Quantification in ordinary language

TL;DR

This paper critiques the standard logical interpretation of natural quantification, highlighting its neglect of generic versus distributive readings, and proposes a proof-theoretic approach to better capture linguistic nuances, including non-first-order quantifiers.

Contribution

It introduces a proof-theoretic framework for natural quantification, addressing limitations of set-theoretic models and extending to non-first-order quantifiers like "the majority of."

Findings

Standard logic overlooks generic/distributive differences.

Proof-theoretic approach captures linguistic nuances.

Extends to non-first-order quantifiers.

Abstract

We firstly show that the standard interpretation of natural quantification in mathematical logic does not provide a satisfying account of its original richness. In particular, it ignores the difference between generic and distributive readings. We claim that it is due to the use of a set theoretical framework. We therefore propose a proof theoretical treatment in terms of proofs and refutations. Thereafter we apply these ideas to quantifiers that are not first order definable like "the majority of".