Addressing the sidebar: We are doing philosophy, not studying philosophy. I don't remember textual reference in Zarathustra. Let's just write.
"Artificial" definitely should be dropped from the vocabulary. Logic and order don't pertain to "the world," they just describe the life of man. The point is exactly that once science provides an "artificial intelligence," man no longer corners the market on logic and order. This also does relate directly to my Antichrist essay. The man/nature dichotomy implies man's power over nature. When that power is exerted over women (the theology of the Antichrist film), we notice the problem.
Regarding the possibility of arriving in Japan and Singapore simultaneously: Reconsider in these terms- My existence is a phenomenon. In 40 seconds, my existence is a phenomenon. By my existence, I mean that my constituent elements are so arranged. Hell, I don't even really mean my existence. I don't mean existence either.
Since the substance of the Earth came from space in eons past, and we physically come from the same material by the rules of consumption and feeding, and return to that material because of decay, all of existence is one substance (like water. Oh, there's some H over here and some O over there and some dusty minerals in this area... shut up; it's water.) So I imagine all of existence as one substance, like Play-doh flattened on a table. You can pinch some of that doh up into a raised form, and you can name it something special, like "mountain," "Jackson," or "lamp." But that distinction doesn't change the fact that when the mountain erodes, or the lamp rust away, or Jackson dies and decomposes, it's analogous to smashing that lump back into the rest of the Play-doh and starting over. It's not that it's gone; it's that it was never separate.
This gives us a means to dissociate the things we want to say about identity from actual descriptions of the world. If it occurs that the phenomenon designated as "Jackson" is smashed back into the putty at the moment that in the zones designated "Japan" and "Singapore" there appear two phenomena that perfectly resemble the previously designated "Jackson," there is no burden of identifying "me." At least, not in any existential way.
The notion that identity is fluid across teleported bodies applies only in third-person, legal-style situations. If the teleporter creates a clone, does a prior standing warrant for arrest apply to both bodies? Our court system would say yes, though I would argue that it raises questions about the motivation of our justice system, which sees fit to execute people who are, very arguably, not the "same person" they were when they committed the crime.
Suppose you step into the teleporter, and two clones come out. Both will have their own first-person identities. Which one was Dan before stepping into the teleporter? Both. Which one is Dan after the teleporter. Neither.
Moral of the story: don't get cloned. It screws up the paperwork. That's the only reason not to get cloned.
Here's why I resist social identity. Social identity is the paperwork. You don't give justification to your claim that surviving effectively requires being a part of social orders. I submit that memories are not essential. I like them, but if I can't form memories about pain (thinking of Memento and the shock test), though I would probably die in a hazardous environment, that doesn't mean I couldn't have a rewarding life of thought if I resided in a safe place.
I guess I'm saying that talking about Lincoln's identity at all is folly, but if we were forced to decide...
Spontaneous Lincoln goes to court to decide whether or not he may collect the pension given to ex-presidents. I think Spontaneous Lincoln could demonstrate enough presidential characteristics in his testimony to have his right to the pension upheld. Likewise, if Spontaneous Lincoln manifested as having suffered the gunshot wound to the head inflicted by Booth, and was summarily treated in a modern hospital, where he made a full recovery except for the loss of his memory, the courts would likely not uphold the pension, not because he's not Lincoln, but because convincing evidence could not be produced.
Regarding your modifications to my earlier statements, is identity so difficult to reduce? We're already supposing the reducibility of the mind. Isn't anything we acknowledge as true reducible? (i.e. we can reduce that which we call the mind, but we don't bother reducing God because we don't acknowledge it as existent. Instead, we reduce the concept of God to logical (and illogical) conclusions of the workings of the brain.)
Your concluding analysis is spot-on. I am trying to circumvent the necessary culture bias inherent in our thought. Except I'm claiming it's UNNECESSARY, because the cultural bias that's keeping you out of the teleporter doesn't hold me back.