Principle of Structural Induction

Theorem
Let $\LL$ be a formal language.

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

Let $\map P \phi$ be a statement (in the metalanguage of $\LL$) about well-formed formulas $\phi$ of $\LL$.

Then $P$ is true for all WFFs of $\LL$ both:


 * $\map P a$ is true for all letters $a$ of $\LL$,

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


 * $\map P \phi$ is true only if $\map P {\phi_1}, \ldots, \map P {\phi_n}$ are all true.

Proof
Let $\phi$ be a WFF of $\LL$.

Then $\phi$ is the result of applying finitely many rules of formation of $\LL$.

If we can show that the result of each rule of formation satisfies $P$, we will have finished.

Suppose now that for a rule of formation $\mathbf R$, all preceding rules have produced WFFs satisfying $P$.

By the bottom-up nature of the formal grammar of $\LL$, $\mathbf R$ either:


 * Introduces a letter as a WFF; or
 * Constructs a new WFF from previously constructed ones.

The two given hypotheses precisely ensure that the WFF resulting from $\mathbf R$ must also satisfy $P$.

For the first rule of formation applied, all preceding rules have vacuously produced WFFs satisfying $P$.

But now we see that any subsequent rule of formation will satisfy this premise.

In particular, it applies to the final rule of formation, and hence $\map P \phi$ is true.

Remark
Although the proof is strongly reminiscent of the Principle of Mathematical Induction, it can be seen to be a finite, algorithmic procedure:


 * Given any WFF $\phi$, we have a definite procedure for verifying $\map P \phi$, which involves only finitely many operations.

The assumptions of the theorem ensure that each of these operations will affirm the truth of $\map P \phi$.

It is therefore justifiable to accept this proof in the metalanguage.