  Essentially, the denial rules for an operator correspond to the assertion rules for the dual operator. Several logics of denial have been investigated, in particular Nelson’s logics of “constructible falsity” motivated first by Nelson (1949) with respect to a certain realizability semantics. The main focus has been on his systems later called N3 and N4 which differ with respect to the treatment of contradiction (N4 is N3 without ex contradictione quodlibet). Using denial any approach to proof-theoretic semantics can be dualized by just exchanging assertion and denial and turning from logical constants to their duals. In doing so, one obtains a system based on refutation (= proof of denial) rather than proof. It can be understood as applying a Popperian view to proof-theoretic semantics.