 Armstrong (1997) works with a different notion of independency, which we may call independency*. A set of facts is independent* iff given any of it subsets \(G\) and any of its member \(x\) not in \(G\), it is possible that the members of \(G\) exist without \(x\), and it is also possible that they exist with \(x\). Independency is strictly stronger than independency*: The independency*, but not the independency, of a set is compatible with there being a member of that set existing in every world (not the same member each time, of course). Armstrong expresses the hope that the set of atomic facts he countenances is independent*, though he recognizes that there may be difficulties to defend the view. We do not know whether he would express the hope that the set of atomic facts he countenances is independent tout court. Anyway, even if he held the view he would not endorse \(\textrm{T}_3\), since he recognizes “totality facts” (cf.  section 2.4.2),  which are not conjunctions of atomic facts.