Thought Experiment

Jon,

I have a few questions about the thought experiment. How long can people stay in the state of being outside the sphere? Also, will there be some sort of just judgment for those folks after a finite period of time? Suppose that all those "outside" the sphere will face a judgment that determines in a completely just manner whether they will stay put or go to the inner sphere. In that case it would seem that the second world is better than the first; for in the second world there are more persons who experience blessedness than those who suffer, and those who suffer do not suffer unjustly. In the first world, however, more folks suffer than those who experience blessedness and it's not clear that the initial suffering is just.

I use this question as an extra credit test question when I cover the problem of evil. I tell the students I'll give them 5 points if they give me a philosophical reason to prefer one world and 10 if they give me philosophical reasons for each. Very few can get the 10 points (and it isn't always the best students), but a lot can get 5.

From the description: "Once in the sphere, individuals are there forevermore."

And no, there's no justice, unless their final fate is what they deserve, which is purely a matter of chance in terms of which of the two worlds they ended up in. There's no going in and out of the sphere. Everyone stays put. The sphere moves to encompass them. Everyone eventually gets itno the sphere and stays there. It's just that at any given time there is in infinite number of people outside it and a finite number in it.

I think one result of this is that the utilitarian calculus gives an answer conflicting with common sense. It says to figure out what the best world is by counting how many people are blessed and how many are despairing. At any given time, the world with the expanding despair has more blessed people, though the number of despairing people increases constantly. If you consider the aggregate of all the states of the world and count person-stages, you get the same result. Each stage of the world contains infinitely many people who are blessed and finitely many who are in despair, so even though the despair is increasing it seems as if the world is better because it has more person-stages even across time who are blessed. The utilitarian calculus results in the expanding despair world as a better world.

What seems to be the more commonsense answer is that each person, even if despairing at first, will eventually be blessed forever in the other world. That means that for any given person the expanding blessedness world is better. So from an egoistic perspective, the expanding blessedness world is clearly the one to prefer. But also, if you aren't simply counting how many people but seeing what each person egoistically would want, you get this answer, because it's true not just for me but for each person.

Ted, the experiment doesn't involve any concept of judgment. That's not part of the case. All you know is the direction things are heading. You're trying to read too much theology into the case, but the idea is to think about a value question independently of any particular concerns about the afterlife.

Speed of inclusion in the sphere doesn't matter to the case either, as far as I can tell. If you want to assume something there, that's fine, but I don't see how it would relevant to the assessment.

Of the first Kvanvig world, Ted wrote:

"What seems to be the more commonsense answer is that each person, even if despairing at first, will eventually be blessed forever in the other world. That means that for any given person the expanding blessedness world is better."

I don't think that's Jon's case, is it? There is only room for finitely many in the joy circle, although there are infinitely many people outside, so I don't think it could be that all those in the latter domain eventually get into the former one. And Jon's description is only that the joy circle grows and not that it comes to include everything that was once a member of the outter set.

Or maybe I'm missing something.

Oops....it was Jeremy I was intending to quote and not Ted. Sorry about that.

Jon,

For what it's worth here's an argument that the second world has more value than the first.

Either the process by which individuals are transferred into the inner sphere is arbitrary from a moral point of view or it is not.

Assume it's arbitrary. In the first world, blessedness and desert come apart, some suffering is undeserved, and more suffer than experience blessedness. In the second world, blessedness and desert come apart, some suffering is undeserved, but more experience blessedness than suffer. Winner: second world.

Assume it's not arbitrary. In the first world, blessedness and desert don't come apart, some suffering is undeserved (the initial suffering), and more suffer than are blessed. In the second world, blessedness and desert don't come apart, all suffering is deserved, and more experience blessedness than suffer. Winner: second world.

Let w1 be the despairing world. Infinitely many people happy, and finitely many hopeless. There is endless movement to their unhappy fate.

Let w2 be the joyful world. Infinitely many despairing and finitely many joyful. There is endless movement to their happy fate.

Which is better? No matter which value is added to which, w1 is better.

V(w1) = oo+ + (finite-) = oo+
V(w2) = oo- + (finite+) = oo-

Case 1: V(w1) + V(w2) = oo+

In this case adding the value of w2 to the value of w1 does not increase or diminish the value of w1. Choose w1: w2 can't be better than w1.

Case 2: V(w2) + V(w1) = oo-

In this case adding the value of w1 to the value of w2 does not increase or decrease the value of w2: Choose w1: w1 can't be worse than w2.

Obviously, ordinary addition and subtraction are not well-defined for transfinite numbers (hence the odd result in cases 1 and 2). But if instead you moved to Robinson's non-standard infinitesimals (where addition and subtraction are well-defined), you'd get an even more convincing conclusion that w1 is the better world.

Excellent, Mike! But now consider: which world would you rather be in? I would say: the hopeful world. And I think everyone would say the same. Wouldn't you rather be in the hopeful world? There, no matter your present condition, you will someday have the best, and then forever. If that's right, how can a correct value theory yield that the best world is one that no rational person would prefer?

Jeremy, I just noticed an important difference between your case and mine. You say that in your example everyone eventually gets in the sphere. That's not the thought experiment I'm imagining. In the one I'm imagining, more and more get in the sphere. In fact, for each individual, there is some finite length of time by which time they will be in the sphere. But it is false that at some point everyone gets in. In fact, for any given moment of time in the sequence, there's always a finite number in the sphere, and an infinite number outside the sphere.

You must all be misreading me. It was sort of strange to see more than one person taking me to say something I explicitly denied, and I had to re-read what I wrote to figure out why. The statement:

E: Everyone gets in the sphere.

is ambiguous between:

E1: For every person, that person gets in the sphere.
E2: At some time, everyone will be in the sphere.

I should have been keen to this, because my students would write something like E, and I wouldn't give them full credit because I didn't know if they had understood the case correctly, depending on which disambiguation they meant.

The case in the post, as I understood it, entails E1. It also precludes E2. I seem to have been taken to be stating E2. What I said, however, is exactly E1. At no time will everyone be in the sphere, no matter how far in the future we get, but each person will at some point eventually be in the sphere. If you look at the sentence right after my statement that everyone will be in the sphere, you can see that in context it can't mean E2, because I would be contradicting it immediately afterward. So in context it must mean E1.

I don't think the question can be answered until you define "better". In what sense is one world better than the other, and why?

Jon,

there appears to be a contradiction in these two statements of yours above:

(K1)Wouldn't you rather be in the hopeful world? There, no matter your present condition, you will someday have the best, and then forever.

(K2)it is false that at some point everyone gets in. In fact, for any given moment of time in the sequence, there's always a finite number in the sphere, and an infinite number outside the sphere.

I am imagining your world like a Dyson Sphere, with the notable alteration that the walls are made of an ethereal "stuff" capable of passing through a region of matter with no adverse effect. During a growth phase, the surface of the sphere balloons out in all three-dimensional directions at the same time and in approximately 24 hour intervals. The effect I have have in mind is that that sphere inflates in bursts, one occurring roughly every day.

Question 1: What if I am such an enormous distance from the surface of the sphere at the beginning of the thought experiment that no matter how much it inflates each day the surface only approaches me asymptotically. So each day the sphere grows a bit, but its surface never reaches me.

Question 2: What if each expansion phase is preceeded by a 'gravity-wave', that pushes certain types of matter away from the surface of the sphere, so that this matter is always more a day's growth away.

Either case is possible given the terms of the experiment.

If it is never the case that everyone gets in, then it is always the case that I might be kept out. In that case my hope would be false. And what could be worse than misplacing our hope or awaiting a salvation that never comes?

The only way I can see to make the expanding-joy world preferable is to add a premise, something to the affect that each individual in the expanding-joy world will at some point move into the sphere, regardless of the infinitely many people that remain outside.

I think it WOULD be rational to pick w1. This is because the probability that I'll ever get stuck in the despair-sphere is 0. For any finite number divided by infinity is 0, so the probability that I'll be in the finite-sized sphere is also zero. Or to put it another way, since the sphere is only a potential infinity (with a finite number of persons) and the number of total persons in the world is an actual infinity, the number of individuals who will NEVER be in the sphere is an actual infinity. If we divide the number of people who will never be in the sphere (infinity) with those who are (a finite number), then the probability is 1. This is because even as the sphere grows, the cardinality of the set of those outside the sphere will be infinitely more than the cardinality of the set of the total population.

Perhaps I am breaking rules by applying division to transfinite numbers. Still, it is intuitive that the probability that I will NEVER be in the despair-sphere is very high given that the total number of people in the world is infinite and those in the sphere is only finite.

(As a note, I'm skeptical of infinity discourse as applied to concrete objects anyway after reading William Lane Craig's work on the Kalam Cosmological Argument, showing numerous worse counterintuitive paradoxes that arise when applying transfinite numbers to concrete reality, but that's a whole other discussion. Craig's option was to just deny that transfinite numbers can be applied to concrete objects.)

Patrick, think of the sphere as a set, if you wish, and model the individuals on the natural numbers. The set begins with a finite number of elements. For each finite unit of time t, add some finite number to the set. Then, for any number, there is some finite span of time until it is a member of the set. Yet, there's always an infinite number remaining outside.

Andrew, you're right that you should divide by infinity, so we can't accept your argument for preferring w1. As far as I can tell, the only reason for being skeptical about infinities here would be if you had a reason for thinking an omnipotent being couldn't create an infinite number of concrete entities. I don't see why that's not possible.

Patrick, think of the sphere as a set, if you wish, and model the individuals on the natural numbers. The set begins with a finite number of elements. For each finite unit of time t, add some finite number to the set. Then, for any number, there is some finite span of time until it is a member of the set. Yet, there's always an infinite number remaining outside.

Andrew, you're right that you should divide by infinity, so we can't accept your argument for preferring w1. As far as I can tell, the only reason for being skeptical about infinities here would be if you had a reason for thinking an omnipotent being couldn't create an infinite number of concrete entities. I don't see why that's not possible.

Okay, I guess I'm really slow: how does it follow from Jon's description that, eventually, every person will be in the, for example, joy circle (even though there will never be a time when all are in)? Both Jon and Ted believe it does but I don't get it.

Here's Jon's original description:

"Here's how it goes. Assume there are infinitely many sentient/cognitive beings, and assume that there is a sphere into which some finite number fit. The sphere expands over time, and we assume that time is unending, so that more and more people fit into the sphere the farther history progresses. Once in the sphere, individuals are there forevermore.

There are two such worlds, we assume. In the first world, the sphere is a sphere of pure joy and satisfaction with one's life, and outside the sphere is despair. As history progresses, more and more people are brought into the realm of pure joy, satisfaction, and fulfillment."

Why isn't it consistent with this description that every second person is admitted to the circle, with all the others never being admitted? So assign each person in line a single number (either at the beginning of the process or at some point during it) that corresponds to some positive integer, with no number being used twice. Now why can't it be a rule in these worlds that only those holding an odd numeral will eventually get in, and those with the even numerals are forever on the outside?

What am I missing? A little help?

Tom,

Let me second your question. As I read the original post I was thinking one’s destiny could either be the inner sphere or the initial sphere. That’s why I asked initially how long one could stay in one’s initial state. If one can stay in the initial state forever then I think the argument I gave earlier shows that the second world has more value than the first. If not, then it’s less clear because of the considerations Jon mentions: no rational person would prefer to be in a world in which their eventual destiny is endless despair.

Okay, I'm hoping to eventually get this blog thing down. But I've once again mis-identified one of the folks I was responding to. I meant to say that it was Jon and *Jeremy* (and not Jon and Ted) who hold that each person in line will eventually get to the inner circle. Ted, it turns out, has the good sense to agree with me on this score (or maybe he just has the misfortune of sharing my unenlightened condition?). (-:

Jon,

Overall w1 is better than w2. But yes a rational agent might prefer w2 to w1. The overall value/disvalue of these worlds is no doubt infinite. But suppose (as seems reasonable) that the overall value of any region for any arbitrarily selected member of these worlds is great, but finite. Suppose the following are true of w1 and w2:
1. Any arbitrarily selected member of w1 has a chance C of moving to the hopeless region and chance C' of remaining in the joyful region and C > C'.
2. Any arbitrarily selected member of w2 has chance C of moving to the joyful region and chance C' of remaining in the hopeless region and C > C'.
3. Being permanently in the joyful region has a value for any arbitrarily selected individual of N+ and being permanently in the hopeless region has a value for any arbitrarily selected individual of N-.
If (1)-(3) are true, then it is rational to select w2 over w1. The rationale is simple.
The expected value of being in w2 is this,
EV(w2)= C(N+) + C'(N-)= some positive value.
But the expected value of being in (w1) is this,
EV(w1)= C(N-) + C'(N+) = some negative value.

Now the situation is admittedly idealized. But I think these sorts of cases are really important. Consider the "it-ought-to-be-a-better-world" objection. The sort of case we're discussing makes room for a response like this "If it were a better world you'd prefer it less". Now that's very interesting.

Jon, we might formalize your example like this:

(K1)model the individuals on the natural numbers.
(K2)The set begins with a finite number of elements.
(K3)For each finite unit of time t, add some finite number to the set.

(K4)Then, for any number, there is some finite span of time until it is a member of the set.

I just don't see how (K4) follows at all. My initial objection, and Tom's and Mike's as well, serve to show that unless we add some futher premise like

(K5) The chance C that any individual I will remain outside the Sphere zeros out over a finite timespan.

It may look like (K4) and (K5) amount to the same thing. But I do not think they do. We have to address the chance, however slim, that some individuals could remain outside the sphere indefinitely.

Only if we adopt a premise like (K5)is world w1 clearly the rational choice.

OK, in response to Tom's worry, let's fill in the details of expansion as follows. Every individual is a given distance from the sphere, and the sphere expands a given distance unit every second, and those within that distance come to be inside the sphere as a result.

Mike, I like what you do here. Instead of making it a certainty that you'll end up in the sphere, all that is needed is a positive chance that you will. That is still enough to make it rational to prefer the hopeful world to the despair world, and it does allow for the truth of the very strange counterfactual you mention: asking for a better world might get you a world it is rational to be averse to.

So are you planning a separate post about how this connects to heaven and hell? Because my first thought was that it doesn't. The interesting features of this example seem to me to be importantly disanalagous to heaven and hell, at least on the traditional conception. There wouldn't be an infinite number of people still to be decided where they would end up. Even if it could go infinitely into the future with more people created as time goes on, they don't all exist now. The state of people not in heaven or hell is neutral with respect to the badness of hell and the goodness of heaven.

It also ignores the element of justness of punishment. Ted kept wanting to import those things into the case, because you flagged it as being relevant to heaven and hell, but the case itself ignores justness of suffering entirely. If the suffering of evildoers is a good, then the very terms of this thought experiment are problematic. It assumes a sort of hedonism about well-being. If everyone deserves the despair, and it would be good if more people received it, then the world with the most suffering is the best one, contrary to most people's intuitive responses to this thought experiment. Or were you planning to sneak up on this very point?

It occurred to me also that what's rational might be affected by this. If Plato is correct, and it's in our best interest to suffer when we deserve punishment, then the rational thing to do would be to seek to be punished. So it seems that you can't answer either the question of which world is better or the question of which world it would be rational to prefer without some truly controversial philosophical presuppositions about well-being.

Jeremy,

Certainly the received view on rational behavior is the decision-theoretic view. It really incorporates the fewest assumptions. Perhaps you want to add to this that reason can tell us something about rational ends or goals--in the spirit of Plato that you suggest. Fine with me, but this is surely where things get more controversial. On the other hand, if you mean that even the decision-theoretic view of rationality is controversial, then of course I agree. But in this sense just about any interesting philosophical thesis is controversial.

With Jon's fine tuning of the case so that it entails that, eventually, each individual goes from the initial area to the inner area, I don't see that there is any real doubt about which world is better. Sure, for each time slice, more people are then happy than miserable, and that's something. But when you realize that in that world each individual will be happy only a finite amount of time and miserable an infinite time, then the other world in which everyone will be happy for an infinite time and miserable only a finite time looks to be clearly the lesser world.

I am inclined to agree with you, Tom. It doesn't matter much that in the despairing-world, there will be a greater quantity of happiness in any given time slice. What matters is that in the hoping-world every individual ends up with lasting personal happiness.

But I have a couple of questions about the psychology of the thought experiment people.

(1) Could people be lastingly happy carrying the memory of utter despair around with them?

(2) Could people truly despair carrying the memory of utter bliss around with them?

It seems like (2) is more plausible than (1).

In the case of of (1), I have in mind people like my grandfather Taylor who fought at the Battle of the Bulge with the 101st Airborne. He cannot go out of the house on the 4th of July; he cannot go outside on winter evenings when it snows; and, though he was an avid outdoorsmen as a young man, he is deathly afraid to go near forests now (even the woods on our family farm). It seems that people like my grandfather, who have really known despair, are never able to leave it behind.

Of course, It could be the case that people in either sphere simply don't remember their lives on the outside.But if this were the case case I don't think it would matter where you went, because you would never know any state but the one you found yourself in at the moment.

Tom, I'm not sure I follow this,
"then the other world in which everyone will be happy for an infinite time and miserable only a finite time looks to be clearly the lesser world"

Do you mean "clearly the better world"? But even then it is not obvious. Take w2, it is perfectly possible that though each person suffers only a finite amount of time in despair, there are always more people suffering than joyful. This might go on infinitely, though every person in the suffering realm is there for a finite time and moves to the joyful realm permanently. This world is then worse than w1. Here's a simple model. Suppose that God always keeps the realm of despair more populated than the realm of joy. But no one stays in the realm of despair for an infinite period of time. For every hundred that leave despair, God creates one hundred and one. This goes on infinitely. Clearly the amount of suffering in this world is worse than the amount in its counterpart where the converse is happening. Every person in w2 winds up in the joyful realm, yet the world is worse overall: there are always more in the despairing realm.

The real world is a combination of both worlds and cannot be otherwise. Joy and despair define each other like space and matter. If you lived in a world of constant joy eventually you would discern between mundane joy and extremely blissful joy and be right back in the real balanced world.

Maybe a more realistic thought experiment would be: World one is defined as the current created world and world two is the uncreated, undifferentiated being "before" the creation. Which world is prefferable?

The former is exciting but prone to fear, sickness and etc. The latter pure perfect peace but maybe boring.

Best regards,
Addison