A proof-theoretical approach to some extensions of first order quantification

TL;DR

This paper explores proof-theoretical frameworks for non-standard quantifiers, including branching quantifiers, using second-order logic to extend traditional first-order quantification in logic and linguistics.

Contribution

It introduces a proof-theoretical approach to non-standard quantifiers, including branching quantifiers, via second-order logic, advancing the understanding of their formal properties.

Findings

Proof-theoretical treatment of first order quantifiers as second order concepts.

Second order translation of branching quantifiers applied to social knowledge scenarios.

Enhanced formal understanding of non-standard quantifiers in logic and linguistics.

Abstract

Generalised quantifiers, which include Henkin's branching quantifiers, have been introduced by Mostowski and Lindstr\"om and developed as a substantial topic application of logic, especially model theory, to linguistics with work by Barwise, Cooper, Keenan. In this paper, we mainly study the proof theory of some non-standard quantifiers as second order formulae . Our first example is the usual pair of first order quantifiers (for all / there exists) when individuals are viewed as individual concepts handled by second order deductive rules. Our second example is the study of a second order translation of the simplest branching quantifier: ``A member of each team and a member of each board of directors know each other", for which we propose a second order treatment.