# Natural Density and The Quantifier 'Most'

## TL;DR

This paper formalizes the semantics of the quantifier 'most' for infinite sets using natural density, proposing a new approach that considers finite approximations and axiomatizes natural density.

## Contribution

It introduces a novel semantics for 'most' based on natural density, addressing limitations of existing cardinal-based interpretations for infinite sets.

## Key findings

- Natural density provides a more sensitive interpretation of 'most' for infinite sets.
- The proposed semantics aligns with statistical intuition of 'most' as majority.
- Axiomatization of natural density supports formal reasoning about infinite set quantification.

## Abstract

This paper proposes a formalization of the class of sentences quantified by \textit{most}, which is also interpreted as {\em proportion of} or {\em majority of} depending on the domain of discourse. We consider sentences of the form "\textit{Most A are B}", where \textit{A} and \textit{B} are plural nouns and the interpretations of $ A $ and $ B $ are infinite subsets of $ \mathbb{N} $. There are two widely used semantics for \textit{Most A are B}: (i) $C(A \cap B) > C(A\setminus B) $ and (ii) $ C(A\cap B) > \dfrac{C(A)}{2} $, where $ C(X) $ denotes the cardinality of a given finite set $ X $. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier {\em most} has a solid cardinal behaviour under the interpretation {\em majority} and has a slightly more statistical behaviour under the interpretation {\em proportional of}, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using {\em natural density} for sentences in which interpretations of their nouns are infinite subsets of $ \mathbb{N} $, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of \textit{most} but define it as applying to finite approximations of infinite sets computed to the limit.