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

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Highlights

  • We characterize the definability of second-order generalized quantifiers:

  • Q1 is definable in 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.

Abstract

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.

Keywords

Second-order generalized quantifiers
Definability
Majority quantifier
Second-order logic
Collective quantification

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