Transporters work by de-assembling something (e.g. you) and re-assembling it somewhere else. What if, when you're dis-assembled, you die, and the re-assembled version of you is essentially a copy? Then every time someone steps onto a transporter, their final thought before death is that they'll end up beamed somewhere else. And the re-assembled version (copy) just thinks that everything went fine and continues on like nothing bad happened.
It goes a lot farther than that, actually. If you have the technology to assemble a person on a molecular level like that, you can basically prevent that person from dying. Captain got killed by an alien on an away mission? Just print another copy of them from their scan from just before they left.
Pretty sure that's why they specifically talk about how signal degradation occurs over time in TNG. You're not supposed to be able to keep somebody in the buffer for an extended period of time, or you'll lose the signal. This has been retconned multiple times though.
You don't need a distant science fiction MacGuffin for this. Every night you lay down and "die" for 8 hours or so, then your consciousness turns back on and you simply trust that it wasn't altered too much in the interim. We know very well that the way we think can change from one day to the other, so who's to say you're really the same person?
I don't think that's quite the same. Sleeping (or intoxication) is a temporary effect clouding your "normal" consciousness. Once whatever caused it goes away - assuming nothing actually changed, as you say - you're back to your old self. While sleep is "discontinuity of consciousness" in one way, a tell is that you can still remember dreams. If you've ever had (deep) general anesthesia, that time you were under can seem like it's "missing" in a way that pure sleep doesn't.
In contrast, the teleporter sort of... obliterates you, shredding you into atoms and rebuilding you later (if the matter doesn't need to "travel", at least the information is limited to light speed). It'd be no different from any other catastrophic damage to the brain.
The determining question for whether or not it's the same is this: Are you the physical matter of your brain, or the electricity running through it? In the first case, sleep isn't death. In the second case, it is. I would argue that you're closer to the electricity than the brain matter, since an unpowered brain is how we define death.
But REALLY it ultimately doesn't matter, if you think about it. An exact clone of you created after any kind of destruction of consciousness is no different than the original you had the destruction never occurred. We just intuitively really do not like that idea.
"No one ever steps in the same river twice, for it is not the same river and they're not the same person." - Heraclitus
What if the feeling that you are the same "person" as you were yesterday, or even as you were 5 seconds ago was illusory all along? What if you "die" and a new you is born many times a second all the time?
There's also a Ship of Theseus thing going on here since your cells are constantly being replaced by new cells. At what point do you stop being you and are simply the train of thought that carried over from your old cells?
Some episodes are written like that (Will Riker/Thomas Riker)
Some episodes you keep your consciousness throughout the "beaming" process, so you're technically you the whole way through (Broccoli in Realm of Fear)
It's fiction and some writers treated it as a suicide machine, and some wrote it as a movement machine. Even the technical details shifted from episode to episode.
I actually had a new thought in this that probably comes from Rick and Morty, what if transportation is possible, but it's cheaper to murder your ass and portal in some schmuck from a parallel universe?
I've often wondered though - say you do die, and get cloned at the other end... there hasn't been a 'gap' in your consciousness as such. So were you ever truly dead? Even if you're not technically the same person, as long as a version of you lives on.
I am my memories, my experiences, and the continuity of my consciousness, not the meat prison my mind is forced to reside in. If my consciousness continues, I am alive. Whether that's this bag of meat, that bag of meat, or the Transmetropolitan-inspired nanite-cloud I wish I could live on as, I'm still me either way. There might be a difference, but from my perspective it doesn't make a difference.
If you die, you're dead. The clone that appears on the other end, while being identical in every way, wouldn't be the same you that you are and wouldn't possess the same consciousness, an identical consciousness yes, but not the same one.
If you arm gets cut off, you and your arm go together to the hospital, and they reattach it, are you a different person?
If you die in the ambulance and they revive you, are you a different person?
Haven't watched Star Trek in any form, but I think I've heard of this thought experiment before anyway.
The way I see it: "the you" at the transmitter will die. There's no place for you to go as your body is torn into atoms, unless you start bringing in "souls" or something. The question in the more philosophical sense is that, suppose you can get a 100% identical clone down to (almost) continuity of consciousness... is that still you? Really you, whatever that means?
Speaking of which... what if you left out the "breaking down" part? Are there just two of you now?
That last part about there being two of you reminds me of The Prestige. I won't go into further detail in case I'd be spoiling it for you, but would encourage you to watch it if you haven't.