Williams describes two versions of a thought-experiment. (A thought-experiment is an example that is supposed to help make a point; Thomson’s violinist and Singer’s drowning child were thought-experiments)
The first version suggests that it is possible for a person to change bodies. It is very much like Locke’s example of the Cobbler and the Prince.
The second version suggests that being the same human animal (or man, in Locke’s terminology) as A is a necessary and sufficient condition of being the same person as A.
The trick is that these are two versions of the same thought-experiment. The second leaves out what happens to another person, B, while the first includes what happens to B. But why should what happens to another person, B, have any bearing on whether A will survive the experiment or not?
Here’s a way of phrasing the matter that is favorable to Williams. A’s ability to have conscious experiences does not depend on any distinctive aspect of A’s personality.
Thus when you take away memories, character traits, and so on, A still sensibly anticipates the pain at the end of the experiment. That is, A thinks that he is the one who will feel it. And that means he thinks he is the person who will inhabit the A-body after the experiment.
Micah and Martin spoke up for the view that the person is identical with his personality: memories, character-traits, and all the rest. I think that what I just said summarizes the pressure others put on them to move down that list on p. 172. And it’s the challenges that they would have to meet in defending their view, should they want to.
Perhaps it can be done! Many philosophers who work on this problem disagree with Williams.