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 mathematical proofs and refutations that exclude inconsistency. We formalise the logic via a bilateral natural deduction system with the desirable proof-theoretic properties of normalisation, subformula property and consistency, together with a base-extension semantics grounded in explicit proofs and refutations. Finally, refutation is shown to coincide with Nelson’s constructive falsity, extending intuitionistic logic for constructive epistemic reasoning.

Abstract We define base-extension semantics ( $$\textsf{BeS}$$ BeS ) using atomic systems based on sequent calculus rather than natural deduction. While traditional $$\textsf{BeS}$$ BeS aligns naturally with intuitionistic logic due to its constructive foundations, we show that sequent calculi with multiple conclusions yield a $$\textsf{BeS}$$ BeS framework more suited to classical semantics. The harmony in classical sequents leads to straightforward semantic clauses derived solely from right introduction rules. This framework enables a Sandqvist-style completeness proof that extracts a sequent calculus proof from any valid semantic consequence. Moreover, we show that the inclusion or omission of atomic cut rules meaningfully affects the semantics, yet completeness holds in both cases.