Principle of Structural Induction

Theorem
Let $\mathcal L$ be a formal language.

Let the formal grammar of $\mathcal L$ be a bottom-up grammar.

Let $P \left({\phi}\right)$ be a statement (in the metalanguage of $\mathcal L$) about well-formed formulas $\phi$ of $\mathcal L$.

Then $P$ is true for all WFFs of $\mathcal L$ iff both:


 * $P \left({a}\right)$ is true for all letters $a$ of $\mathcal L$

and, for each rule of formation of $\mathcal L$, if $\phi$ is a WFF resulting from WFFs $\phi_1, \ldots, \phi_n$ by applying that rule, then:


 * $P \left({\phi}\right)$ is true as soon as $P \left({\phi_1}\right), \ldots, P \left({\phi_n}\right)$ are all true.