 Closely connected to belief is the notion of propositional content. Within possible worlds semantics,  propositions  can be defined as functions from worlds to truth values, or as sets of worlds: a proposition is the set of worlds at which it is true. The account is notoriously too coarse-grained (Barwise 1997). Intuitively distinct impossible propositions (that swans are blue and not blue; that Fermat’s Last Theorem is false; that Charles is a married bachelor) all hold at precisely no possible worlds. And we have a dual problem with (unrestrictedly) necessary propositions, which are all identified with the set of all possible worlds. Treating propositions as set-theoretic constructions out of possible worlds leads to a very coarse individuation of propositions, and because of this it has been subject to seemingly devastating attacks, for instance, by Scott Soames 1987. However, impossible worlds allow for fine-grained distinctions unavailable in standard possible worlds semantics. An impossible proposition need not be equated with the empty set of worlds, for it may be a set which includes (only) impossible worlds. We can have an impossible world \(w_1\) with impossibly coloured swans, a distinct impossible world \(w_2\) at which Fermat’s Last Theorem is false, and a further impossible world \(w_3\) at which bachelors are married (but at which swans and Diophantine equations behave correctly). Ripley (2013) argues that an account along these lines is a better strategy for addressing the coarse-grainedness problem than resorting to  structured propositions.