Rigorous definition of the set of forms of a propositional formula

Question

Consider the propositional formula $\neg \neg p$. Associated with it is a set of three forms: $A$, $\neg A$, and $\neg \neg A$. Now, consider the propositional formula $\neg (\neg p \land \neg q)$. Associated with it is a set of six forms: $A$, $\neg A$, $\neg(A \land B)$, $\neg(\neg A \land B)$, $\neg(A \land \neg B)$, and finally $\neg(\neg A \land \neg B)$. By giving these examples, it should be clear what I intuitively mean by a form. But how does one rigorously define what the set of forms of a propositional formula is? For instance, every formula is of the form $A$. Anyway, I would like a precise definition of my intuitive notion.

Answer 1

What you call forms are often called formula schemes with $A, B$ meta variables serving as placeholders for arbitrary formulas, and the concrete formulas are called instances.

$\phi$ is a substitution instance of $\psi$ iff there is a function from meta variables in $\psi$ to formluas in $\phi$ such that replacing in $\psi$ each occurrence of a schema symbol with the corresponding instance symbol yields $\phi$.

The "forms" of a formula are then the set of formula schemes (modulo naming of the metavariables) of which the formula is a substitution instance.