Bilateralist base-extension semantics with incompatible proofs and refutations
Victor Barroso-Nascimento, Maria Osório Costa, Elaine Pimentel
TL;DR
This work develops a bilateral base-extension semantics (BeS) for a constructive logic of proofs and refutations, BPR, which treats assertion and denial as independent yet incompatible acts. By extending a bi-intuitionistic foundation ($N2Int^*$) with paired proof/refutation rules and coordination rules $PR(+)$/$PR(-)$, it achieves normalisation and the subformula property, guaranteeing consistency. The BeS framework imposes epistemic constraints and explicit base-consistency, proving soundness and completeness of BPR relative to the semantic model and clarifying the relationship between constructive falsity and Nelson's strong negation. Overall, the paper advances a coherent, ecumenical approach to constructive epistemic reasoning where proofs and counterexamples are handled explicitly without explosion, aligning with Nelson's constructive falsity while preserving intuitionistic foundations.
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.