Definition:Scope (Logic)

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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:

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