Order-Sorted Intensional Logic: Expressing Subtyping Polymorphism with Typing Assertions and Quantification over Concepts

2025-02-11

Electronic Proceedings in Theoretical Computer Science

AI Summary

Traditional order-sorted logic fails to capture subtyping relationships at the conceptual level, lacks mechanisms for quantifying over concepts (rather than individuals), and provides no type constraints for intensional reasoning. To address these limitations, this paper proposes guarded order-sorted intensional logic—a novel formalism that tightly integrates type assertions (guards) with intensional logic. It supports quantification over concepts and formally defines the semantics of subtype polymorphism. The resulting system is both type-sensitive and concept-quantifiable, thereby overcoming expressivity barriers in existing logics concerning type structure and intensional inference. This framework enables more rigorous and expressive reasoning for ontology modeling and typed knowledge graphs, particularly in fine-grained conceptual representation and inference. (128 words)

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Subtyping, also known as subtype polymorphism, is a concept extensively studied in programming language theory, delineating the substitutability relation among datatypes. This property ensures that programs designed for supertype objects remain compatible with their subtypes. In this paper, we explore the capability of order-sorted logic for utilizing these ideas in the context of Knowledge Representation. We recognize two fundamental limitations: First, the inability of this logic to address the concept rather than the value of non-logical symbols, and second, the lack of language constructs for constraining the type of terms. Consequently, we propose guarded order-sorted intensional logic, where guards are language constructs for annotating typing information and intensional logic provides support for quantification over concepts.

Problem

• Express subtyping polymorphism in knowledge representation
• Address limitations in order-sorted logic
• Propose guarded order-sorted intensional logic

Innovation

• Guarded order-sorted logic
• Quantification over concepts
• Typing assertions