A deductive construction of canonical spaces and coalgebras leads to completeness results. These give a proof-theoretic characterisation of the semantic consequence relation for the logic of any measurable polynomial functor as the least deduction system satisfying Lindenbaum's Lemma. It is also the only Lindenbaum system that is sound.
The theory is additionally worked out for Kripke polynomial functors, on the category of sets, that have infinite constant sets in their formation.