(1) How do we know that anything continues to exist when we aren't perceiving it?
(2) How do we know that there is anything external to us?
That leaves the third question:
(3) How do we know that there is anything that doesn't depend on our minds?
Shepherd gives five ways in which we know that there is a continuing, external world that is independent of us.
First, we can extend the reasoning that was used for discussed the externality of the external world. Shepherd doesn't develop this at great length, because it is not her primary approach to the topic, but the basic ideas is that causal principles show us that the causes of our sensations are external to that internal world which we call our minds, understood as the capacity for sensation in general. But if these causes, when not being sensed, continue to exist and are wholly external our minds when they are doing so, then they must be independent of our minds.
Second, we can consider again the five organs of the senses and motion (which Shepherd elsewhere says is like a sixth sense) and how they relate to the world. There are objects that are ready to appear through our sensory organs; we previously recognized that they continue to exist even when not sensed. But of these ready-to-appear objects, we find many that have changed some of their qualities or properties while they were continuing to exist unperceived. The grass is a little longer, perhaps, than when we saw it last time; the color of something has faded or bled a bit; the mailbox was knocked down. Since we didn't observe those changes, and they took place outside our minds and unperceived, the processes and operations that made them must be independent of our minds.
In many ways, Shepherd's third approach is the most interesting. One of the early modern philosophers who come closest to holding that the whole external world is dependent on our mind is George Berkeley. He doesn't quite get there because he thinks that there are actually many minds, and that the external world is just the collection, so to speak, of all the sensations of these minds insofar as these sensations can be signs of each other. But to get this far, he has to argue that we have reason to think that other minds exist. This he does in A Treatise Concerning Principles of Human Knowledge, section 145:
From what has been said, it is plain that we cannot know the existence of other spirits otherwise than by their operations, or the ideas by them excited in us. I perceive several motions, changes, and combinations of ideas, that inform me there are certain particular agents, like myself, which accompany them and concur in their production. Hence, the knowledge I have of other spirits is not immediate, as is the knowledge of my ideas; but depending on the intervention of ideas, by me referred to agents or spirits distinct from myself, as effects or concomitant signs.
Shepherd agrees that this is the primary way we know that other minds exist, but she also argues that Berkeley's argument for other minds can be generalized to become an argument for the mind-independence of the external world. On her view, we cannot know the existence of the external world otherwise than by the ideas (sensations) excited in us by them; Shepherd thinks we only know the external world by way of effects or concomitant signs. We perceive several motions, changes, and combinations of sensations that inform us that there are particular causes effecting them. These causes are like ourselves in being continuously existing causes but that differ from us in ways proportionate to the effects they causes.
Fourth, we can use this kind of argument for other minds in a different way. Let's use it to conclude that there are other minds. Then let's think about what happens when five different minds perceive one pond. If there were nothing about the pond that was independent of the mind, then this situation would be impossible: five different minds would perceive five different ponds. In order for their to be one pond seen by all five minds, there has to be something, continuing to exist when unperceived, that in its unperceived state is independent of, and external to, each of the five, so that the pond can be ready to appear to any of the five. She recognizes that Berkeley has a response to this, but she doesn't find it convincing; I won't go into any details of why here.
Fifth, and this is the point at which Shepherd most sharply deviates from Berkeley (who rejects the claim that there are abstract ideas), we can run, with minor modifications, many of the arguments we've looked at for abstract truths and relations. We find the abstract relations of mathematics as ready to appear as any sensible form, and by the same reasoning we have given, the only conclusion to be drawn is that in some way these relations, even when unperceived by the intellect, continue to exist independently of, and external to, our minds. Our discovery of a truth does not cause it to be true; rather, our minds discover it to be true because it is true. Our discovery of an abstract relation in, say, mathematics, does not make that abstract relation to exist; rather, the existence of that abstract relation makes its discovery possible, and is part of the explanation of the discovery. When physicists discover a certain mathematical law in the world around us, for instance, it can only be because there is some continuing, external, independent cause that makes the phenomena behave according to that mathematical principle. It is more abstract, but as straightforward, as the argument that some kind of continuing, external, independent cause makes us see green on leaves.
Given this, we can form an abstract idea of existence. Every sensation springs up and passes away, but what begins to exist must have a cause. On the basis of this, Shepherd argues, we can conclude that there is some existence that is neither a particular sensation (which is in the mind), nor a mere capacity for sensation in general (which would be the mind), which can be a cause of particular sensations in the mind. Thus we can formulate the idea of an indefinite unknown existence -- as we might put it, the idea of a something-or-other that exists. Berkeley, of course, rejects the possibility of such an idea, but Shepherd is unimpressed with his arguments, and says in response:
...we can always separate or abstract the most general quality of an object from the rest, whether that quality be supposed among them by the imagination, known to be among by the senses, or concluded to be among them by reason, as a result from their mutual bearings. (EPEU 85)
All this is somewhat brief, and Shepherd herself goes over some parts very quickly when she discusses independence as such, although she discusses many of these parts in more detail elsewhere. But if any of these arguments work, Shepherd has established what she set out to show: we know that there is an external world (continuing, external, and independent); we know it by reason and not (e.g.) by mere assumption or by a trick of the mind; and this rational inference is of a sort that it can be quite extensive -- even children have some grasp of it. If you asked an ordinary person why she thinks that the world is not wholly dependent on her mind, she might well appeal to its externality, to the shared experiences of the human race, and so forth; and on Shepherd's account such arguments, however primitive, are quite good when the essential causal principles underlying them are recognized.
This is not the whole of Shepherd's account of how and why we perceive there to be an external universe. There is a very important test case for whether any account of our knowledge of the external world is viable -- any such account must be able to distinguish between dreams and waking life. But that will take another post.