The generalized quantifiers of natural language are predicatively definable

Abstract

This paper studies the definability of natural language generalized quantifiers. The semantics of generalized quantifiers are provided by a collection of subsets of the underlying domain. However, the generalized quantifiers appearing in natural language are definable either by first-order quantification or by cardinality notions. This paper provides an explanation for this observed phenomenon. The explanation is that the famous constraints of domain independence and conservativity, when extended to Henkin models, suffice to ensure low-level definability, namely $Δ^1_1$-definability or at least $Σ^1_1$-definability; and in most cases this definability can be made to be bounded. This is basically a consequence of Feferman's Preservation Theorem, which Marker has provided a short model-theoretic proof of. Further, we verify that the paradigmatic cardinality quantifiers are indeed $Δ^1_1$-definable for a reasonable choice of background theory. Finally, in many other cases, we show that this definability can be lowered to first-order definability.