A couple of personal problems with this scenario:
1) The idea of non-continuity in sleep. I know it's rare - probably very rare - but it is possible to stay lucid while falling asleep and dreaming. Conscious awareness for most people pauses (or, more appropriately, tunes out the senses) every night; but the few cases where it doesn't tune out should be enough to make people understand that the self remains continuous and dynamic even through sleep.
2) In spite of the popular materialist mantra that we are physically totally replaced every X period of time, that's actually untrue. Apparently, the vast majority of our brain cells are, in fact, NOT replaced over our lifetimes, and there may even be some apparent and significant effects upon our personas in those cases where brain cells ARE replaced. So, while I doubt we can read this much into this fact, it is possible that some element in those non-replaced brain cells may be vitally responsible for our self-awareness, and we may not be able to duplicate (or replace) brain cells without losing self awareness.
That's a bit of a stretch, but there it is.
3)In spite of thought experimenter (like Blackmore) trying to convince you to consider this experiment without considering the failure case scenarios, the fact is, you have to consider those very scenarios to get the full awareness of what is going on. Their mediocre attempts to get you to 'ignore the man behind the curtain' should tell any critical thinker that the position being expounded is weak in some way. I had this out with Interesting Ian.
The fact is, if the 'transporter' fails and two of you suddenly exist, if the potential for that to happen even exists (which, obviously, it does), this proves that, no matter how you want to define it, you
2 is emphatically NOT the same person as you, and you would walk into the transporter, not to enjoy life on Saturn, but to die. Period, point blank, end of story.
The transporter is a duplicator with a built-in illusion - nothing more.
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I can't really enumerate the remaining points, as I'm losing focus...
The twin argument should be pretty damning, too. Twins don't share experiences. They don't feel what each other feels (except in some unconfirmed, anecdotal cases). They don't (often) feel themselves to be two instances of the same person. Of course, even very similar identical twins are not fully, absolutely identical; but we've never seen evidence that a person shares more 'sense of self' the closer they are to another genetically similar person, after all.
The 'transporter' makes a twin for you at another spacetime location. How is that 'you' transporting? It's not. Congratulations, you've got a brother - not a new you.
If you have sex with your duplicate, is it masturbation? Homosexuality? Incest? Could you just imagine the laws that would appear governing that behavior?
Suppose the 'teletrans' did its job except - oops! Substituted one gender for another. Sorry, Bill, you're now a Betty. Try explaining that one to the wife.
I'm not claiming to be a materialist - I kind of have to support Ian's contention that the transporter would do nothing but create a dead lump of flesh at the receiving end. But from a materialist POV, it's still idiotic to think of this thing as nothing but a murderous duplication ray.