Premature death paradox

by bfinn2 min read13th Apr 202021 comments

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There seems to be a paradox about premature death. (Or at least, I am confused about it - perhaps due to ignorance of philosophy of death, or some related paradoxes, e.g. about time.)

Thinking e.g. of COVID-19, what makes potentially fatal illnesses bad is not that they kill people, or increase the chance of dying. For throughout the past and future, everyone has always died, and always will die, precisely 1.0 times (a contingent fact, but nonetheless the most accurately known physical constant in the universe, somehow). So nothing can increase the chance of dying (which is already 100%), and we all die eventually of something.

What seems bad, though, is when an illness causes premature death - dying too soon, before one's time. But when is that?

Everyone dies prematurely

Life expectancy tables report that at birth, in the UK (or US) you have a life expectancy of 81 years. So dying before 81 would be premature; if you died aged 20 or 50, people would say you'd died young, before your time.

But when you get to 81, it's not your time either - for you then have a life expectancy of as much as 8.5 more years, living to 89.5. And if you reach 89.5 years, you can then expect to live on to 94, etc. Even at age 100 you have a life expectancy of 2.1 further years.

So whatever age you die at, your death is always premature - for your life expectancy at that age is invariably greater than the time you continue to live for (= 0 years).

Maybe this is part of the reason death seems like a raw deal. For however old you are when you die, you certainly could have lived longer (by at least a nanosecond). Because there is no upper limit for age: no age which you have a non-zero probability of reaching, but a zero probability of exceeding (another contingent fact about the universe we oddly seem to be certain of).

But the raw deal is even worse. Suppose you die at age x; and the life expectancy at that age is e further years. Then of all the people in the world aged x, you are the very first to die - for you die immediately, whereas they all continue living (for at least a nanosecond), and indeed on average they live e further years that you missed out on.

Moreover, all those who are younger than x also continue living, and even those older than x do too (for at least a nanosecond). So you not only died before everyone the same age as you, but also before everyone younger, and even everyone older! You're the unluckiest person alive! (Or at least, the unhealthiest.)

No-one dies prematurely

Perhaps the confusion is about the concept of life expectancy: e doesn't tell you how long you will live, but how long people in general aged x live, on average. So it's not personalized to your situation. The more we know about your health & circumstances, the more we can improve on this estimate to make it specific to you. And if we knew everything about you, we could make an extremely accurate, perhaps even exact, tailored prediction.

So let's try it: at the age in question - x (your death) - your tailored prediction (which oddly we can make without knowing anything about you) is that your life expectancy is precisely 0.0 years.

But then in what sense did you die prematurely? Given this actual, individual life expectancy, it's clear that you haven't missed out at all - even if you were aged only 20. You lived exactly the expected number of years, and died when your time came - not a nanosecond before, nor after.

So at first it looked like people always die prematurely; now it looks like they never do.

Is the confusion then just that we didn't know enough to make an individually tailored life expectancy prediction for you in advance, so rely instead on life expectancy tables? But they're so unreliable; when your time comes, why do the tables invariably overestimate? (No wonder death seems too soon!) Can't we just correct them?


I'm partly joking, but there seems to be a real paradox here. Is it a known one? Or more than one paradox? It feels a bit like one of Zeno's; and also a bit like the unexpected hanging/test.

[added:] Also it feels like there may be a confusion between credence and chance (i.e. subjective & objective probability) when calculating life expectancy, and/or confusion about the appropriate reference class of people.

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Math has nothing to do with it. It's the violated expectations. You expect that your elderly grandmother might get pneumonia and die some day soon. You expect an occasional young person to die after catching the flu. You don't expect the carnage that is going on now, with a lot of people dying before they reach their a priori life expectancy. In that way the deaths seem premature for a lot of people, as opposed to premature for some and overly mature for others, with the average being where you expect it.

Yes... but what is their a priori life expectancy?

Consider an atom of uranium 238. It has a constant probability of per unit time of emitting an alpha particle. Unlike people, it does not get tired and frail. The probability that it goes ping in the next second remains constant, however long it has survived already. That probability is extremely small. The half-life is more than 4 billion years, similar to the age of the earth. Whenever it happens, it was extraordinarily unlikely to happen just then. It has decayed 4 billion years ahead of its expected remaining lifetime.

But in a quarter kilogram of U238, it will happen 4 million times a second.

Ok - I see the analogy. Though not sure it points to a clear solution to the paradox.

If you die at age 90, you died prematurely relative to what we'd expect a month before you died, but (postmaturely? it should be a word) relative to what we'd expect and bet on 80 years before your death (i.e., at age 10).

Now, you may still think there's a paradox in the following sense: let's say the median lifespan expected at birth is 70. That means that the 50% of people who died before 70 died prematurely relative to all predictions made throughout their lives, while for the remaining 50% some of the predictions were too pessimistic (those made early in their lives) but some were optimistic. Isn't there still a skew towards being surprised that people died early?

The imbalance disappears if we count not people, but people-seconds. I.e., if we predict how long everyone is going to live at every second of their lives, the average prediction will not be either pre- or post-mature. The people who live longer will accumulate more pessimistic early death predictions through the sheer fact that they live more seconds and so more predictions are made about them. A person who lives to 100 may accumulate 95 years of too-pessimistic predictions and only 5 years of too-optimistic ones.

Thanks for this. This certainly sounds plausible, though whether the premature and post-mature predictions would exactly cancel out isn't obvious. (Not sure without trying the maths, and it may or may not be tricky - this is what actuaries are for.)

Also I'm not sure whether it requires the life expectancy predictions to be made on the same basis for everyone (e.g. life expectancy tables) or if it would still work for individually tailored predictions.

I can see this could dissolve most of the paradox, but (as I threw as many confusions as I could think of into the post!) I sense one may still remain about tailored predictions. At the extreme, in a deterministic world, God could calculate our individual time of death when we're born. All his life expectancy predictions made at any time would be correct, and he wouldn't be surprised when we die; but nonetheless dying age 20 would still be premature. (And not just because we can't make perfect predictions.) Possibly 'premature' means slightly different things in different contexts.

IIUC, you say "people always died --> therefore dying is not the object of our caring". as an intuition pump, that's fine. but it's not a robust argument; observing a moral harm shouldn't update us that it's not a moral harm (that's the just world bias). it might be that finite lives are meaningless. (also, I'll just assume the 100% is a high probability rounded up)

"premature death" entangles both beliefs and values, but those are orthogonal. I think people only "want" to live for how long they expect to live in a superficial way (just world bias). I doubt anyone's morality for how much life is good is linked with how much life they expect.

the concept of "premature death" is only useful to evaluate interventions to know how much life they would add counterfactually

Thanks, good points.

So I agree that this paradox is quite interesting as a statistical puzzle. But I'm not sure it shows much about the ethical question of whether and when death is bad. I think the relevant notion of "premature death" might not be a descriptive notion, but might itself have a normative component. Like, "premature death" doesn't mean "unusually early death" (which is a fully descriptive notion) but something else. For example, if you assume Thomas Nagel's "deprivation account" of the badness of death, then "premature death" might be cashed out roughly as: dying while there's still valuable life ahead of you to live, such that you're deprived of something valuable by dying. In other words, you might say that death is not bad when one has lived a "full life," and is bad when one dies before living a full life. (Note that this doesn't beg the question against the transhumanist "death is always bad" sort of view, for one might insist that a life is never "full" in the relevant sense, and that there's always more valuable life ahead of you.) Trying to generalize this looks objectionably circular: death is bad when it's premature, and it's premature when it's bad. But at any rate it seems to me like the notion of premature death is trying to get at more than just the descriptive notions of dying before one is statistically predicted to die, or dying before Laplacean demon who had a perfect physical model of the world would predict one to die.

Anyway, low confidence in this, and again, I agree the statistical puzzle is interesting in its own right.

Yes, I think Steve White in that livestream was making a similar point (more briefly). I can certainly see this as a sense of premature death; in that you can imagine someone living to a reasonable old age and feeling like they've achieved all they wanted and are ready to die (e.g. I think Einstein ended up like this); so someone in the opposite condition (in the middle of important work and not wanting to die at all, as well as maybe dying significantly younger than their cohort) would be said to have died prematurely or 'too soon'.

(And many people die in an in between state, still doing stuff and not wanting to die, but fairly elderly and not particularly in the middle of important projects. So not clearly premature or not. Which merely shows that the concept, like many, is somewhat vague.)

That livestream also showed me there are many different philosophical angles on death. Really my post was just about the statistical puzzle, rather than the wider issue of premature death, which I've never given much thought to before!

This is getting discussed right now on livestream by a couple philosophers including Agnes Callard (recording will be available after): https://www.crowdcast.io/e/k7viaqzp

Thanks for this. I listened to quite a lot of it but couldn't find the specific discussion in the recording. (A similar question is listed in the text questions that were asked, but it doesn't seem to link to the right point in the discussion for the answer.) Anyway.

It was the very first thing they discussed. Start from the beginning of the stream and you'll get most of it.

Ah OK - thanks!

The statistics you are using are biased to refer only to the people who have survived until now. To refer to premature death, you need to compare an individual's lifetime with that of all people of roughly the same birthdate and birth conditions, including the negative life expectancy of those who are already dead.

Yes, though what the right reference class of people is is tricky. For it's possible for a whole cohort to die prematurely (e.g. from plague or war).

Maybe it's just that dying prematurely is dying 'earlier than expected' (as another comment suggests), but various different things can influence what people expect; it's not as clear-cut concept as it seems.

Life expectancy tables may overestimate on your death day, but they underestimate some people's lives on some other days, so it's not like they always overestimate. It seems like you've explained it all pretty well, I don't see any paradox left.

Well, what does dying 'prematurely' mean? Is it just dying younger than the life expectancy your cohort had at the time when you were born? What's so special about that life expectancy, rather than later estimates, or ones more specific to your own health & circumstances? Which are surely more relevant.

But if we use what seems to be the most relevant life expectancy, viz. one incorporating all relevant information about you, no-one ever dies prematurely.

(And dying prematurely doesn't mean dying before the average of your cohort either, because it's possible for most/all of a whole cohort to die young, e.g. from plague or war.)

It means dying before the age of 50 or so.

Maybe this is obvious, but:

Say life expectancy at age X is Y further years, and life expectancy at age (X + Y) is Z further years. I think at least part of the reason why Z isn't 0 is that if you're following someone along their life, the fact that they lived Y further years is more evidence to update on. If X means "lived to at least X years", then P(X+Y+Z|X+Y) > P(X+Y+Z|X), because living to X+Y years indicates certain things about, say, your health, genetics, etc that aren't indicated by just living to X years.

Yes indeed. Though the curiosity remains that at the time of death, that evidence is woefully wrong, because you're the least healthy person alive; an extreme outlier. Which means, life expectancy tables are only useful if you're not actually near death. Their % prediction error approaches infinity as you approach death (and the error is systematically in one direction).