Proofs, Disproofs, and Their Duals

Bi-intuitionistic logic, also known as Heyting-Brouwer logic or subtractive logic, is extended in various ways by a strong negation connective used to express commitments arising from denials. These logics have been introduced and investigated in [48]. In the present paper, an inferentialist semantics in terms of proofs, disproofs, and their duals is developed. Whereas the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic uses just the notion of proof as primitive, and López-Escobar’s inferentialist interpretation of Nelson’s logics with strong negation utilizes only the notions of proof and disproof as primitive, the inferentialist interpretation of bi-intuitionistic logic with strong negation employs the four notions of proofs, disproofs, dual proofs, and dual disproofs as primitive concepts.