Now, when I was reading it, I noticed a parallel between the problem Armstrong is addressing and the problem of transworld identity. Armstrong wants to know how different things can share properties, and the problem of transworld identity is about how different possible worlds can share inhabitants. Some issues in one debate have corresponding issues in the other, and this gives rise to analogies between the positions. I think at one point Armstrong even calls the thing he’s trying to explain ‘generic identity’.
Transworld identity gets explained by the worlds having either strictly identical things in them, or suitably similar things in them. Generic identity gets explained by particulars having strictly identical universals in them or suitably similar tropes. The ‘no explanation needed’ positions are magical modal realism, which reifies worlds but not possibilia, and ostrich nominalism, which reifies objects but not properties. (Both were named by their opponents.) Another parallel is that people disagree about whether particulars are bundles of properties or something besides their properties, and they also disagree about whether worlds are fusions of their inhabitants or something besides their inhabitants. The bundle theory of particulars is typically most plausible to trope theorists, and the fusion theory of worlds is typically most plausible to counterpart theorists.
When you spot a parallel between two debates it can help you in at least two ways. One is that the analogy can help us understand the more mysterious debate better. Our thinking about time became clearer when we realised it was a bit like space, and our thinking about modality became clearer when we realised it was a bit like time. I’m not sure whether we’ll be helped much in this way here: space and time really are quite like each other, whereas the relationship between a world and its inhabitants seems quite unlike the relationship between a thing and its properties, at least if the bundle theory isn't right.
The other way analogies between debates can help is more promising though. There’s some pressure to hold analogous positions in analogous areas, because a good argument for one position will often correspond to a good argument for the other. This way of killing a large number of birds with a small number of stones is particularly useful for people like me who like to make their minds up about things. For example, shortly after I came round to counterpart theory about de re modality I came round to stage theory about persistence. I’ve never really had a view about property ontology before, but maybe now I should have another look at trope theory.