The first thesis is that, just as with questions about the identity of a nation or a ship, questions about personal identity may sometimes have no determinate answer. Some situations may be "gray cases," where it's not completely right to say "This is the same person," and not completely right to say "This is a different person."
If forced to give an answer, Parfit will go with a Psychological Continuity plus No Competitors Theory (like Proposal #6). In later work, he inclines more towards a Brain-Based Psychological Continuity Theory (like Proposal #9).
But--and here is Parfit's second thesis--he also thinks that personal identity is not what we really value and care about. What we really value and care about are the psychological connections, like quasi-memory and sameness of personality and intentions and values, that the Psychological Continuity theories appeal to. Ordinarily, these psychological connections go together with personal identity, but in problem cases it's possible for them to come apart.
For instance, in an example we discussed earlier, we take some of Lisa's cells and create a clone Cleo, and we copy Lisa's neural patterns over to Cleo's brain. We do all of this without harming Lisa in any way. It would be clear then that Cleo is not identical to Lisa. However, Cleo would have the same personality as Lisa, she would have quasi-memories of Lisa's childhood, and so on. (She would even think she is Lisa, unless we told her how she was created.) On Parfit's view, Cleo has most of what we really value and care about. If Lisa were to die, and Cleo were to continue on, that would be almost as good--even for Lisa!--as if Lisa herself were to continue living.
Consider a community of people who have this genetic quirk which makes them always have identical twins when they bear children. Suppose these people also have a psychological quirk which makes it very important for their childhood development not to be an only child. Once in a while, one of the twins is in an accident and so the other one has to grow up an only child, and this causes severe psychological problems later in life. This community will think: it's very important that every child have a twin.
But now suppose that something changes, and some of the people start to bear triplets instead of twins. Now the children that are born don't have twins. (If you're a triplet, then neither of your siblings counts as "your twin.") Should the community be upset about this? No, they shouldn't. They should realize that's what important is not really that every child have a twin, but rather that every child have at least one sibling. Triplets would be just as good as twins. (Perhaps even better, since now there's less chance that an accident will leave behind only one child.) This community was just confused, because as they were at the start of the story, having a twin was the ordinary way of getting what's really important.
Parfit thinks our situation with respect to personal identity is analogous. What should matter for me isn't that there be someone alive in the future who's identical to me. Rather, what should matter is just that there be at least one person around who's psychologically continuous with me. (In Parfit's later work, he adds: "and who has enough of my brain.") It's just that, given our present technology, having someone around who's identical to me is the ordinary way of getting what's really important. So I'm liable to confuse personal identity with what's really important; just as the community in the previous example confused having a twin with what's really important.
Parfit introduces a notion of "surviving" that doesn't require the original person to be identical to the person that she "survives" as:
We can suggest that I survive as two different people, without implying that I am these people. ("Personal Identity", p. 203)
To avoid confusion, it may be easier to call this notion "quasi-surviving," on the model of "quasi-memory." Quasi-surviving is just a matter of having someone around in the future who is psychologically continuous with you. You can quasi-survive as more than one person at once. At most one, and perhaps none, of the people you quasi-survive as will be identical to you.
Parfit thinks that quasi-survival gives us all or most of what we really value and care about in personal identity:
[In cases of fission] the relation of the original person to each of the resulting people contains all that interests us--all that matters--in any ordinary case of survival. ("Personal Identity", p. 206)
Judgments of personal identity have a great importance. What gives them their importance is the fact that they imply psychological continuity. ("Personal Identity", p. 207)
That is, the identity itself isn't so importance to us; it's just that the psychological connections--which are what we really care about and value--tend to come along for the ride. In problem cases, though, those psychological connections can continue, even if there's no one around anymore who's identical to us.
A consequence of Parfit's view is that you ought to pay as much, or nearly as much, to steer some future torture away from one of your quasi-survivors, as you'd pay to steer it away from yourself. And if given a choice between really surviving (having someone around who's identical to you), and merely quasi-surviving, you ought to regard these as equally worthwhile, or nearly so. You ought not to pay very much to ensure that you really survive, as opposed to merely quasi-survive.
We've seen that, on Parfit's view, we can have the psychological connections that really matter to future people who aren't identical to us, and that we can have them to more than one future person. Parfit thinks it's also important to recognize that these psychological connections can be a matter of degree. Over the course of a long life, you may eventually come to be very different from the child you started out as. Your personality will become very different, you won't remember many of the events he or she experienced, and so on. So there won't be many direct psychological connections between you and that child. You will still be psychologically continuous with the child, because you'll be part of a chain, earlier stages of which do have direct psychological connections to the child. But you won't yourself, in your later life, have many direct connections to the child you once were.
Parfit regards the direct psychological connections as being what's most valuable. So if you had a choice between:
The movie Memento explores what it would be like if we lost these direct psychological connections more rapidly than we actually do. Would we have much reason to care about the people we will later become, if we they won't remember much of our present life, or share our beliefs and values? Would we have any responsibility for, or interest in, providing for them? Would they bear responsibility for the choices and actions we make today?