It depends on what you define as "you". Honestly, I don't think the question is that interesting. It's like asking whether Linux 2.2.17 can be considered the same project as Linux 2.6.28. All we can say objectively is that they share a history and a name. Whether they are the same project is a subjective opinion.
The point is that -- given that you will in objective fact not exist tomorrow -- if you can find a reason to care about the impostor who tomorrow will take your place in society and pretend to be you, you should be able to apply the same reasoning to caring about your community as it might exist after you pretend to "die". That you are related to both is objectively confirmable.
In short: if you care about one, logically you should care about the other.
The "you" now shares a common history with the "you" a second ago. Other individuals do not. Your conclusion only follows if there is no objective difference between your past self and other individuals, and there clearly is.
I'm talking about the personal history an individual perceives through his memories. The history that connects the "me" of the now to the "me" of the past.
So, you would feel sensible in caring about (having altruistic feelings for) someone who could remember being you. Yes?
Would you feel sensible in caring about the future-you if (the present) you knew that he was going to have amnesia?
Me: "I'm going to drain my bank account and run wild, having a good time, because I know that tomorrow (and forever after) I won't remember having done such a dastardly thing to myself."
It's not about what _I_ feel sensible about caring about. We're talking about a hypothetical individual who cares only about versions of himself that share a common ancestor or descendant state.
If such an individual were given advanced knowledge of an irreversable amnesia, then logically he would commit acts that benefit him and cost the individual inhabiting his body in a day's time. So yes, it makes perfect sense.
But given the same circumstances, I'd behave differently, as I'm not a sociopath.
