This paper investigates an abstract duality existing between paracomplete and para- consistent logics, with the suggestion that these can be understood as co-constructive logics of proofs and refutations. In the current literature, dual-logics (including self-dual logics) are typically held together by some form of coherence principle, where, roughly, if a formula is a theorem (or provable) in one system then the dual sentence will be a counter-theorem (or refutable) in the dual system. We formalize this by means of a Ga- lois connection between dual logics, and show that coherence holds for: classical logic; Greg Restall’s [2] inferentialist approach to logic by means of assertion and denial; gen- eral logics of proofs and refutations (e.g. [3, 4, 5]); Dummett’s [1] consideration of a dual falsificationism logic to verificationism; Urbas’ [6] analysis of dual-intuitionistic logic. In so doing, it is also simple to see why the coherence principle renders such systems fairly uninteresting (for example, bi-intuitionistic logic, which combines intuitionistic and dual-intuitionistic logic contains theorems which are constructively unacceptable). By syntactically separating dual calculi for intuitionistic and co-intuitionistic logic, we then investigate structures where the coherence principle does not hold unrestrictedly, and which generate non-trivial inferentialist semantics. Philosophically, we understand the relation between dual calculi in terms of a dialogue between “prover” and “refuter”, allowing for both potential and conclusive proofs (refutations).