Who is Jack, really?

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This was a philsophy assignment question giving to students, well something along these lines. If Jack was to get his thoughts,memories and linked feeling erased and placed into Person B, then who is Jack? Person A or Person B? Does a person's thoughts, memories, and feeling identify as to who they are, or it is a bystanders opinions of that person, identify who that Person is. (e.g. A person with memory loss would still be known to other people as their "original name', even if they had no recollection of that being their name.)