Natural Language Quantifiers over Countably Infinite Domains

Abstract

Natural language quantifiers is generally thought to be restricted to finite domains. This restriction is mostly in place as a generalization to shield quantifier theory from the oddities of infinite domains. However we have intuitions about the interpretation of natural language quantifiers even when their domains are countably infinite. These intuitions can be captured in entailments. We expect an entailment between quantifiers over finite domains to be preserved over countably infinite domains. We will show that with straightforward expansion, this is not the case for proportional quantifiers. Therefore, we introduce the notion of stability for quantifiers over finite domains. Given this definition, we show that we can extend stable quantifiers to countably infinite domains. This extension preserves the entailments that hold over finite domain and abides by the natural language constraints of extension, conservativity and permutation invariance.