Other Wagers

While no one believes the ridiculous position that only atheists go to heaven, not even Richard Carrier himself, many people do believe things which allow for the construction of wagers in favor of things besides Christianity.

Other real religions would be a typical example. Thus Muslims probably generally believe that you are more likely to go heaven if you become a Muslim, and some of them believe that all non-Muslims go to hell. So they can argue that you should choose to believe that Islam is true, in order to increase your chances of going to heaven, and to avoid going to hell.

Similarly, some Catholics who hold a Feeneyite position hold not only that you must be a Catholic, externally and literally, in order to be saved, but that you also cannot be saved unless you accept this position. Since there is some chance that they are right, they might argue, you should choose to accept their position, in order to be more sure of getting to heaven and avoiding hell.

Likewise, a Catholic could argue that the Church teaches that the religious life is better than married life. This probably means that people choosing to embrace religious life are more likely to go to heaven, since it is unlikely that the chances are completely equal, and if getting married made you more likely to go to heaven, it would be better in an extremely important way. So if you are a single Catholic, you should choose to believe that you have a religious vocation, in order to maximize your chances of going to heaven.

It is instructive to consider these various wagers because they can give some sort of indication of what kinds of response are reasonable and what kinds are not, even to Pascal’s original wager.

A Christian would be likely to respond to the Islamic wager in this way: it is more likely that Christianity is true than Islam, and if Christianity is true, the Christian would increase his chances of going to hell, and decrease his chances of going to heaven, by converting to Islam, and especially for such a reason. This is much like Richard Carrier’s response to Pascal, but it is reasonable for Christians in a way in which it is not for Carrier, because Christianity is an actually existing religion, while his only-atheists-go-to-heaven religion is not.

A Catholic could respond to the Feeneyite wager in a similar way. The Feeneyite position is probably false, and probably contrary to the teaching of the Church, and therefore adopting it would be uncharitable to other people (by assuming that they are going to hell), and unfaithful to the Catholic Church (since the most reasonable interpretation of the Church’s teaching does not allow for this position.) So rather than increasing a person’s chances of going to heaven, adopting this position would decrease a person’s chances of this result. Again, the possibility of this answer derives from the fact that there are two already existing positions, so this answer does not benefit someone like Carrier.

It is more difficult for a Catholic to reject the religious vocation wager in a reasonable way, because the Church does teach that the religious life is better, and in fact this most likely does imply that Catholics embracing this form of life are more likely to be saved.

A Catholic could respond, “But I don’t actually have a religious vocation, and so I shouldn’t choose to believe that I do.” This is no different from a unbeliever saying, “But Christianity is not actually true, and so I shouldn’t choose to believe that it is.” And this response fails in both cases, because the arguments do not purport to establish that you have a vocation or that Christianity is true. They only intend to establish that it is better to believe these things, and such a response does not address the argument.

The Catholic could insist, “But if I don’t actually have a religious vocation, God does not want me to do that. So if I choose to believe in a vocation and follow it, I will be doing something that God does not want me to do. So I will be less likely to be saved.” This seems similar to the unbeliever responding, “If God exists, he does not want people believing things that are false, and Christianity is false. So I will be less likely to get anything good from God by believing.”

Both responses are problematic, basically because they are inconsistent with the principles that are assumed to be true in order to consider the situation. In other words, the response here by the Catholic is probably inconsistent with the teaching of the Church on religious vocations, and likewise the response by the unbeliever is obviously inconsistent with Christianity (since it denies it.) The reason the Catholic response is likely inconsistent with the teaching of the Church on vocations is that given that someone asserts that he has a religious vocation, the Church forbids other people from dissuading him from it on the supposed grounds that he does not have a real vocation. But if it were true that someone who chooses to believe that he has a vocation, and then acts on it, is less likely to be saved given that he did not really have a vocation, then it would be extremely reasonable for people to dissuade him. Since the Church forbids such dissuasion, it seems to imply that this answer is not correct.

Both responses can be modified so that they will be consistent with the situation under consideration. The Catholic can respond, “I very much do not want to live the religious life. Granted that the religious life in general would make someone more likely to be saved, I would be personally extremely miserable in it. This would tempt me to various vicious things, either as consequences of it, as snapping angrily at people, or in order to relieve my misery, as engaging in sinful pleasures. So despite the general situation, I would be personally less likely to be saved if I lived the religious life.”

It is not clear how strong this response is, but at least it is consistent with accepting the general teaching of the Church on vocations. Likewise, the unbeliever can modify his response to the following: “Even if Christianity is true, it seems false to me, and from my point of view, choosing to believe it would be choosing to believe something false. The Bible makes clear that people rejecting the light and choosing darkness are doing evil, so that would mean God wouldn’t be pleased by this kind of behavior. So choosing to believe would make me less likely to go to heaven, even given that Christianity is true.”

Once again, it is not clear how strong this response is, but it is consistent with the claims of Christianity, and no less reasonable than the response of the single Catholic to the argument regarding vocations.

Scott Alexander, in the comment quoted previously, is dissatisfied with such arguments because they do not provide a general answer that would apply in all possible circumstances (e.g. if you were guaranteed that God did not object to your choosing to believe something you think to be false), and it seems to him that he personally would want to reject the wager in all circumstances. He concludes that his desire to reject it is objectively unreasonable. However, he does this under the assumption that the value of getting to heaven and avoiding hell is actually infinite. As we have seen, it is not infinite in the way that matters. However, he also assumes in his comment that the probability of Christianity is astronomically low. If this were correct, then since the value of salvation is not numerically infinite, it would be right to reject the wager in all circumstances. But since it is not correct to assign such a low probability, this does not follow.

The implication is that it is not possible to give a reasonable response to the wager which implies that it would never be reasonable to accept it. Likewise, the implication of the possible responses is that it is not possible to propose the wager in a form which ought to be compelling to anyone who is reasonable and under all circumstances. In this sense, Scott Alexander is right to suppose that his desire to reject the wager under all possible circumstances is irrational. Real life is complicated, and in real life a person could reasonably accept such a wager under certain circumstances, and a person could reasonably reject such a wager under certain circumstances.

Richard Carrier Responds to Pascal’s Wager

Richard Carrier attempts to respond to Pascal’s Wager by suggesting premises which lead to a completely opposite conclusion:

The following argument could be taken as tongue-in-cheek, if it didn’t seem so evidently true. At any rate, to escape the logic of it requires theists to commit to abandoning several of their cherished assumptions about God or Heaven. And no matter what, it presents a successful rebuttal to any form of Pascal’s Wager, by demonstrating that unbelief might still be the safest bet after all (since we do not know whose assumptions are correct, and we therefore cannot exclude the assumptions on which this argument is based).

If his response is taken literally, it is certainly not true in fact, and it is likely that he realizes this, and for this reason says that it could be taken as “tongue-in-cheek.” But since he adds that it seems “so evidently true,” it is not clear that he sees what is wrong with it.

His first point is that God would reward people who are concerned about doing good, and therefore people who are concerned about the truth:

It is a common belief that only the morally good should populate heaven, and this is a reasonable belief, widely defended by theists of many varieties. Suppose there is a god who is watching us and choosing which souls of the deceased to bring to heaven, and this god really does want only the morally good to populate heaven. He will probably select from only those who made a significant and responsible effort to discover the truth. For all others are untrustworthy, being cognitively or morally inferior, or both. They will also be less likely ever to discover and commit to true beliefs about right and wrong. That is, if they have a significant and trustworthy concern for doing right and avoiding wrong, it follows necessarily that they must have a significant and trustworthy concern for knowing right and wrong. Since this knowledge requires knowledge about many fundamental facts of the universe (such as whether there is a god), it follows necessarily that such people must have a significant and trustworthy concern for always seeking out, testing, and confirming that their beliefs about such things are probably correct. Therefore, only such people can be sufficiently moral and trustworthy to deserve a place in heaven–unless god wishes to fill heaven with the morally lazy, irresponsible, or untrustworthy.

But only two groups fit this description: intellectually committed but critical theists, and intellectually committed but critical nontheists (which means both atheists and agnostics, though more specifically secular humanists, in the most basic sense).

His second point is that the world is a test for this:

It is a common belief that certain mysteries, like unexplained evils in the world and god’s silence, are to be explained as a test, and this is a reasonable belief, widely defended by theists of many varieties.

His next argument is that the available evidence tends to show that either God does not exist or that he is evil:

If presented with strong evidence that a god must either be evil or not exist, a genuinely good person will not believe in such a god, or if believing, will not give assent to such a god (as by worship or other assertions of approval, since the good do not approve of evil). Most theists do not deny this, but instead deny that the evidence is strong. But it seems irrefutable that there is strong evidence that a god must either be evil or not exist.

For example, in the bible Abraham discards humanity and morality upon God’s command to kill his son Isaac, and God rewards him for placing loyalty above morality. That is probably evil–a good god would expect Abraham to forego fear and loyalty and place compassion first and refuse to commit an evil act, and would reward him for that, not for compliance. Likewise, God deliberately inflicts unconscionable wrongs upon Job and his family merely to win a debate with Satan. That is probably evil–no good god would do such harm for so petty a reason, much less prefer human suffering to the cajoling of a mere angel. And then God justifies these wrongs to Job by claiming to be able to do whatever he wants, in effect saying that he is beyond morality. That is probably evil–a good god would never claim to be beyond good and evil. And so it goes for all the genocidal slaughter and barbaric laws commanded by God in the bible. Then there are all the natural evils in the world (like diseases and earthquakes) and all the unchecked human evils (i.e. god makes no attempt to catch criminals or stop heinous crimes, etc.). Only an evil god would probably allow such things.

He concludes that only atheists go to heaven:

Of the two groups comprising the only viable candidates for heaven, only nontheists recognize or admit that this evidence strongly implies that God must be evil or not exist. Therefore, only nontheists answer the test as predicted for morally good persons. That is, a morally good person will be intellectually and critically responsible about having true beliefs, and will place this commitment to moral good above all other concerns, especially those that can corrupt or compromise moral goodness, like faith or loyalty. So those who are genuinely worthy of heaven will very probably become nontheists, since their inquiry will be responsible and therefore complete, and will place moral concerns above all others. They will then encounter the undeniable facts of all these unexplained evils (in the bible and in the world) and conclude that God must probably be evil or nonexistent.

In other words, to accept such evils without being given a justification (as is entailed by god’s silence) indicates an insufficient concern for having true beliefs. But to have the courage to maintain unbelief in the face of threats of hell or destruction, as well as numerous forms of social pressure and other hostile factors, is exactly the behavior a god would expect from the genuinely good, rather than capitulation to the will of an evil being, or naive and unjustified trust that an apparently evil being is really good–those are not behaviors of the genuinely good.

It is not completely clear what he thinks about his own argument. His original statement suggests that he realizes that it is somewhat ridiculous, taken as a whole, but it is not exactly clear if he understands why. He concludes:

Since this easily and comprehensively explains all the unexplainable problems of god (like divine hiddenness and apparent evil), while other theologies do not (or at least nowhere so well), it follows that this analysis is probably a better explanation of all the available evidence than any contrary theology. Since this conclusion contradicts the conclusion of every form of Pascal’s Wager, it follows that Pascal’s Wager cannot assure anyone of God’s existence or that belief in God will be the best bet.

This might express his failure to see the largest flaw in his argument. He probably believes that it is actually true that “this analysis is probably a better explanation of all the available evidence than any contrary theology.” But this cannot be true, even assuming that his arguments about good and evil are correct. The fact that very many people accept a Christian theology, and that no one believes Carrier’s suggested theology, is in itself part of the available evidence, and this fact alone outweighs all of his arguments, whether or not they are correct. That is, a Christian theology is more likely to be true as a whole than his proposed theology of “only atheists go to heaven”, regardless of the facts about what good people are likely to do, of the facts about what a good God is likely to do, and so on.

It is a common failure on the part of unbelievers not to notice the evidence that results from the very existence of believers. This is of course an aspect of the common failure of people in general to notice the existence of evidence against their current beliefs. In this sense, Carrier likely does in fact actually fail to notice this evidence. Consequently he has a vague sense that there is something ridiculous about his argument, but he does not quite know what it is.

Nonetheless, although his argument is mistaken as a whole, there are some aspects of it which could be reasonably used by an unbeliever in responding to Pascal’s wager in a truly reasonable way. Such a response would go something like this, “My current beliefs about God and the world are largely a result of the fact that I am trying to know the truth, and the fact that I am trying to know the truth is a part of the fact that I am trying to be a good person. Choosing to believe would be choosing to abandon significant parts of my effort to be a good person. If there is a good God, I would expect him to take these things into account.”

Numbering The Good

The book Theory of Games and Economic Behavior, by John Von Neumann and Oskar Morgenstern, contains a formal mathematical theory of value. In the first part of the book they discuss some objections to such a project, as well as explaining why they are hopeful about it:

1.2.2. It is not that there exists any fundamental reason why mathematics should not be used in economics. The arguments often heard that because of the human element, of the psychological factors etc., or because there is allegedly no measurement of important factors, mathematics will find no application, can all be dismissed as utterly mistaken. Almost all these objections have been made, or might have been made, many centuries ago in fields where mathematics is now the chief instrument of analysis. This “might have been” is meant in the following sense: Let us try to imagine ourselves in the period which preceded the mathematical or almost mathematical phase of the development in physics, that is the 16th century, or in chemistry and biology, that is the 18th century. Taking for granted the skeptical attitude of those who object to mathematical economics in principle, the outlook in the physical and biological sciences at these early periods can hardly have been better than that in economics, mutatis mutandis, at present.

As to the lack of measurement of the most important factors, the example of the theory of heat is most instructive; before the development of the mathematical theory the possibilities of quantitative measurements were less favorable there than they are now in economics. The precise measurements of the quantity and quality of heat (energy and temperature) were the outcome and not the antecedents of the mathematical theory. This ought to be contrasted with the fact that the quantitative and exact notions of prices, money and the rate of interest were already developed centuries ago.

A further group of objections against quantitative measurements in economics, centers around the lack of indefinite divisibility of economic quantities. This is supposedly incompatible with the use of the infinitesimal calculus and hence (!) of mathematics. It is hard to see how such objections can be maintained in view of the atomic theories in physics and chemistry, the theory of quanta in electrodynamics, etc., and the notorious and continued success of mathematical analysis within these disciplines.

This project requires the possibility of treating the value of things as a numerically measurable quantity. Calling this value “utility”, they discuss the difficulty of this idea:

3.1.2. Historically, utility was first conceived as quantitatively measurable, i.e. as a number. Valid objections can be and have been made against this view in its original, naive form. It is clear that every measurement, or rather every claim of measurability, must ultimately be based on some immediate sensation, which possibly cannot and certainly need not be analyzed any further. In the case of utility the immediate sensation of preference, of one object or aggregate of objects as against another, provides this basis. But this permits us only to say when for one person one utility is greater than another. It is not in itself a basis for numerical comparison of utilities for one person nor of any comparison between different persons. Since there is no intuitively significant way to add two utilities for the same person, the assumption that utilities are of non-numerical character even seems plausible. The modern method of indifference curve analysis is a mathematical procedure to describe this situation.

They note however that the original situation was no different with the idea of quantitatively measuring heat:

3.2.1. All this is strongly reminiscent of the conditions existent at the beginning of the theory of heat: that too was based on the intuitively clear concept of one body feeling warmer than another, yet there was no immediate way to express significantly by how much, or how many times, or in what sense.

Beginning the derivation of their particular theory, they say:

3.3.2. Let us for the moment accept the picture of an individual whose system of preferences is all-embracing and complete, i.e. who, for any two objects or rather for any two imagined events, possesses a clear intuition of preference.

More precisely we expect him, for any two alternative events which are put before him as possibilities, to be able to tell which of the two he prefers.

It is a very natural extension of this picture to permit such an individual to compare not only events, but even combinations of events with stated probabilities.

By a combination of two events we mean this: Let the two events be denoted by B and C and use, for the sake of simplicity, the probability 50%-50%. Then the “combination” is the prospect of seeing B occur with a probability of 50% and (if B does not occur) C with the (remaining) probability of 50%. We stress that the two alternatives are mutually exclusive, so that no possibility of complementarity and the like exists. Also, that an absolute certainty of the occurrence of either B or C exists.

To restate our position. We expect the individual under consideration to possess a clear intuition whether he prefers the event A to the 50-50 combination of B or C, or conversely. It is clear that if he prefers A to B and also to C, then he will prefer it to the above combination as well; similarly, if he prefers B as well as C to A, then he will prefer the combination too. But if he should prefer A to, say B, but at the same time C to A, then any assertion about his preference of A against the combination contains fundamentally new information. Specifically: If he now prefers A to the 50-50 combination of B and C, this provides a plausible base for the numerical estimate that his preference of A over B is in excess of his preference of C over A.

If this standpoint is accepted, then there is a criterion with which to compare the preference of C over A with the preference of A over B. It is well known that thereby utilities, or rather differences of utilities, become numerically measurable. That the possibility of comparison between A, B, and C only to this extent is already sufficient for a numerical measurement of “distances” was first observed in economics by Pareto. Exactly the same argument has been made, however, by Euclid for the position of points on a line in fact it is the very basis of his classical derivation of numerical distances.

It is important to note that the the things being assigned values are described as events. They should not be considered to be actions or choices, or at any rate, only insofar as actions or choices are themselves events that happen in the world. This is important because a person might very well think, “It would be better if A happened than if B happened. But making A happen is vicious, while making B happen is virtuous, so I will make B happen.” He prefers A as an outcome, but the actions which cause these events do not line up, in their moral value, with the external value of the outcomes. Of course, just as the person says that A happening is a better outcome than B happening, he can say that “choosing to make B happen” is a better outcome than “choosing to make A happen.” So in this sense there is nothing to exclude actions from being included in this system of value. But they can only be included insofar as actions themselves are events that happen in the world.

Von Neumann and Morgenstern continue:

The introduction of numerical measures can be achieved even more directly if use is made of all possible probabilities. Indeed: Consider three events, C, A, B, for which the order of the individual’s preferences is the one stated. Let a be a real number between 0 and 1, such that A is exactly equally desirable with the combined event consisting of a chance of probability 1 – a for B and the remaining chance of probability a for C. Then we suggest the use of a as a numerical estimate for the ratio of the preference of A over B to that of C over B.

So for example, suppose that C is an orange (or as an event, eating an orange). is eating a plum, and is eating an apple. The person prefers the orange to the plum, and the plum to the apple. The person prefers a combination of a 20% chance of an apple and an 80% chance of an orange to a plum, while he prefers a plum to a combination of a 40% chance of an apple and a 60% chance of an orange. Since this indicates that his preference changes sides at some point, we suppose that this happens at a 30% chance of an apple and a 70% chance of an orange. All the combinations giving more than a 70% chance of the orange, he prefers to the plum; and he prefers the plum to all the combinations giving less than a 70% chance of the orange. The authors are suggesting that if we assign numerical values to the plum, the apple, and the orange, we should do this in such a way that the difference between the values of the plum and the apple, divided by the difference between the values of the orange and the apple, should be 0.7.

The basic intuition here is that since the combinations of various probabilities of the orange and apple vary continuously from (100% orange, 0% apple) to (0% orange, 100% apple), the various combinations should go continuously through every possible value between the value of the orange and the value of the apple. Since we are passing through those values by changing a probability, they are suggesting mapping that probability directly onto a value. Thus if the value of the orange is 1 and the value of the apple is 0, we say that the value of the plum is 0.7, because the plum is basically equivalent in value to a combination of a 70% chance of the orange and a 30% chance of the apple.

Working this out formally in the later parts of the paper, they show that given that a person’s preferences satisfy certain fairly reasonable axioms, it will be possible to assign values to each of his preferences, and these values are necessarily uniquely determined up to the point of a linear transformation.

I will not describe the axioms themselves here, although they are described in the book, as well as perhaps more simply elsewhere.

Note that according to this system, if you want to know the value of a combination, e.g. (60% chance of A and 40% chance of B), the value will always be 0.6(value of A)+0.4(value of B). The authors comment on this result:

3.7.1. At this point it may be well to stop and to reconsider the situation. Have we not shown too much? We can derive from the postulates (3:A)-(3:C) the numerical character of utility in the sense of (3:2:a) and (3:1:a), (3:1:b) in 3.5.1.; and (3:1:b) states that the numerical values of utility combine (with probabilities) like mathematical expectations! And yet the concept of mathematical expectation has been often questioned, and its legitimateness is certainly dependent upon some hypothesis concerning the nature of an “expectation.” Have we not then begged the question? Do not our postulates introduce, in some oblique way, the hypotheses which bring in the mathematical expectation?

More specifically: May there not exist in an individual a (positive or negative) utility of the mere act of “taking a chance,” of gambling, which the use of the mathematical expectation obliterates?

The objection is this: according to this system of value, if something has a value v, and something else has the double value 2v, the person should consider getting the thing with value v to be completely equal with a deal where he has an exactly 50% chance of getting the thing with value 2v, and a 50% chance of getting nothing. That seems objectionable because many people would prefer a certainty of getting something, to a situation where there is a good chance of getting nothing, even if there is also a chance of getting something more valuable. So for example, if you were now offered the choice of $100,000 directly, or $200,000 if you flip a coin and get heads, and nothing if you get tails, you would probably not only prefer the $100,000, but prefer it to a very high degree.

Morgenstern and Von Neumann continue:

How did our axioms (3:A)-(3:C) get around this possibility?

As far as we can see, our postulates (3:A)-(3:C) do not attempt to avoid it. Even that one which gets closest to excluding a “utility of gambling” (3:C:b) (cf. its discussion in 3.6.2.), seems to be plausible and legitimate, unless a much more refined system of psychology is used than the one now available for the purposes of economics. The fact that a numerical utility, with a formula amounting to the use of mathematical expectations, can be built upon (3:A)-(3:C), seems to indicate this: We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate. Since (3:A)-(3:C) secure that the necessary construction can be carried out, concepts like a “specific utility of gambling” cannot be formulated free of contradiction on this level.

“We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate.” In other words, the reason for the strange result is that calling a value “double” very nearly simply means that a 50% chance of that value, and a 50% chance of nothing, is considered equal to the original value which was to be doubled.

Considering the case of the $100,000 and $200,000, perhaps it is not so strange after all, even if we think of value in the terms of Von Neumann and Morgenstern. You are benefited if you receive $100,000. But if you receive $100,000, and then another $100,000, how much benefit do you get from the second gift? Just as much? Not at all. The first gift will almost certainly make a much bigger change in your life than the second gift. So even by ordinary standards, getting $200,000 is not twice as valuable as getting $100,000, but less than twice as valuable.

There might be something such that it would have exactly twice the value of $100,000 for you in the Von Neumann-Morgenstern sense. If you care about money enough, perhaps $300,000, or $1,000,000. If so, then you would consider the deal where you flip a coin for this amount of money just as good (considered in advance) as directly receiving $100,000. If you don’t care enough about money for such a thing to be true, there will be something else that you do consider to have twice the value, or more, in this sense. For example, if you have a brother dying of cancer, you would probably prefer that he have a 50% chance of survival, to receiving the $100,000. This means that in the relevant sense, you consider the survival of your brother to have more than double the value of $100,000.

This system of value does not in fact prevent one from assigning a “specific utility of gambling,” even within the system, as long as the fact that I am gambling or not is considered as a distinct event which is an additional result. If the only value that matters is money, then it is indeed a contradiction to speak of a specific utility of gambling. But if I care both about money and about whether I am gambling or not, there is no contradiction.

Something else is implied by all of this, something which is frequently not noticed. Suppose you have a choice of two events in this way. One of them is something that you would want or would like, as small or big as you like. It could be having a nice day at the beach, or $100, or whatever you please. The other is a deal where you have a virtual certainty of getting nothing, and a very small probability of some extremely large reward. For example, it may be that your brother dying of cancer is also on the road to hell. The second event is to give your brother a chance of one in a googolplex of attaining eternal salvation.

Of course, the second event here is worthless. Nobody is going to do anything or give up anything for the sake of such a deal. What this implies is this: if a numerical value is assigned to something in the Von Neumann-Morgenstern manner, no matter what that thing is, that value must be low enough (in comparison to other values) that it won’t have any significant value after it is divided by a googolplex.

In other words, even eternal salvation does not have an infinite value, but a finite value (measured in this way), and low enough that it can be made worthless by enough division.

If we consider the value to express how much we care about something, then this actually makes intuitive sense, because we do not care infinitely about anything, not even about things which might be themselves infinite.

Pascal, in his wager, assumes a probability of 50% for God and for the truth of religious beliefs, and seems to assume a certainty of salvation, given that you accept those beliefs and that they happen to be true. He also seems to assume a certain loss of salvation, if you do not accept those beliefs and they happen to be true, and that nothing in particular will happen if the beliefs are not true.

These assumptions are not very reasonable, considered as actual probability assignments and actual expectations of what is going to happen. However, some set of assignments will be reasonable, and this will certainly affect the reasonableness of the wager. If the probability of success is too low, the wager will be unreasonable, just as above we noted that it would be unreasonable to accept the deal concerning your brother. On the other hand, if the probability of success is high enough, it may well be reasonable to take the deal.

Erroneous Responses to Pascal

Many arguments which are presented against accepting Pascal’s wager are mistaken, some of them in obvious ways. For example, the argument is made that the multiplicity of religious beliefs or potential religious beliefs invalidates the wager:

But Pascal’s argument is seriously flawed. The religious environment that Pascal lived in was simple. Belief and disbelief only boiled down to two choices: Roman Catholicism and atheism. With a finite choice, his argument would be sound. But on Pascal’s own premise that God is infinitely incomprehensible, then in theory, there would be an infinite number of possible theologies about God, all of which are equally probable.

First, let us look at the more obvious possibilities we know of today – possibilities that were either unknown to, or ignored by, Pascal. In the Calvinistic theological doctrine of predestination, it makes no difference what one chooses to believe since, in the final analysis, who actually gets rewarded is an arbitrary choice of God. Furthermore we know of many more gods of many different religions, all of which have different schemes of rewards and punishments. Given that there are more than 2,500 gods known to man, and given Pascal’s own assumptions that one cannot comprehend God (or gods), then it follows that, even the best case scenario (i.e. that God exists and that one of the known Gods and theologies happen to be the correct one) the chances of making a successful choice is less than one in 2,500.

Second, Pascal’s negative theology does not exclude the possibility that the true God and true theology is not one that is currently known to the world. For instance it is possible to think of a God who rewards, say, only those who purposely step on sidewalk cracks. This sounds absurd, but given the premise that we cannot understand God, this possible theology cannot be dismissed. In such a case, the choice of what God to believe would be irrelevant as one would be rewarded on a premise totally distinct from what one actually believes. Furthermore as many atheist philosophers have pointed out, it is also possible to conceive of a deity who rewards intellectual honesty, a God who rewards atheists with eternal bliss simply because they dared to follow where the evidence leads – that given the available evidence, no God exists! Finally we should also note that given Pascal’s premise, it is possible to conceive of a God who is evil and who punishes the good and rewards the evil.

Thus Pascal’s call for us not to consider the evidence but to simply believe on prudential grounds fails.

There is an attempt here to base the response on Pascal’s mistaken claim that the probability of the existence of God (and of Catholic doctrine as a whole) is 50%. This would presumably be because we can know nothing about theological truth. According to this, the website reasons that all possible theological claims should be equally probable, and consequently one will be in any case very unlikely to find the truth, and therefore very unlikely to attain the eternal reward, using Pascal’s apparent assumption that only believers in a specific theology can attain the reward.

The problem with this is that it reasons for Pascal’s mistaken assumptions (as well as changing them in unjustified ways), while in reality the effectiveness of the wager does not precisely depend on these assumptions. If there is a 10% chance that God exists, and the rest is true as Pascal states it, it would still seem to be a good bet that God exists, in terms of the practical consequences. You will probably be wrong, but the gain if you are right will be so great that it will almost certainly outweigh the probable loss.

In reality different theologies are not equally probable, and there will be one which is most probable. Theologies such as the “God who rewards atheism”, which do not have any actual proponents, have very little evidence for them, since they do not even have the evidence resulting from a claim. One cannot expect that two differing positions will randomly have exactly the same amount of evidence for them, so one theology will have more evidence than any other. And even if it did not have overall a probability of more than 50%, it could still be a good bet, given the possibility of the reward, and better than any of the other potential wagers.

The argument is also made that once one admits an infinite reward, it is not possible to distinguish between actions with differing values. This is described here:

If you regularly brush your teeth, there is some chance you will go to heaven and enjoy infinite bliss. On the other hand, there is some chance you will enjoy infinite heavenly bliss even if you do not brush your teeth. Therefore the expectation of brushing your teeth (infinity plus a little extra due to oral health = infinity) is the same as that of not brushing your teeth (infinity minus a bit due to cavities and gingivitis = infinity), from which it follows that dental hygiene is not a particularly prudent course of action. In fact, as soon as we allow infinite utilities, decision theory tells us that any course of action is as good as any other (Duff 1986). Hence we have a reductio ad absurdum against decision theory, at least when it’s extended to infinite cases.

As actually applied, someone might argue that even if the God who rewards atheism is less probable than the Christian God, the expected utility of being Christian or atheist will be infinite in each case, and therefore one will not be a more reasonable choice than another. Some people actually seem to believe that this is a good response, but it is not. The problem here is that decision theory is a mathematical formalism and does not have to correspond precisely with real life. The mathematics does not work when infinity is introduced, but this does not mean there cannot be such an infinity in reality, nor that the two choices would be equal in reality. It simply means you have not chosen the right mathematics to express the situation. To see this clearly, consider the following situation.

You are in a room with two exits, a green door and a red door. The green door has a known probability of 99% of leading to an eternal heaven, and a known probability of 1% of leading to an eternal hell. The red door has a known probability of 99% of leading to an eternal hell, and a known probability of 1% of leading to an eternal heaven.

The point is that if your mathematics says that going out the red door is just as good as going out the green door, your mathematics is wrong. The correct solution is to go out the green door.

I would consider all such arguments, namely arguing that all religious beliefs are equally probable, or that being rewarded for atheism is as probable as being rewarded for Christianity, or that all infinite expectations are equal, are examples of not very serious thinking. These arguments are not only wrong. They are obviously wrong, and obviously motivated by the desire not to believe. Earlier I quoted Thomas Nagel on the fear of religion. After the quoted passage, he continues:

My guess is that this cosmic authority problem is not a rare condition and that it is responsible for much of the scientism and reductionism of our time. One of the tendencies it supports is the ludicrous overuse of evolutionary biology to explain everything about life, including everything about the human mind. Darwin enabled modern secular culture to heave a great collective sigh of relief, by apparently providing a way to eliminate purpose, meaning, and design as fundamental features of the world. Instead they become epiphenomena, generated incidentally by a process that can be entirely explained by the operation of the nonteleological laws of physics on the material of which we and our environments are all composed. There might still be thought to be a religious threat in the existence of the laws of physics themselves, and indeed the existence of anything at all— but it seems to be less alarming to most atheists.

This is a somewhat ridiculous situation.

This fear of religion is very likely the cause of such unreasonable responses. Scott Alexander notes in this comment that such explanations are mistaken:

I find all of the standard tricks used against Pascal’s Wager intellectually unsatisfying because none of them are at the root of my failure to accept it. Yes, it might be a good point that there could be an “atheist God” who punishes anyone who accepts Pascal’s Wager. But even if a super-intelligent source whom I trusted absolutely informed me that there was definitely either the Catholic God or no god at all, I feel like I would still feel like Pascal’s Wager was a bad deal. So it would be dishonest of me to say that the possibility of an atheist god “solves” Pascal’s Wager.

The same thing is true for a lot of the other solutions proposed. Even if this super-intelligent source assured me that yes, if there is a God He will let people into Heaven even if their faith is only based on Pascal’s Wager, that if there is a God He will not punish you for your cynical attraction to incentives, and so on, and re-emphasized that it was DEFINITELY either the Catholic God or nothing, I still wouldn’t happily become a Catholic.

Whatever the solution, I think it’s probably the same for Pascal’s Wager, Pascal’s Mugging, and the Egyptian mummy problem I mentioned last month. Right now, my best guess for that solution is that there are two different answers to two different questions:

Why do we believe Pascal’s Wager is wrong? Scope insensitivity. Eternity in Hell doesn’t sound that much worse, to our brains, than a hundred years in Hell, and we quite rightly wouldn’t accept Pascal’s Wager to avoid a hundred years in Hell. Pascal’s Mugger killing 3^^^3 people doesn’t sound too much worse than him killing 3,333 people, and we quite rightly wouldn’t give him a dollar to get that low a probability of killing 3,333 people.

Why is Pascal’s Wager wrong? From an expected utility point of view, it’s not. In any particular world, not accepting Pascal’s Wager has a 99.999…% chance of leading to a higher payoff. But averaged over very large numbers of possible worlds, accepting Pascal’s Wager or Pascal’s Mugging will have a higher payoff, because of that infinity going into the averages. It’s too bad that doing the rational thing leads to a lower payoff in most cases, but as everyone who’s bought fire insurance and not had their house catch on fire knows, sometimes that happens.

I realize that this position commits me, so far as I am rational, to becoming a theist. But my position that other people are exactly equal in moral value to myself commits me, so far as I am rational, to giving almost all my salary to starving Africans who would get a higher marginal value from it than I do, and I don’t do that either.

While a far more reasonable response, there is wishful thinking going here as well, with the assumption that the probability that a body of religious beliefs is true as a whole is extremely small. This will not generally speaking be the case, or at any rate it will not be as small as he suggests, once the evidence derived from the claim itself is taken into account, just as it is not extremely improbable that a particular book is mostly historical, even though if one considered the statements contained in the book as a random conjunction, one would suppose it to be very improbable.

Pascal’s Wager

Blaise Pascal, in his Pensees, proposes his wager:

Let us now speak according to natural lights.

If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is. This being so, who will dare to undertake the decision of the question? Not we, who have no affinity to Him.

Who then will blame Christians for not being able to give a reason for their belief, since they profess a religion for which they cannot give a reason? They declare, in expounding it to the world, that it is a foolishness, stultitiam; and then you complain that they do not prove it! If they proved it, they would not keep their word; it is in lacking proofs that they are not lacking in sense. “Yes, but although this excuses those who offer it as such and takes away from them the blame of putting it forward without reason, it does not excuse those who receive it.” Let us then examine this point, and say, “God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not, then, reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. “That is very fine. Yes, I must wager; but I may perhaps wager too much.” Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; where-ever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.

For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainly of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.

“I confess it, I admit it. But, still, is there no means of seeing the faces of the cards?” Yes, Scripture and the rest, etc. “Yes, but I have my hands tied and my mouth closed; I am forced to wager, and am not free. I am not released, and am so made that I cannot believe. What, then, would you have me do?”

True. But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavour, then, to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith and do not know the way; you would like to cure yourself of unbelief and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness. “But this is what I am afraid of.” And why? What have you to lose?

But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.

The end of this discourse.–Now, what harm will befall you in taking this side? You will be faithful, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognise that you have wagered for something certain and infinite, for which you have given nothing.

Pascal is not arguing that here God exists, or that Christian doctrines are true. He is making a moral argument that it is better to believe these things, whether they are true or not. But in this process he states or suggests a number of things which are not in fact the case.

First, he suggests that the probability that God exists is 50%. It is unlikely that this is a reasonable subjective probability for someone to assign, however he understands the idea of God. If God is understood as the first cause, a reasonable probability could be much higher than this. If God is understood in a significantly more concrete sense, a reasonable probability might be lower than this. But it would be unlikely for the reasonable probability to be exactly 50%, however God is understood.

He also implies that this probability of 50% is also the probability that the body of Christian and Catholic doctrine taken as a whole is true. This is basically inconsistent with the previous point, because if the existence of God has a certain probability, the probability that “God exists and these other statements are true as well,” will necessarily be lower than the probability of God.

He is also implying an extremely strong version of the doctrine of extra Ecclesiam nulla salussince he presumes that the infinite gain will necessarily be lost if you do not believe. The most reasonable reading of Catholic teaching is not consistent with this interpretation, which is a flaw in his argument, given that he is arguing specifically for people to become Catholics.

Despite these errors, however, there is a moral argument here which should be taken seriously, something that is rarely done.