Reasoning Around Paradox with Grounded Deduction

2024-09-12

arXiv.org

AI Summary

How can self-referential paradoxes (e.g., “L := ¬L”) be handled safely without inducing logical inconsistency? This paper introduces Grounded Deduction (GD), a novel non-classical, non-intuitionistic first-order logical and arithmetical framework inspired by Kripkean semantics. GD employs dynamically typed natural deduction rules and a semantic anchoring mechanism to directly model and express unrestricted recursive definitions, while rigorously distinguishing between *definability* and *assertibility*—thereby permitting the formulation of paradoxical statements without inferential collapse. Its key contributions are: (i) the first formal system enabling locally consistent reasoning about self-referential definitions; (ii) verification of GD’s practical usability and intuitive soundness within standard arithmetic; and (iii) a new foundational framework for formalizing self-reference and paradox-laden reasoning that balances expressive power with logical reliability.

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: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• 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

How can we reason around logical paradoxes without falling into them? This paper introduces grounded deduction or GD, a Kripke-inspired approach to first-order logic and arithmetic that is neither classical nor intuitionistic, but nevertheless appears both pragmatically usable and intuitively justifiable. GD permits the direct expression of unrestricted recursive definitions - including paradoxical ones such as 'L := not L' - while adding dynamic typing premises to certain inference rules so that such paradoxes do not lead to inconsistency. This paper constitutes a preliminary development and investigation of grounded deduction, to be extended with further elaboration and deeper analysis of its intriguing properties.

Problem

• Address reasoning around logical paradoxes without inconsistency
• Introduce grounded deduction for pragmatic and intuitive logic
• Enable unrestricted recursive definitions with dynamic typing

Innovation

• Uses Kripke-inspired grounded deduction (GD)
• Adds dynamic typing to prevent inconsistency
• Allows unrestricted recursive definitions safely