Proposal #1: A is the same person as B iff A and B have the same soul. (This proposal is presented by Sam Miller on pp. 6-7 of the Perry dialogue.)
Gretchen's argument has three stages. She begins by asking Sam Miller to explain how, on his view of personal identity, one is supposed to acquire reason to believe that A and B are the same person.
Sam's first suggestion is: our judgments about personal identity are based on the hypothesis that if A and B have the same body, then they probably have the same soul (p. 8).
However, Gretchen points out that having the same body does not entail having the same soul. If souls exist, then it seems like it would be possible in principle for a given body to come to be inhabited by a different soul. Nor does having the same soul entail having the same body. If souls exist, then it seems like it would be possible in principle for a given soul to inhabit more than one body, in succession. So what entitles us to the hypothesis that, just because A and B have the same body, they must have the same soul? For example, how do we know that the body's soul wasn't replaced every time the body went to sleep? We have no way of testing the hypothesis that souls always "stick" to bodies in the ways that we think persons do.
This argument is summarized in the following passage:
Gretchen: ...Since you can never, so to speak, bite into my soul, can never see or touch it, you have no way of testing your hypothesis that sameness of body means sameness of self.(Some respond to this argument by claiming that there's a special theory about souls, which one can come to know by religious study, or by deep philosophical reflection. Perhaps this theory tells us that sameness of body implies sameness of soul. One problem with this response is that, at best, it only explains how certain experts--those who do the religious study, or those who do the deep philosophical reflection--might be justified in believing the hypothesis that sameness of body implies sameness of soul. What about all us non-experts? What justifies us in accepting that hypothesis?)
Sam: I daresay you are right. But now I'm a bit lost. What is supposed to follow from all of this?
Gretchen: If, as you claim, identity of persons consisted in identity of immaterial unobservable souls, then judgments of personal identity of the sort we make everyday whenever we greet a friend or avoid a pest are really judgments about such souls.
Gretchen: But if all such judgments were really about souls, they would all be groundless and without foundation...
Sam: That seems fair.
Gretchen: But our judgments about persons are not simply groundless and silly, so we must not be judging of immaterial souls after all. (pp. 11-12)
Sam Miller then tries a second suggestion. This time, he says that our judgments about personal identity are based on a more complicated set of hypotheses: first, we observe whether a given body displays the same psychological characteristics as it did formerly; second, we reason that if the psychological characteristics are the same, then the body is probably inhabited by the same soul. Hence, since on his view sameness of person just consists in sameness of soul, we can conclude that the body is occupied by the same person (p. 12).
Gretchen objects: what entitles us to believe that, just because a body exhibits the same psychological characteristics over a stretch of time, it has continued to be inhabited by a single soul during that stretch of time? For all we know, the body might have been inhabited by a series of short-lived, numerically distinct souls, all of which are psychologically similar to each other. (The later souls in this series have "memories" of events that the earlier souls experienced.) All together this series of souls would present the appearance of a single, extended life. But according to the theory that we are identical to souls, the body would in fact have been occupied by a multitude of persons. Unless we have some way to rule these sorts of possibility out, we can't be justified in believing that the a single person has inhabited the body, over the stretch of time we're considering.
This argument is summarized in the following passage:
Gretchen: ...My point is this. For all you know, the immaterial soul which you think is lodged in my body might change from day to day, from hour to hour, from minute to minute, replaced each time by another soul psychologically similar. You cannot see it or touch it, so how would you know?
Sam: Are you saying I don't really know who you are?
Gretchen: Not at all. You are the one who says personal identity consists in sameness of this immaterial, unobservable, invisible, untouchable soul. I merely point out that if it did consist in that, you would have no idea who I am. Sameness of body would not necessarily mean sameness of person. Sameness of psychological characteristics would not necessarily mean sameness of person. I am saying that if you do know who I am then you are wrong that personal identity consists in sameness of immaterial soul. (pp. 14-15)
Sam makes one last effort to defend the theory that we are identical to souls, but that we can nonetheless have justified beliefs about personal identity. Sam suggests that he can establish a correlation between sameness of body and sameness of soul in his own case. He observes that his soul and his body are always found together. Perhaps this gives him some small reason to expect that in other people's cases, too, sameness of body goes along with sameness of soul.
Gretchen makes two objections here.
First, that seems a very hasty extrapolation, using observations of only a single body and a single soul to support a hypothesis about what is true of all bodies and all souls (p. 15).
Secondly, how does Sam know that his body has only been occupied by a single soul? How can Sam rule out the possibility that his own body has been occupied by a series of psychologically similar souls, all seeming to remember events experienced by the earlier souls? She argues:
Gretchen: ...I grant you that a single person has been associated with your body since you were born. The question is whether one immaterial soul has [also] been so associated--or more precisely, whether you are in a position to know it. You believe that a judgment that one and the same person has had your body all these many years is a judgment that one and the same immaterial soul has been lodged in it. I say that such judgments concerning the soul are totally mysterious, and that if our knowledge of sameness of persons consisted in knowledge of sameness of immaterial soul, it too would be totally mysterious...Gretchen then goes on to list possible grounds for believing that hypothesis, and explains why she takes them to be inadequate:
Sam: You have simple asserted, and not established, that my judgment that a single soul has been lodged in my body these many years is mysterious.
Gretchen: Well, consider these possibilities. One is that a single soul, one and the same, has been with this body I call mine since it was born. The other is that one soul was associated with it until five years ago and then another, psychologically similar, inheriting all the old memories and beliefs, took over. A third hypothesis is that every five years a new soul takes over. A fourth is that every five minutes a new soul takes over. The most radical is that there is a constant flow a souls through this body, each psychologically similar to the preceding, as there is a constant flow of water molecules down the Blue [River]. What evidence do I have that the first hypothesis, the "single soul" hypothesis is true, and not one of the others?"
...What evidence do I have that the first hypothesis, the "single soul" hypothesis is true, and not one of the others? Because I am the same person I was five minutes or five years ago? But the issue in question is simply whether from sameness of person, which isn't in doubt, we can infer sameness of soul. Sameness of body? But how do I establish a stable relationship between soul and body? Sameness of thoughts and sensations? But they are in constant flux. By the nature of the case, if the soul cannot be observed, it cannot be observed to be the same... (pp. 16-17)
For instance, on Lockean picture of the relation between souls and persons, being the same person over time doesn't require having the same soul. So on that picture, you can have justified beliefs about your own identity, without needing to first establish that the same soul has always been associated with your body.