Recall the Lewisian definitions of branching and of divergence; two objects branch if they share a common initial part, and they diverge if they have intrinsically exactly similar but numerically distinct initial parts. Saunders and Wallace [2008] are keen to stress that their semantics is compatible with both a branching metaphysics and a diverging metaphysics:

*The salient distinction, between similar and identical initial segments, is invisible at the level of syntax and logical form.*

Saunders and Wallace [2008], p.297

However, as Saunders [forthcoming] recognises, the distinction between branching and divergence is not invisible at the level of metaphysics. He concludes that in the Lewisian sense, Macroscopic Pairing is a theory of diverging worlds.

I think that this is incorrect; Macroscopic Pairing and Space-time Point Pairing are strictly speaking neither theories of branching worlds nor theories of diverging worlds. In the context of Everettian quantum mechanics, a more helpful distinction is that between overlapping and non-overlapping worlds. I will argue that both Macroscopic Pairing and Space-time Point Pairing tell in favour of taking Everettian worlds as non-overlapping.

Take Macroscopic Pairing first, and consider the situation depicted in Figure 1. According to Macroscopic Pairing, there are two worlds present: <A, A> and <B, B>. There are also two distinct material objects present, <A, C> and <B, C>. So what do we have here? Is <A, C> identical to <B, C>, or are they intrinsically exactly similar? The answer appears to be ‘neither’. <A, C> cannot be identical to <B, C>, for the obvious reasons that the former includes A as a element while the latter does not, and that the latter includes A as a element and the former does not. But equally, and for the same reasons, it appears that <A, C> cannot be intrinsically exactly similar to <B, C>. So, according to the Lewisian definitions of branching and diverging, worlds <A, A> and <B, B> neither branch nor diverge.

However, there is a natural alternative sense of ‘exactly similar’ according to which <A, C> and <B, C> are exactly similar. Everything which is knowable about <A, C>at some time when it is present is also knowable about <B, C>: no possible observation can distinguish whether we are observing <A, C> or <B, C> up until the interaction that separates the worlds <A, A> and <B, B>. This sense of ‘exactly similar’ is an epistemic one: it could be explained as ‘observationally exactly similar’. If we appeal to this notion of exact similarity, rather than the notion of intrinsic exact similarity which is part of the Lewisian definition of divergence, then we can recover a sense in which worlds <A, A> and <B, B> diverge.

Another sense of ‘exactly similar’ which could recover divergence of worlds could be defined as follows, along the lines of the parthood* relation discussed above: two world-continuants x and y are exactly-similar* iff they share a second element. <A, C> and <B, C> are exactly-similar*: so if we define divergence by a having exactly-similar* initial segments, then worlds A and B do diverge. If you are prepared to countenance parthood* as a replacement for parthood in the context of world-continuants, then you might equally be prepared to countenance exact-similarity* as a replacement for exact similarity in the context of world-continuants. Such a replacement would once again render Everettian worlds as diverging.

Similar comments apply to the case of Space-time Point Pairing. Fusions of pairs <A, pC> (where pC means ‘a point-like part of C’) cannot be either identical or (strictly speaking) intrinsically exactly similar to fusions of pairs <B, pC>. However, if we are prepared to extend the notion of exact similarity to that of observational exact similarity, or to exact-similarity*, then we can recover senses in which the world which is a maximal fusion of pairs <A, pC> and the world which is a maximal fusion of pairs <B, pC> diverge from one another.

Despite the failure of Lewis’ distinction between branching and divergence to apply directly to the worlds delivered by Macroscopic Pairing and Space-time Point Pairing, it remains the case that these worlds are non-overlapping. The distinction between diverging (in the strict Lewisian sense) and branching worlds is not obviously more important than the distinction between overlapping and non-overlapping worlds. And in terms of this latter distinction, it seems clear that the Everettian should think of worlds as non-overlapping rather than as overlapping. This conclusion, somewhat unexpectedly, vindicates the application of the popular phrase ‘parallel worlds’ to Everettian quantum mechanics.

What is left of the intuitive thought that Everettian quantum mechanics involves branching worlds? The Space-time Point Pairing account which I have been urging involves no branching at the level of worlds; however, it does still incorporate branching at an underlying level, the level of Metalanguage 2. The structure of branches, which provide the ‘raw material’ for the elements of the pairs identified with space-time points, is a branching structure; different branches genuinely do have different parts in common. The difference between the view I am defending, and the straightforward picture, is that I would not identify branches directly with worlds. Building worlds up out of branches through the Space-time Point Pairing procedure gives us the resources to explain ignorance of the future and to explain probability. The straightforward view lacks these resources.

The picture we are left with has much in common with the Modal Realism of David Lewis (with the exception, of course, that the laws of quantum mechanics hold in every Everettian world). Both Everettian quantum mechanics and Modal Realism give us distinct disconnected non-overlapping space-times populated with distinct disconnected non-overlapping macroscopic objects and events.