<p>Robert Brandom has described “implication space” semantics, first put forward in Daniel Kaplan’s dissertation, as the “holy grail” of inferentialist semantics. However, when one tries to work through the details of the semantic framework, it is very hard to make intuitive sense of what the semantic clauses actually say. In this paper, I articulate a novel interpretation of implication space semantics as a bilateral successor to Brandom’s earlier “incompatibility semantics.” On this interpretation of the framework, semantic values are assigned, in the first instance, to “positions” consisting in assertions and denials, and the semantic value of such a position is the set of positions that are incompatible with it. I show how this interpretation renders otherwise obscure features of this semantic framework intuitively intelligible in terms of notions previously articulated by Brandom, most notably, that of incompatibility entailment.</p>