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

2025-08-19

AI Summary

This paper addresses the limitation of Russell’s theory of definite descriptions, which applies only to individual reference, by systematically extending it to second-order logic to formalize unique relational reference between objects. Building upon a strongly complete fragment of second-order logic under Henkin’s generalized semantics, we introduce a second-order definite description operator and formalize it within a cut-free sequent calculus framework. Our contributions are threefold: (1) the first theory of second-order definite descriptions for relational reference; (2) a proof of strong completeness of this theory with respect to Henkin semantics; and (3) a decidable proof system enabling effective deduction of relational referential statements. The results provide a novel formal tool for higher-order referential semantics and formal ontology.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Extends Russellian definite descriptions to second-order relations
• Develops theory within Henkin's general models framework
• Formalizes using cut-free sequent calculus for completeness

Innovation

• Second-order definite descriptions for relations
• Henkin's general models for completeness
• Cut-free sequent calculus formalization