well-formed formula of $\mathcal L_0$.

The **scope** of an occurrence of $\circ$ in $\mathbf W$ is 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

Let $\circ$ be a binary logical connective in a compound statement $p \circ q$.

The **scope** of $\circ$ is $p$ and $q$.

Consider the statement:

- $\left({p \land \left({q \lor r}\right)}\right) \implies \left({s \iff \neg \, t}\right)$

- The scope of $\land$ is $p$ and $\left({q \lor r}\right)$.
- The scope of $\lor$ is $q$ and $r$.
- The scope of $\implies$ is $\left({p \land \left({q \lor r}\right)}\right)$ and $\left({s \iff \neg \, t}\right)$.
- The scope of $\iff$ is $s$ and $\neg \, t$.
- The scope of $\neg$ is $t$.

## Sources

- 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 5$: Using Brackets