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

• We characterize the definability of second-order generalized quantifiers:
• Q1Q1 is definable in MSO(Q2,+)MSO(Q2,+) iff Q1⁎ is definable in FO(Q2⁎,+,×).
• We use our characterization to proof new definability results, e.g.:
• The monadic second-order majority quantifier is non-definable in SO.
• We discuss consequences for the linguistic semantics of collective quantifiers.

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q1Q1 is definable in terms of another quantifier Q2Q2, 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 Most1Most1 is not definable in second-order logic.