 This is a very strong principle. Under reasonable assumptions, P1 entails the strong principle W4. For instance, this is so under the assumption that every set of worlds is the truth-set of some proposition. This assumption trivially holds if sets of worlds are taken to be propositions, but such an assumption is not needed. For let \(V\) be any set of worlds. Assume that No Twins is true. To each world \(w\) associate the conjunction \(c_w\) of all propositions true in \(w\) (we suppose it exists), and consider the disjunction \(d_V\) of all \(c_w, w\in V\) (we suppose it exists). Then \(\TS(d_V) = V\).