Abstract

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a characterizing frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics.