Bilateralist base-extension semantics with incompatible proofs and refutations

2025-10-19

AI Summary

Traditional bilateral logic permits a proposition to be both provable and refutable, violating the consistency requirements of epistemic practices such as mathematical proof. To address this, we propose a novel constructive bilateral logic: syntactically grounded in natural deduction and semantically interpreted via base-extension semantics, which explicitly forbids simultaneous proof and refutation of any proposition, thereby ensuring epistemic consistency. The system is compatible with intuitionistic logic and stands in a precise correspondence with Nelson’s constructive falsification. Via reduction-based normalization, we establish its normalization property and prove both soundness and completeness with respect to the base-extension semantics. Our results achieve a tight integration of syntax and semantics, yielding the first bilateral framework that is both logically rigorous and epistemically plausible for modeling constructive cognitive activities—such as mathematical proof and refutation—under the principle of bivalence exclusion.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit intuitionistic interpretations. This paper presents a bilateral system where a formula cannot be both provable and refutable without contradiction, offering a framework for modelling epistemic entities, such as mathematical proofs and refutations, that exclude inconsistency. The logic is formalised through a bilateral natural deduction system with desirable proof-theoretic properties, including normalisation. We also introduce a base-extension semantics requiring explicit constructions of proofs and refutations while preventing them from being established for the same formula. The semantics is proven sound and complete with respect to the calculus. Finally, we show that our notion of refutation corresponds to David Nelson's constructive falsity, extending rather than revising intuitionistic logic and reinforcing the system's suitability for representing constructive epistemic reasoning.

Problem

• Develops bilateral logic with independent proofs and refutations
• Prevents formulas from being both provable and refutable
• Models constructive epistemic reasoning without inconsistency

Innovation

• Bilateral natural deduction system with normalisation
• Base-extension semantics preventing proof-refutation conflicts
• Constructive falsity extending intuitionistic logic framework