In his 1938 book, Hans Reichenbach imagines a "cubical world" whose inhabitants are prevented from approaching the sides. Outside the world, birds fly, whose silhouettes show on the translucent ceiling of the world. A "friendly ghost" has arranged lights and mirrors so that identical silhouettes also appear on one wall of the world, so that any time a silhouette on the ceiling flaps its wings so also, in perfect correspondence, a silhouette on the wall flaps its wings, etc.
Here's Reichenbach's diagram:
The inhabitants of this world, Reichenbach says, will eventually come to infer that something exists beyond the cubical boundary that causes the shadows on the ceiling and wall. So also likewise, he says, can we infer, from the patterns of relationship among our experiences, that something exists beyond those experiences, causing them.
It is crucial to Reichenbach's argument that the inhabitants of this world ("cubists", let's call them) infer the existence of something beyond the walls that is the common cause of the pairs of corresponding silhouettes. If the cubists could reasonably believe that only the shadows existed, with laws of relation among them, no external world would follow; and so correspondingly in the experiential case there might only be laws of relationship among our experiences with no external common cause beyond.
Unfortunately, it's obscure why Reichenbach thinks the cubists couldn't instead reach the conclusion that the shadows on the ceiling directly affect the shadows on the wall or vice versa, e.g., by the transmission of invisible and unblockable waves through the interior of the cube or simply by action at a distance. (In Reichenbach's mirror set-up, height has no influence on the bird's ceiling position but it does influence position on the wall; and the reverse holds for horizontal position; but direct-causers can posit hidden-variable explanations or similar.) Reichenbach addresses this worry with a single sentence: Within the confines of cubical world, he says, the cubists will have found that "Whenever there were corresponding shadow-figures like spots on the screen, there was in addition, a third body with independent existence", so they'll reasonably regard it as likely that the same is true on their walls (p. 123).
There are two serious problems with this response. First, it cannot be straightforwardly adapted to the sensory-experience/external-world case, which is of course the real aim of Reichenbach's argument. Second, it is false anyway: We can readily construct cases where one spot on a screen causes another on a separate screen without a common cause behind them, e.g. by using a mirror to reflect light from one screen onto another or by making the first screen sufficiently translucent and staging the second screen directly behind it; this is no less natural than the friendly ghost's arrangement.
Sober imagines sitting on the beach, noticing the correlation between visual experiences of waves breaking on the beach and auditory experiences of crashing waves. The two types of experience cannot be related as cause and effect because he can stop one while the other continues: When he closes his eyes he still hears the crashing; when he stops his ears he still sees the breakers. Presumably, then, there's a common cause of both.
So far, so good. But to establish an external world beyond the realm of experience, we must establish that this common cause is something outside the realm of experience. Sober responds to this concern by considering one solipsistic alternative: the intention to go to the beach. He then argues that this intention cannot serve as an adequate common cause because the visual and auditory experiences are correlated beyond what would follow simply from taking the intention into account. So he challenges the solipsist to produce a more adequate common cause. He suggests that this challenge cannot be met.
But it can be met! Or so I think. The common cause could be my first beach-like experience. This experience, whether auditory or visual or both, then causes subsequent beach-like experiences. That takes care of the correlation. If I have an experience as of closing my eyes, the auditory experience at time 1 causes the auditory experience at time 2 and also the visual experience at time 2 conditionally upon my having an experience as of opening my eyes; analogously if I stop my ears. The solipsist can either play this out with the first experience causing all the subsequent ones until conditions change, or she can have each experience cause the next in a chain. On the chaining version, if I have my eyes and ears simultaneously closed, my opinion that I will soon have beach-like experiences then does the causal work. (There are imperfections in these regularities, of course, e.g., I might seem to myself to have booted up an an audiorecording of waves, but to take advantage of those imperfections is contrary to the spirit of the toy example and would cause trouble for Sober's model too.)
I agree in spirit with what Reichenbach and Sober are trying to do -- and Bertrand Russell, and Jonathan Vogel. The most reasonable explanation of the patterns in my experience is that there is an external world behind those experiences. But the argument isn't quite as easy as it looks! That's why you need to read Alan Moore's and my paper on the topic. (Or for short blog-post versions of our arguments, see here, here, and here.)
Revised 4:42 pm.