Suppose God gives to each person the greatest equal probability of being saved. It is true, suppose, that there are two, and only two, groups of people. The members of one group will all be saved and the members of the other group will all be damned. The good news is that one of the groups is twice as large as the other. God gives each person the greatest equal probability of being saved only if he saves every member of the larger group. The epistemic probability that you are among the saved is then about .67 or 2/3.
Now suppose God offers to tell everyone whether she is in the larger group or the smaller group. Would it be rational to accept this information? If everyone learns which group she is in, then the greatest equal epistemic probability that each person is saved diminishes to .5 or 1/2. God must now flip a coin to decide which group is saved, the smaller or the larger. That is the only way to give each person the greatest equal probability of being saved. What should you do?
It is a strange problem since, if we refuse the information, many more people get saved! It is also strange since, you are already in one or the other of those groups. The information doesn’t affect which group you’re in.
"If everyone learns which group she is in, then the greatest equal epistemic probability that each person is saved diminishes to .5 or 1/2."
Mike, why is this? I'm having a hard time seeing why the new information changes the epistemic probabilities to .5 for everyone, rather than 1 for the saved and 0 for the damned.
Are we assuming here that nothing you do or believe determines which group you're in but that it's purely chance? If not, I think there will be complicating factors and a need to distinguish between this method of finding out you're saved and the ordinary one of seeing the evidence in your life that you are one of the faithful.
Christian,
Certainly, the objective probability that the members of the larger group is saved is 1. 0 for the smaller. But the epistemic probability that you are saved depends on what you know. Suppose all I know is that God saves the larger group and the larger group is twice the size of the smaller. If I had to set probabilities on which group I'm in, it would be 2/3 I'm in the larger, 1/3 the smaller. Given my lack of knowledge, I have a 2/3 probability of being saved. That is the greatest equal probability of being saved that God can give to all.
But as soon as I learn which group I'm in, the greatest equal probability of being saved goes to 1/2. God flips a coin or something. Recall, some group has ot be damned, and some group is saved. So, he cannot give all a certain chance of being saved. He cannot put the two groups together and flip a coin, all saved if heads, none saved if tails. That way, all could be saved again. The best he can do is give each a .5 chance of being save. Large group if heads, small group is tails.
Jeremy,
I'm assuming that, whatever else is true, God acts in a way to give each the greatest equal chance of being saved, and that some are saved and some not. I guess we could assume that no one is worthy of being saved and that God graciously saves better than half of us. Would that avoid the worries you have?
Suppose only one person accepts the information, and everyone else rejects it. This is sufficient for the change to coin flipping.
I think you're right, David. You could give that person only a 0 or 1 probability of being saved, by choosing the larger group. Either way it's unequal. Of course, God could give each person the greatest probability of surviving consistent with everyone else having their greatest probability. Choosing the larger group then gives most people a 2/3 probability of surviving and (if he too is in the larger group) it gives him his greatest probability of surviving.
Mike, I think that would do it, provided that one's lifestyle and beliefs don't provide evidence of whether one is saved or damned.
If God simply flips a coin and the results will be that heads is saved and tails are damned then there is as you suggest a 50-50 chance of being saved. In the actually of flipping of the coin 67% of the flips turns out to be heads. God now knows that 67% are (not will be) saved, and who these people are. You now know that there is a probability of 1 that you will are saved if you got heads when your name came up for the coin toss and you got heads or damned if it came up tails. If God tells you that he choose the outcome for each of us by flipping a coin with heads being saved, etc., this assures us that there is a equal chance of being saved or damned. But once the flips are made there is no chance of the outcome being different then what it actually is. If God tells us that the actual outcome is 67/33 the outcome is a probability of 1 that we are in the group of 67 or 33 in so far as we are in one of these groups. If I am in the group that is damned there is no chance that I am also in the group that is saved.
Sorry John, I just can't follow that. In the case i describe, if God chooses to save everyone in the larger group, you have a 67% chance of being saved. That part of the problem has nothing to do with flipping a coin.
Jeremy,
What makes me resist getting informed by God about which group I'm in is the fact that (i) I'm probably in the larger group and (ii) if I learn which group I'm in and I'm in the larger group, my chances of being saved go from 2/3 to 1/2.
If God decides to save and damn at a rate of 2 to 1 then yes we have a 67% chance of being saved before we are assigned a group. But once we are in a group then the probability that we are in that group is 1. That cannot change unless God is going to do the assignments all over. If he uses a flip of a coin then the chance of being saved or damned changes to 50-50. but again once we are in a group the probability that we are in that group is 1. Membership in the group we are in has nothing to do with what we know - our knowledge does not change the probability that we are in the group that we are in. Even if we know that God wants a certain result percentagewise, say 67-33, we do not know the outcome regarding which group we are in prior to the distribution actually being made.
I was assuming that the 'greatest equal probability' is 50-50. If that is the case then God has to use a mechinism for choosing that results in a 50-50 chance that I will be saved. He could line us up at random (I assume He wants to be fair)and have us call off by 2's with the 1's being saved and the 2's being damned. But that will result in a distribution at worse that is 51-49 (if there is an odd number of beings to be saved), not 67-33. Flipping a coin on the other hand gives us a 50-50 chance per flip that could result in a distribution greater then 51-49 depending on the actual outcome of all the flips. But that distribution is not knowable, predictable, foreseeable, prior to the flip, but dependant upon the outcome of the flips. This is true as long as I do not know if I am in in that group of people at the end of the flipping (whatever) where the effect does not change as the number of people yet to make a choice have been reduced to a point where the outcome cannot be changed. But it is not my knowledge that affects the outcome, it is the number of people left to make choices that affects the outcome.
What am I missing?
Membership in the group we are in has nothing to do with what we know - our knowledge does not change the probability that we are in the group that we are in.
After everyone is divided into the larger and smaller group it is true that (i) if you do not know which group you are in, then if God chooses the larger group, then you have a 2/3 chance of beng saved (ii) if you do know what group you are in, then if God chooses the larger group, then you have a 0 or 1 chance of being saved.
So, if you do not know which group you are in, and are asked whether you'd like to know, you have to consider whether you'd be willing to give up a 2/3 chance of being saved for a 1/2 chance of being saved.
Importantly, all of this assumes that everyone is already divided into groups. The knowledge you acquire affects the probability of being saved.
"After everyone is divided into the larger and smaller group it is true that (i) if you do not know which group you are in, then if God chooses the larger group, then you have a 2/3 chance of beng saved (ii) if you do know what group you are in, then if God chooses the larger group, then you have a 0 or 1 chance of being saved."
If God chooses the larger group and you are in that group you have a 100% chance of being saved. If I am already in the group that is saved there is no possibility that I am in the group that is damned therefore there is a probability of 1` that I am in the group that I am in. Once I am in the group that probability does not change even if from my epistemic perspective the chances appear to be different (I admit I know not understand how this could happen). If I am saved then I am saved, I must be saved or not saved, and I cannot be both saved and not saved at the same time. If I am already in a group it is established that I am what I am. My epistemic point of view does not affect the probability of which group I am in It makes no difference if I know which group I am in or not. I am already in the group I am in. That has already been settled.
I guess I do not understand what it is that you are trying to demonstrate by your thought experiment.
If God chooses the larger group and you are in that group you have a 100% chance of being saved.
That's not true, John. You have a 100% objective probability of being saved. But you also have a 67% epistemic probability of being saved. Given your epistemic position (i.e. not knowing which group you're in) you should conclude that your chances of being saved are 2/3.
The puzzle considers the question of whether it would be rational for you to let God inform you of which group you're in. It looks like it's better to remain ignorant of that fact, since your chances of being saved are better if you don't know. There is no deep theological point; it's just something fun to think about.
Mike: I agree that this is fun. I do enjoy our interactions on this blog. I hope I do not come across in any manner that suggests otherwise. If I do, then I apologize.
I wonder whether there is an argument that it is rational to accept the information, along the lines of Rawls's maximin argument for his difference principle. Someone may think, "If I turn out to be in the smaller group, I can raise the probability that I will be saved from 0 to 50-50 by accepting the information." I think that the plausibility of this argument depends on the relative size of the smaller group. Suppose, e.g, that out of a total population of 1,000,000, God decides to save everyone except one person. It doesn't seem reasonable to accept the information on the grounds that if you turn out to be the one unfortunate, you will increase the probability of your salvation.
David,
I htink the Rawlsian idea is interesting, since I think the moral thing to do is get the information. Morally, everyone should be given the same objective chance at salvation. But in terms of self-interest or prudential rationality, I think you should refuse the information. So long as +V and -U are finite, it looks like you should prudentially reason,
No information = 2/3(+V) + 1/3(-U) = X
Information = 1/2(+V) + 1/2(-U) = Y
And certainly X > Y. On the other hand, refusing relevant information does have the feel of ostridge reasoning!
John,
Glad to hear that!
I'm sorry, I know this is a dead thread, but I am just baffled by why this is a question of rationality.
Unless you think that you should maintain a belief that you are 2/3 likely to be saved instead of finding out whether you are saved, John is right.
Mike, you're right that your epistemic probability of being saved in the original scenario is 2/3. What I don't understand is why finding out whether you are in the large or small group will change your actual probability. Presumably, this is the fear. It would be irrational of you to seek out information that would lower your objective chance of being saved. But there is no reason to think that finding out whether you will be saved will change this. Your objective probability of being saved is either 1 or 0, because God has already decided.
If you find out, then you will have the epistemic probability of 1 or 0 that you will be saved. Because God is the most reliable source of information his telling you that you will or will not be saved is sufficient for your epistemic probability in regards to saving to be 1 or 0.
So, you have the option of matching your epistemic probability to the objective probability. There is a 1/3 chance that your epistemic probability will go from 2/3 to 0 and a 2/3 chance that your epistemic probability will go from 2/3 to 1.
Whether there is a rational choice to make here isn't clear to me.
Am I missing something? Perhaps I'm incorrectly assuming that group membership is fixed throughout the process?
Hi Kathryn, you write,
What I don't understand is why finding out whether you are in the large or small group will change your actual probability.
Suppose you do not request the information. In that case, God gives each the greatest equal probability of being saved only if he chooses the largest group. You have a 2/3 chance of being saved. Suppose we request the information, and find out who is in which group. In that case the greatest equal probability of being saved that God can offer each is 1/2. He now must flip a coin or use some other randomizer (recall, by stipulation, one group is saved and one isn't, so he can't save all).
So, I know that if I request the information and I'm in the larger group (probably I am) my chances of being saved go from 1 to 1/2. But if I am in the smaller group (probably I'm not) and I request the information my chances of being saved go from 0 to 1/2. And finally I know my probability right now, w/o information, of being in the larger group are 2/3.
Okay, I understand now. You are using Greatest Equal Probability to be Epistemic probability. So, in this case, finding out which group you're in would cause God to need to redistribute assignments to the groups to allow everyone equal belief that they are saved.
But in this case, it is obvious that it is irrational to find out which group you are in. Because, assuming you are randomly assigned, your first group assignment is more likely to be in the large group. And you have stipulated that the large group is the saved group under ignorance. Under knowledge of whether one is in the large group or not God is forced to re-assign destinations to the group in order to keep people ignorant about their salvation (in order to meet his obligation to give us all greatest equal Epistemic probability). So our chances of being saved have gone down.
But once this is clear, why would anyone seek out the information about whether she is in the large or small group. It is of no use whatsoever, because God needs you to not know if you are saved or damned. So you will never up your odds of being saved by finding out if you're in the large group or small group. You have a very small chance that you are in the small group and of causing the small group to be made the saved group by the coin toss. But this probability is much lower than the chance that you make the large group the damned group when you're in the large group.
Obviously you should not find out. The only reason this seems like a problem is the equivocation between Greatest Equal Probability and Greatest Equal EPISTEMIC Probability. Once this is clarified, it becomes obvious that finding out which group you are in yields no good information and only serves to decrease your chances of being saved.
Okay, I understand now. You are using Greatest Equal Probability to be Epistemic probability.
Well, not exactly. You know that you're in one group or the other. So, you know that you have both an epistemic probability of being saved and an objective probability (or, chance) of being saved. Your chances of being saved are either 0 or 1, you don't know which, but you can find out. But your epistemic probability of being saved are 2/3. Now the worry comes in, I think, for those who think that you should measure epistemic probability to chance. Lewis, for instance, suggests that this is what a rational person should do. But if I go about measuring my epistemic probability of being saved to my chance of being saved, I seem thereby to reduce the (epistemic) probability of being saved. So, strangely, it looks false that a rational person measures his epistemic probabilities to his chances.