Natural Density and The Quantifier 'Most'

This paper proposes a formalization of the class of sentences quantified by $\mathit{\text{most}}$, which is also interpreted as {\em proportion of} or {\em majority of} depending on the domain of discourse. We consider sentences of the form “$\mathit{\text{Most}}\mathit{\text{A}}\mathit{\text{are}}\mathit{\text{B}}$”, where $\mathit{\text{A}}$ and $\mathit{\text{B}}$ are plural nouns and the interpretations of $A$ and $B$ are infinite subsets of $\mathbb{N}$. There are two widely used semantics for $\mathit{\text{Most}}\mathit{\text{A}}\mathit{\text{are}}\mathit{\text{B}}$: (i) $C(A \cap B) > C(A\setminus B) $and(ii)$ C(A\cap B) > \dfrac{C(A)}{2} $,where$ C(X) $denotesthecardinalityofagivenfiniteset$ 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 $\mathit{\text{most}}$ but define it as applying to finite approximations of infinite sets computed to the limit.