Assume someone's invented a teleporter that works like this: Picard steps into the machine on Earth. The machine scans all his atoms. The machine vaporizes his body on Earth. His scanned data is sent to a machine on Mars. The machine there creates an exact replica based on the scanned data, and a Picard appears on Mars.
Would you use such a machine?
We're so used to sci-fi that this might seem perfectly OK at first. I don't think anyone watching Star Trek is yelling at the TV "NO, Jean-Luc, don't get on the transporter pad, it's gonna kiiilll you!!!". Mars Picard has all the memories of the original Picard, and appears to be totally psychologically continuous with the individual who stepped into the machine. This is a compelling way to define personal identity.
But, now suppose that the vaporizer on the machine breaks. Earth Picard steps into the teleporter machine, sits for a few minutes, and then steps out, thinking "man this was a hoax". But, he has been scanned and an exact replica has been created on Mars. In this case, we have an Earth Picard and a Mars Picard. Which is the same individual as the Picard that stepped into the machine? Certainly they can't both be the same person as the original Picard... can they? Who has to pay the original Picard's bills? Who gets to come home to the original Picard's property and sleep with the original Picard's wife?
It certainly seems like Earth Picard has a more legitimate claim than Mars Picard. He stepped into a machine, was scanned, and then stepped out. Why should some replication happening on Mars change whether this man is the same man he was a few minutes ago? It seems we must conclude that Mars Picard is NOT the same person as original Picard, but just a replica.
But why should it matter to Mars Picard's identity whether the body on Earth is vaporized or not? It seems like, if the vaporization works as intended, Picard hasn't been teleported to Mars, he's been killed, and then a copy made on Mars!
The question is, would you use a teleporter, assuming the vaporization works and your original body is destroyed?
If I ask myself "what would it be like to use the teleporter", I'm scared. It seems, intuitively, that it would feel like dying. I would go unconscious as my body is vaporized, and then someone else would wake up on Mars thinking he was me! Sure, he's got all my memories and such, but he's not me, he's a copy! I'd be dead! If you look at it this way, every episode of Star Trek is pretty tragic: every time someone teleports a person dies, and a new person is created who has no idea, and that person unwittingly kills themself by beaming up shortly thereafter.
My position is that this intuitive idea is just wrong. I think I would use the teleporter. I'd be damn scared though.