Abstract: A full semantic theory for natural language needs to at least partially explain a lot of speaker intuitions: intuitions of entailment, of incompatibility, of synonymy, of truth conditions, and so on. Different approaches to semantic theory might take one of these to be primary, and attempt to explain the others derivatively. For example, much mainstream semantic theorizing takes truth conditions as primary, using them to explain all of entailment, incompatibility, and synonymy. In this talk, I will pursue a different option. Developing ideas from Greg Restall (and, earlier, Christine Ladd-Franklin and Peter Strawson), I will put forward a theory on which incompatibility is taken as primary, and the other notions are seen to be derivative. It's been argued that such an approach cannot make good sense of entailment in particular, essentially because incompatibility is about ruling things *out* while entailment is about ruling things *in*. I will show that this is not the case. By developing a notion of implicit assertion, I will argue that an incompatibility-based approach can give us a solid and useful understanding of entailment.