Definition:Scope (Logic)

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 $\LL_0$ be the language of propositional logic.

Let $\circ$ be a connective of $\LL_0$.

Let $\mathbf W$ be a well-formed formula of $\LL_0$.

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:

$\paren {p \land \paren {q \lor r} } \implies \paren {s \iff \neg \, t}$

1. The scope of $\land$ is $p$ and $\paren {q \lor r}$.
2. The scope of $\lor$ is $q$ and $r$.
3. The scope of $\implies$ is $\paren {p \land \paren {q \lor r} }$ and $\paren {s \iff \neg \, t}$.
4. The scope of $\iff$ is $s$ and $\neg \, t$.
5. 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
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): scope