Personally, it makes more sense to me if they are able to remember their old name. When they are reborn, they know how to walk, and talk, they don't relearn these things. They already know. It is probably the same with a name; it is your identity, so it persists through death. Unless we forget who we are each time we die.
English
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Actually, cases of severe amnesia will often cause the victim to forget anything relating to identity, but they will remember basic functions like math, reading, walking, and other functions.
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Well, this depends on how much memoey persists through death.
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Lore says that Guardians maintain fragments of memory. Either they don't gain memory, or they slowly obtain it. They don't lose it.