It is well known that classical inferentialist semantics runs into problems regarding abnormal valuations [3, 6, 10]. It is equally well known that the issues can be resolved if we construct the inference relation in a multiple-conclusion sequent calculus. The latter has been prominently developed in recent work by Greg Restall [13], with the guiding interpretation that the valid sequent Γ \vdash ∆ says that the simultaneous assertion of all of Γ with the denial of all of ∆ is incoherent. However, such structures face significant interpretive challenges [14, 19, 20], and they do not provide an adequate grasp on the machinery of the duality of assertions and denials that could (a) provide an abstract account of inferential semantics; (b) show why the dual treatment is semantically superior. This paper explores a slightly different tack by considering a dual-calculus framework consisting of two, single-conclusion, inference relations dealing with the preservation of assertion and the preservation of denial, respectively. In this context, I develop an abstract inferentialist semantics, before going on to show that the framework is equivalent to Restall’s, whilst providing a better grasp on the underlying proof-structure.

The final publication is available at Springer via http://dx.doi.org/10.1007/s10516-014-9263-6