A characterization of definability of second-order generalized quantifiers with applications to non-definability

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q1 is definable in terms of another quantifier Q2, the base logic being monadic second-order logic, reduces to the question if a quantifier Q1⋆ is definable in FO(Q2⋆,<,+,×) for certain first-order quantifiers Q1⋆ and Q2⋆. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. We also show that the monadic second-order majority quantifier Most1 is not definable in second-order logic