Think of the Lévy hierarchy in set theory, or higher-order logic more generally. Arithmetic and proof structures recursed over a first-order base can have surprisingly new characteristics, like you can prove different things in different ways as you go up, e.g. the well-ordering lemma can be used to derive the axiom of choice but not vice versa. Or in one family of intuitionistic logic the choice axiom allows deriving the law of excluded middle.

Subtly different structural inputs with substantially different outputs of content.