On a Second-Order Version of Russellian Theory of Definite Descriptions

TL;DR

This paper introduces a second-order extension of Russellian theory for definite descriptions, focusing on relations rather than objects, and formalizes it within a complete logical fragment.

Contribution

It proposes a novel second-order framework for definite descriptions that refer to relations, extending Russell's original object-focused theory.

Findings

Develops a second-order theory within Henkin's fragment

Formalizes the theory using a cut-free sequent calculus

Shows the logical completeness of the proposed framework

Abstract

Definite descriptions are first-order expressions that denote unique objects. In this paper, we propose a second-order counterpart, designed to refer to unique relations between objects. We investigate this notion within the framework of Russell's theory of definite descriptions. While full second-order logic is incomplete, its fragment defined by Henkin's general models admits completeness. We develop our theory within this fragment and formalize it using a cut-free sequent calculus.