Some proof theoretical remarks on quantification in ordinary language

TL;DR

This paper discusses proof-theoretical perspectives on quantification in natural language, emphasizing Hilbert's epsilon and tau operators to better align semantic representations with linguistic data.

Contribution

It introduces a proof-theoretic approach to quantification, proposing new guidelines for designing proof rules that better reflect natural language quantifiers.

Findings

Hilbert's epsilon and tau operators aid in constructing natural language-like semantic representations

Proof-theoretic methods can bridge the gap between formal quantification and empirical linguistic data

Guidelines for proof rule design improve the modeling of generalized quantifiers

Abstract

This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's oper- ator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to natural language, in particular with quantified noun phrases as individual terms. We also define guidelines for the design of the proof rules corresponding to generalised quantifiers.