Spoiler alert!
In Time's Arrow, Part 1 everyone is concerned for Data's safety after they
find his severed head which appears to be from the future.
I thought it was a bit strange how everyone was sad because of this discovery. Geordi even said it's like finding out that a friend has terminal cancer; it's not really as all they found out is
he was not immortal. Data even pointed out he could still live for centuries.
Why does the crew not see it that way?
Because humans are really bad at weighing risk.
Data takes the computationally thoughtful approach; he knows he can function for centuries, so the event causing his head to be removed from his body must happen between T-0 and T-. There's a billion more things for Data to worry about; as he believes that time is immutable, worrying about "which event" will cause this disaster will just lead to confusion and worry, so he won't even bother worrying about it.
This comes up in dialog quite a bit; Cast Member A is all "but it could be TOMORROW! You should stay in bed!" Data points out that staying in bed could be what causes it, that he doesn't have any data he didn't already have (that at some time in the future he would become inoperable), so he doesn't see a reason to change anything.
We humans are bad at managing risk, however! And that's kinda the point of this episode, at the end of the day. If I were to receive a newspaper snipout with my obituary on it from the future, I'd be a nervous wreck. "It's on a newspaper, and those are going out of style, so it could be tomorrow! There's no details about how I died, and 50% of all accidents happen within a mile of home! I should take a trip! But I don't know WHICH newspaper carried this headline, it could be that trip that kills me! It doesn't say I'm survived by my mom, and she's 55, so ohmygosh does that mean she's dying tomorrow too?!"
We would try to fit that new piece of world-shattering data (no pun intended) into our plans for the future, because that's how we keep alive; we hear there's a panther in the woods, we don't take our annual bacon-wrapped swimsuit hike through there. Data is, however, being imminently logical about death: it's gonna happen, and he doesn't have any useful data available to know how to avoid it, so until he DOES get that data, he's not going to worry about it.
I find it way more interesting how calm Data is when he realizes that he's caught up to death; he doesn't scream, he doesn't complain or whine, he doesn't transmit letters to all his friends. He basically goes "Ahh, so this is when it happens" and goes forward with what needs to be done. These two events show the humanity and super-humanity of the character of Data; even given the "completely human" option of worrying about the future and doing everything to avoid it, he instead accepts that he obviously made a decision to sacrifice himself for good reasons, and simply moves on.
Because Data is now guaranteed to "die" a violent death at any moment.
They're not worried that Data is effectively mortal. Everyone is mortal, yet we still think the people around us will be here for a long time. Troi's allusion to "terminal illness" shows that they are now worried he will die "soon", or at the very least within their lifetime. They also know he will be killed in such a way that his head is forcibly removed from his body. You can see this concern when Picard refuses to put Data on the next away team.
DATA: Captain, there is no rational justification for this course.
PICARD: Then I'll be irrational.
DATA: It is possible, sir, that the events leading to my death will not occur for years, even centuries.
PICARD: I hope that's true, Mister Data. Nevertheless, this investigation began with your death. I am simply trying to see that it doesn't end that way.
DATA: I appreciate your concern, Captain, but, to employ an aphorism, one cannot cheat fate.
PICARD: Cheat fate? Perhaps we can't, Mister Data. But at least we can give it a try.
Actually it turns out they were wrong for a different reason...
Even a severed head doesn't mean "death" for Data, and the next episode shows it can easily be reattached and reactivated.
