We consider a family 𝔲 of finite universes. The second order existential quantifier Qℜ, means for each U∈ 𝔲 quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.
"A dichotomy in classifying quantifiers for finite models." J. Symbolic Logic 70 (4) 1297 - 1324, December 2005. https://doi.org/10.2178/jsl/1129642126