 Logics in which the Necessitation rule is valid are called normal modal logics. But historically, not all of the modal logics of interest are normal logics. C.I. Lewis’s systems \(\mathbf{S2}\) and \(\mathbf{S3}\) are non-normal logics, for example. These logics are weaker than normal modal logics, in the sense that they support fewer valid inferences. To give a worlds-based semantics for such logics, we need to look beyond possible worlds.