“Always More,” p. 223–229 in Logica Yearbook 2009, edited by Michal Pelis, College Publications, 2010.
A possible world is a point in logical space. It plays a dual role with respect to propositions. (1) A possible world determines the truth value of every proposition. For each world w and proposition p, either at w, p is true, or at w, p is not true. (2) Each set of possible worlds determines a proposition. If S, a subset of W is a set of worlds, there is a proposition p true at exactly the worlds in S.
In this paper, I construct a logic, extending classical logic with a single unary operator, which has no complete Boolean algebras as models. If the family of propositions we are talking about in (1) and (2) has the kind of structure described in that logic, then (1) and (2) cannot jointly hold. I then explain what this might mean for theories of propositions and possible worlds.