Richard Swinburne's thesis starts with the claim that everyone alive has two important parts, their body and their soul (which I shall call their 'mind', since for present purposes it doesn't matter which we call it, and I don't want to take advantage of any irrelevantly controversial implications of the term 'soul'.) The difference between these two parts is that the mind is the part of a person which has his or her mental properties, these for Richard being the properties that at any one time are best known to the person whose mind it is: typically, states of consciousness.
Given these assumptions, Richard then claims that these two parts of an embodied person can exist separately. Now with half of this claim we can all agree: namely that Richard's body could exist without a mind. For when he dies that is what his corpse is - a body without a mind - and this we can all agree will not be Richard. For Richard to survive it is not enough for his body to survive.
The question then is whether the other part of Richard, his mind, can exist apart from his body; and whether, if it does so, that part of him is Richard, or is something else again. Richard says that is possible; I say it isn't, in any serious sense of possibility.
Richard's case for the possibility of our surviving as disembodied minds rests on an argument about our identity over time. This argument takes the form of a thought experiment in which a person - Richard, for example - is duplicated in the way he describes. Richard then assumes that there must be an answer to the question of which of the two people who emerge from this experiment will be him: the possible answers being that one of them is and the other isn't, or that neither of them is. And from this he concludes that there must be something, not entailed by any facts about Richard's body, which determines whether Richard has continued to exist and, if he does, which of the two later people is him.
I think several things are right about this argument. I'm sure that anyone faced with Richard's awful prospect will think that there is an answer to these questions even if it isn't obvious what the answer is. In this particular case I'm inclined to think that neither of the two people resulting from Richard's duplication is Richard. Take Richard's Hesperus/Phosphorus analogue, and imagine that planet being split into two parts, each of which then grows by accretion until there are two planets just like the original. We would then face the same question: what, if anything, tells you which if either of the two planets you now have is the one you started with?
It seems to me as obvious here as in Richard's duplication example that nothing about the original entity, whether it be a planet or a person, tells us the answer to that question. Yet the issue in the planet case cannot be settled by crediting the planet with an identity-carrying mind. Why then should that be what settles it when people are duplicated; since surely the answer, whatever it is, must be the same in both cases.
Note also that Richard only gave us three options in his duplication example: namely, that after the operation either one of the two embodied characters is Richard, or the other one is, or Richard ceases to exist. Why did he not raise the possibility, if neither of those two embodied people is Richard, that Richard has floated off on his own in some disembodied way? The reason surely is that nothing in his story even begins to suggest that you can have a person who isn't embodied. There may well be hard questions about which past embodied person a presently embodied person is, but that's a question about what determines the identity over time of embodied people. The difficulty of these questions does nothing to show that we can exist without being embodied, which is what Richard needs in order to show that we could survive the death and destruction of our bodies.
And once we see this, it seems clear to me that all the evidence we do have suggests that our minds depend on our bodies (including our brains). There are many obvious interactions between our mental and our bodily properties. These interactions admittedly go both ways, since conscious experiences can have physical effects, as when embarrassment makes you blush, just as physical causes can have mental effects, as when photons entering your eyes cause you to see me. You don't have to be a materialist - i.e. to think that all mental states are physical - to admit that our mental properties depend on our physical ones. You can think, as I do, that we have properties which do not reduce to physical properties, and still acknowledge that not only the nature but the very existence of our conscious experiences depends on our having bodies. All the evidence of our own and other people's lives overwhelmingly implies this.
Next I'd like to comment briefly on Richard's unusually narrow characterisation of mental properties. I don't think this matters much to his argument, but it may affect its appeal. The point is that some properties which most people would call mental are not necessarily best known to the people whose properties they are. Think for example of the subconscious or unconscious states of mind postulated by Freudian and other psychologists. Or take the many less controversial states which are normally counted as mental but which also need not fit Richard's definition, such as your beliefs, your desires, your intentions.
Several theories of the mind strongly suggest that other people may know as well as you do whether you are in some of these seemingly mental states. And on these theories we need a body to be in many of these states, since they entail dispositions to behave in ways in which only bodies can behave. (How for example can you be disposed to go home after this meeting if you have no body to take you there?) In short, as mentality is generally understood, it covers many traits that are important to our character, and to our human capacities, which you couldn't have without a body: not just as a matter of causation but because you need a body in order to have those traits.
My other comment on Richard's definition of mental properties is this. It may be necessary to our concepts of some mental states that only their owners can know directly that they are in them. But we may still have a perfectly good physical explanation of this so-called privileged access which embodied people have to those states. For we can, and I believe do, have internal senses which make us aware of states that we are in which other people cannot detect in us, or at least not detect in the same way. That moreover is a feature which could be built into something that isn't mental at all: self-scanning devices within a machine which monitor its internal states and whose output is used by the machine itself (e.g. to repair damage) but is not physically accessible to anything outside that machine. I see no mystery, nor anything essentially mental, in such privileged access.
None of these however is the main point I wish to make, which is simply that I see nothing in Richards's argument which shows that mental properties, even of his limited sort, can be possessed by anything which does not have a body.
Two final comments. First, the fact that you can imagine a mind without a body doesn't show that it's possible for minds to exist without bodies. It's too easy to imagine (or think we imagine) things that we know to be impossible. For example, (I think) I can imagine there being a greatest prime number - i.e. a number which can only be divided by itself and one and such that no greater number can also be divided only by itself and one. Yet I know and understand the well-known proof that there is no such number. So conceivability, or apparent conceivability, does not seem to me a strong argument even for logical possibility.
And second, even if we grant that it's logically possible for our minds to survive bodily death, just as it's logically possible for pigs to fly, that doesn't show that it's physically possible. It may still follow from laws of nature that in order to have any of Richard's mental properties, you need a physical body. That seems to me strongly implied by all the evidence we have about how our mental states depend on our bodily states. In other words, even if it is not merely apparently conceivable but logically possible for us to survive bodily death, we still have every reason to think that it is not physically possible for us to do so.