Bilateralism with incompatible proofs and refutations
For logicians and philosophers, it provides a constructive bilateral framework that models incompatible proofs and refutations, reinforcing intuitionistic logic rather than revising it.
This paper presents a bilateral logic system where a formula cannot be both provable and refutable, formalized via natural deduction with normalisation and a base-extension semantics that is sound and complete. The refutation notion aligns with Nelson's constructive falsity, extending intuitionistic logic for epistemic reasoning.
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.