Definition:Subordinate

Definition

Let $\mathbf A$ be a WFF of propositional logic.

Let $\circ$ and $\ast$ be connectives.

Then $\circ$ is subordinate to $\ast$ (in $\mathbf A$) if and only if the scope of $\circ$ is a well-formed part of the scope of $\ast$.

Example

Consider the WFF of propositional logic:

$\mathbf A := \paren {\paren {P_0 \land P_1} \implies \paren {\paren {P_2 \lor \neg P_3} \land \paren {P_4 \iff P_5} } }$

It is left as an exercise to the reader to demonstrate that $\mathbf A$ is well-formed.

The following statements are also left as exercises to prove:

• The main connective of $\mathbf A$ is $\implies$. Therefore the scope of $\implies$ is $\mathbf A$.

• The substring:

$\mathbf B := \paren {\paren {P_2 \lor \neg P_3} \land \paren {P_4 \iff P_5} }$

is a well-formed part of $\mathbf A$.

• The main connective of $\mathbf B$ is $\land$. Therefore the scope of $\land$ (the second occurrence of it in $\mathbf A$, of course) is $\mathbf B$.

• The connective $\iff$ is subordinate to the occurrence of $\land$ in $\mathbf B$, but not to $\lor$ or $\neg$.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $1$ Formation Rules