Domain Closure

Induction for finite languages is trivial. We simply use the Domain Closure rule of inference. For a language with object constants \(\sigma_1, \cdots, \sigma_n\),

\[ \begin{aligned} \phi[\sigma_1] \\ \cdots \,\,\, \\ \phi[\sigma_n] \\ \hline \\ \forall \nu.\phi[\nu] \end{aligned} \]

If we believe a schema is true for every instance, then we can infer a universally quantified version of that schema.

More at Introduction to Logic.