Like most people, you rely on the intuition that there's a "true identity" that survives the process of spending a lifetime embedded in a human brain, and then you question whether that "true identity" can survive various other processes.
This questioning of identity preservation also happens in the real world in cases of personality alteration or memory loss, but only when it's unusually rapid or extreme.
For example, not many people experience the intuition that their ten-year-old child "isn't the same person" as the infant they gave birth to. (That is, they assume that "true identity" has been preserved through this radical transformation.) But if their parent with Alzheimer's Disease undergoes a no-more-radical transformation, they are more likely to feel that intuition challenged -- to feel that their parent "just isn't the same person anymore."
Supposing I died and were reconstructed, and supposing that afterwards I didn't feel significantly different than I do right now, you might still assert that my personal identity had not been preserved, I can't think of a compelling argument I could use to convince you otherwise.
Similarly, if I meet someone who doesn't believe I'm the same person I was when my body was born, I can't think of a compelling argument I could use to convince them otherwise.
But... so what?
It's all just dueling intuitions, and those don't tend to lead to much progress. Hell, maybe I don't have the same "true identity" I was born with... how would I know? Or maybe I suffer from a tragic degenerative "true identity" disease, and my "true identity" will degrade completely by the time I'm 45, with no observable consequences.
My own response to this is to discard the notion of "true identity" as hopelessly confused, and resolve not to care about it.
There must be a level of knowledge at which one intuition or the other (or perhaps neither) is vindicated by the facts. Not caring about true identity may be expedient in the present, but some of these questions must have answers, answers that require information we just don't have yet. Some other questions will have more to do with choices, values perhaps, choices about what to identify with and what aspects of yourself to value. I don't think that, even with a shared ontology, everyone will feel the same about the desirability of making a personal backup...
I'm currently trying to convince my parents to sign up for cryonics.
The problem is that they are completely opposed to the any form of life extension and/or immortality (and that’s without even mentioning something as "strange" as cryonics). Unfortunately, being their child, I have the intrinsic property that I can never know more about life then they do. The only thing they will believe are scientific studies from respectable scientists (a respectable scientist being someone who only says what they want to hear and is not me)
I have the arguments I gathered from Less Wrong and the Alcor Library. I’m focusing on my mother since my dad is impossible to convince without her support.
Her argument is that when you live for a very long time/forever wars are almost guaranteed to occur at least once in your lifetime and she doesn’t want to live through those. I asked her when, given that we could perfectly predict the future, we would know a war would break out tomorrow she would commit suicide today. Her answer was yes, as she couldn’t bear losing any of us and doesn’t want experience a war. I pointed out how I would feel if she died but she just dismissed the entire thing as crazy.
My parents aren’t religious at all, so that’s one less bridge to cross but for all the rest I would greatly appreciate anything that might help convince them.