Pascal’s wager · Pascal’s wager. 1. There have been miracles. 2. If there have been miracles,...

Pascal’s wager

1. There have been miracles.2. If there have been miracles, God exists._____________________________________________

C. God exists.

The argument from miracles Hume’s principle about testimony.

We should not believe that M happened on the basis of the testimony unless the following is the case:

The probability of the testimony being false < the probability of M occurring.

Hume’s idea seems to be this. When we are trying to figure out the probability of some event happening in certain circumstances, we ask: in the past, how frequently as that event been observed to occur in those circumstances? Our answer to this question will give us the probability of the relevant event.

This, Hume thinks, is enough to show us that we ought never to believe testimony regarding miraculous events:

One conclusion: testimony is one, but not the only, source of evidence which we should usewhen forming a belief. Testimony is relevant because it has a (relatively) high probabilityof being true. But, like any evidence, this can be overridden by other sources of evidence(like, for example, contrary testimony) which have give a high probability to the negationof the proposition in question.

2.2 Testimony about miracles

We now need to apply these general points about testimony and evidence to the case ofmiracles. One conclusion seems to follow immediately:

“That no testimony is su!cient to establish a miracle, unless the testimonybe of such a kind, that its falsehood would be more miraculous, than the fact,which it endeavors to establish . . . ” (77)

The problem for the believer in miracles is that miracles, being departures from the lawsof nature, seem to be exactly the sorts of events which we should not expect to happen.As Hume puts it:

“A miracle is a violation of the laws of nature; and as a firm and unalterableexperience has established these laws, the proof against a miracle, from thevery nature of the fact, is as entire as any argument from experience canpossibly be imagined . . . There must be a uniform experience against everymiraculous event, otherwise the event would not merit that appellation.” (76-7)

The implied question is: could testimony ever provide strong enough evidence to overrideour massive evidence in favor of nature’s following its usual course (which is also evidenceagainst the occurrence of the miracle)?

2.3 The relevance of religious diversity

In §II, Hume adds another reason to be skeptical about testimony about miracles, whenhe writes

“there is no testimony for any [miracles] . . . that is not opposed by an infi-nite number of witnesses; so that not only the miracle destroys the credit oftestimony, but the testimony destroys itself. To make this better understood,let us consider, that, in matters of religion, whatever is di"erent is contrary. . . Every miracle, therefore, pretended to have been wrought in any of thesereligions (and all of them abound in miracles) . . . has the same force . . . tooverthrow every other system.” (81)

Is this best construed as a separate argument against miracles, or as part of the argumentsketched above? If the latter, how does it fit in?

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Hume’s point is that miracles are always departures from the ordinary laws of nature. But the ordinary laws of nature are regularities which have been observed to hold 100% of the time. Of course, we have not observed testimony to be correct 100% of the time. Hence, the probability of testimony regarding a miracle being false will always be greater than the probability of the miraculous event; and then it follows from Hume’s principle about testimony that we should never accept the testimony.

Last time we discussed Hume’s principle about testimony: one should never believe testimony about some event unless the probability of that testimony being false is less than the probability of the event occurring. This raised the question: how do we determine the relevant probabilities?


C. God exists.






Hence Hume’s conclusion:

Pascal situates the question of miracles within (one part of) the Christian tradition. Butthe question we want to answer is more general: can miracles play this kind of centralrole in justifying religious belief of any sort?

We will focus on the question of whether miracles can justify the religious beliefs of peoplewho have not themselves witnessed miracles.

2 Hume’s argument against belief in miracles

Hume thinks that they cannot, and indeed that no rational person would base belief inGod on testimony that miracles have occurred. He says:

“. . . therefore we may establish it as a maxim, that no human testimony canhave such force as to prove a miracle, and make it a just foundation for anysystem of religion.” (88)

This is Hume’s conclusion. We now need to understand his argument for it, which beginswith some premises about the role of perceptual evidence and testimony in the formingof beliefs.

2.1 Testimony and evidence

Hume’s first claim is that we should base belief on the available evidence:

“A wise man, therefore, proportions his belief to the evidence. . . . He weighsthe opposite experiments: He considers which side is supported by the greaternumber of experiments: To that side he inclines, with doubt and hesitation;and when at last he fixes his judgement, the evidence exceeds not what weproperly call probability.” (73-4)

The general moral seems to be correct: when deciding whether to believe or disbelievesome proposition, we should weigh the evidence for and against it to see whether it makesthe proposition or its negation more probable.

How does this sort of general principle fit with our practice of basing beliefs on testimony?Hume has a very plausible answer:

“we may observe, that there is no species of reasoning more common, moreuseful, and even necessary to human life, than that which is derived from thetestimony of men, and the reports of eye-witnesses and spectators. . . . I shallnot dispute about a word. It will be su!cient to observe, that our assurancein any argument of this kind is derived from no other principle than ourobservation of the veracity of human testimony, and of the usual conformityof facts to the reports of witnesses.” (74)

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C. God exists.






On this reading, Hume’s argument rests on some principle of the following sort:

If some event has never been observed to occur before, then the probability of it occurring is 0%.

This, plus Hume’s principle about testimony, is clearly enough to show that one ought never to believe in miracles on the basis of testimony.

Let’s call this the zero probability principle.


C. God exists.





This, plus Hume’s principle about testimony, is clearly enough to show that one ought never to believe in miracles on the basis of testimony.

Interestingly, it also seems to be enough to establish a stronger claim: one is never justified in believing in the existence of miracles, even if one is (or takes oneself to be) an eyewitness.

After all, perceptual experiences of the world, like testimony, don’t conform to the facts 100% of the time. So, the probability of a miraculous event M occurring will always, given the above principle about probabilities, be less than the probability of one’s perceptual experience being illusory. Hence, it seems, one would never be justified in believing in the existence of a miracle, even on the basis of direct perceptual experience.

This might at first seem like a good thing for Hume’s argument: it shows not just that one an never believe in miracles on the basis of testimony, but also that one can never believe in them for any reason at all! But in fact this points to a problem for the zero probability principle.

The zero probability principle:


C. God exists.






Consider the following sort of example:

You are a citizen of Pompeii in AD 79, and there is no written record of the tops of mountains erupting and spewing forth lava. Accordingly, following the zero probability principle, you regard the chances of such a thing happening as 0%. On the other hand, you know that your visual experiences have been mistaken in the past, so you regard the chances of an arbitrary visual experience being illusory as about 1%. Then you have a very surprising visual experience: black clouds and ash shooting out of nearby Mt. Vesuvius. What is it rational for you to believe?

This sort of case seems to show that the zero probability principle is false. Other such examples involve falsification of well-confirmed scientific theories.

So, if Hume’s argument depends on the zero probability principle, it is a failure. Can we come up with another principle, which would avoid these sort of counterexamples while still delivering the result that Hume wants?


C. God exists.






So, if Hume’s argument depends on the zero probability principle, it is a failure. Can we come up with another principle, which would avoid these sort of counterexamples while still delivering the result that Hume wants?

It seems that we can. All Hume’s argument needs, it would seem, is the following trio of assumptions:

(a) If some event has never been observed to occur before, then the probability of it occurring is at most X%.

(b) The probability of a piece of testimony being false is always at least Y%.

(c) Y>X

Suppose, for example, that the probability of an event of some type which has never before been observed is at most 1%, and that there is always at least a 10% chance of some testimony being false. If we assume Hume’s principle about testimony, would this be enough to deliver the conclusion that we are never justified in believing in miracles on the basis of testimony?


C. God exists.




(a) If some event has never been observed to occur before, then the probability of it occurring is at most X%.(b) The probability of a piece of testimony being false is always at least Y%.(c) Y>X

Suppose, for example, that the probability of an event of some type which has never before been observed is at most 1%, and that there is always at least a 10% chance of some testimony being false. If we assume Hume’s principle about testimony, would this be enough to deliver the conclusion that we are never justified in believing in miracles on the basis of testimony?

Only if we assume that we only ever have testimony from a single witness. Suppose that we have three witnesses, each of whom are 90% reliable, and each independently reports that M has occurred. Then the probability of each witness being wrong is 10%, but the probability of all three being wrong is only 0.1%. This, by the above measure, would be enough to make it rational to believe that M happened.

So it seems that the possibility of multiple witnesses shows that (a)-(c) are not enough to make Hume’s argument against justified belief in miracles on the basis of testimony work. (This is true no matter what values we give to “X” and “Y”.)

(One cautionary note: it is important to distinguish between having testimony from multiple witnesses and having testimony from a single witness who claims there to have been multiple witnesses.)


C. God exists.

The argument from miracles

This principle can sound sort of obvious; but it isn’t, as some examples show. First, what do you think that the probability of the truth of testimony from the writers of the South Bend Tribune would be?

Let’s suppose that you think that it is quite a reliable paper, and that its testimony is true 99.9% of the time, so that the probability of its testimony being false is 0.1%.

Now suppose that you read the following in the South Bend Tribune:

“The winning numbers for Powerball this weekend were 1-14-26-33-41-37-4.”

What are the odds of those being the winning numbers for Powerball? Well, the same as the odds of any given combination being correct, which is 1 in 195,249,054. So the probability of the reported event occurring is 0.0000005121663739%.

So, if Hume’s principle about testimony is correct, one is never justified in believing the lottery results reported in the paper, or on the local news, etc. But this seems wrong: one can gain justified beliefs about the lottery from your local paper, even if it is the South Bend Tribune.

Hume’s principle about testimony.



So far, we have been assuming that Hume’s principle about testimony is true, and asking what assumption could be added to this principle to make Hume’s argument work. But there is reason to doubt whether the principle about testimony is itself true.

One might wonder how, if at all, Hume’s principle could be modified to avoid these counterexamples.


C. God exists.

The argument from miracles

This is a long way from showing that the argument from miracles is a success: for that purpose, we would have to consider specific examples of miracles, and the sorts of evidence given for their occurrence.

We would also, as the example of Powerball shows, have to get a bit clearer about when testimony is and is not sufficient for justified belief.

We would also have to answer the question of when we are justified in believing that some event which is contrary to the usual natural order has supernatural causes.

What our discussion today shows is something much more limited: that one prominent attempt to show that the argument from miracles can’t succeed is, as it stands, unconvincing.

So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not “Does God exist?” but “Should we believe in God?” What is distinctive about Pascal’s approach to the latter question is that he thinks that we can answer it without first answering the former question.

Here is what he has to say about the question, “Does God exist?”:

“Let us then examine this point, and let us say: ‘Either God is or he is not.’ But to which view shall we be inclined? Reason cannot decide this question. Infinite chaos separates us. At the far end of this infinite distance a coin is being spun which will come down heads or tails. How will you wager? Reason cannot make you choose either, reason cannot prove either wrong.”

Pascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.

“Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which offers you the least interest. 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. . . . Since you must necessarily choose, your reason is no more affronted by choosing one rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

This is quite different than the sorts of arguments we have discussed so far for belief in God. Each of those arguments made a case for belief in God on the basis of a case for the truth of that belief; Pascal focuses on the happiness that forming the belief might bring about.

But, he notes, this does not remove the necessity of our choosing to believe, or not believe, in God:


This is quite different than the sorts of arguments we have discussed so far for belief in God. Each of those arguments made a case for belief in God on the basis of a case for the truth of that belief; Pascal focuses on the happiness that forming the belief might bring about.

But what is Pascal’s argument that belief in God will lead to greater happiness? It seems to be contained in the last two sentences of this passage. Pascal is saying that if you believe in God and God exists (“you win”), you win eternal life (“you win everything”), whereas if God does not exist, it doesn’t matter whether you believe in God (“you lose nothing”).

Pascal was one of the first thinkers to systematically investigate the question of how it is rational to act under certain kinds of uncertainty, a topic now known as “decision theory.” We can use some concepts from decision theory to get a bit more precise about how Pascal’s argument here is supposed to work.


Pascal was one of the first thinkers to systematically investigate the question of how it is rational to act under certain kinds of uncertainty, a topic now known as “decision theory.” We can use some concepts from decision theory to get a bit more precise about how Pascal’s argument here is supposed to work.

We are facing a decision in which we have only two options: belief or nonbelief. And there is one unknown factor which will determine the outcome of our choice: whether or not God exists. So, pairing each possible choice with each possible outcome, there are four possibilities. Pascal is here drawing attention to an interesting feature of the way these choices and outcomes line up in the case of belief in God.

This can be illustrated by an analogy with a simple bet.


We are facing a decision in which we have only two options: belief or nonbelief. And there is one unknown factor which will determine the outcome of our choice: whether or not God exists. So, pairing each possible choice with each possible outcome, there are four possibilities. Pascal is here drawing attention to an interesting feature of the way these choices and outcomes line up in the case of belief in God.

This can be illustrated by an analogy with a simple bet.

Suppose I offer you the chance of choosing heads or tails on a fair coin flip, with the following payoffs: if you choose heads, and the coin comes up heads, you win $10; if you choose heads, and the coin comes up tails, you win $5. If you choose tails, then if the coin comes up heads, you get $2, and if it comes up tails, you get $5.

We can represent the possibilities open to you with the following table:

Should you choose heads, or tails? It seems that neither option is better; each gives youthe same odds of winning and losing, and the relevant amounts are the same in each case.

Now suppose that you are given a slightly stranger and more complicated bet: if youchoose heads, and the coin comes up heads, you win §10; if you choose heads, and thecoin comes up tails, you win $5. However, if you choose tails, and the coin comes upheads, you win §0; but if you choose tails, and the coin comes up tails, you win $5. Yourchoices can then be represented as follows:

Courses of action Possibility 1:Coin comes up heads

Possibility 2:Coin comes up tails

Choose ‘heads’ Win $10 Win $5Chose ‘tails’ Win $2 Win $5

If given this bet, should you choose heads, or choose tails? Unlike the simpler bet, itseems that here there is an obvious answer to this question: you should choose heads.The reason why is clear. There are only two relevant ways that things could turn out: thecoin could come up heads, or come up tails. If it comes up heads, then you are better o!if you chose ‘heads.’ If it comes up tails, then it doesn’t matter which option you chose.One way of putting this scenario is that the worst outcome of choosing ‘heads’ is as goodas the best outcome of choosing ‘tails’, and the best outcome of choosing ‘heads’ is betterthan the best outcome of choosing ‘tails’. When this is true, we say that choosing ‘heads’superdominates choosing ‘tails.’

It seems clear that if you have just two courses of action, and one superdominates theother, you should choose that one. Superdominance is the first important concept fromdecision theory to keep in mind.

The next important concept is expected utility. The expected utility of a decision is theamount of utility (i.e., reward) that you should expect that decision to yield. Recall themore complicated coin bet above. If you choose ‘heads’, the bet went, there were twopossible outcomes: either the coin comes up heads, and you win $10, or the coin comesup tails, and you win $5. Obviously, this is a pretty good bet, since you win either way.Now suppose that you are o!ered the chance to take this bet on a fair coin toss, but haveto pay $7 to make the bet. Supposing that you want to maximize your money, shouldyou take the bet?

The answer here may not seem obvious — it is certainly not as obvious as the fact thatyou should, if given the choice, choose ‘heads’ rather than ‘tails.’ But this is the kind ofquestion that calculations of expected utility are constructed to answer. To answer thequestion:

Should I pay $7 to have the chance to bet ‘heads’?

we ask

Which is higher: the expected utility of paying $7 and betting ‘heads’, or theexpected utility of not paying, and not betting?

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Obviously, you should choose heads. One way to put the reason for this is as follows: there is one possibility on which you are better off having chosen heads, and no possibility on which you are worse off choosing heads. This is to say that choosing heads dominates choosing tails.

It seems very plausible that if you are choosing between A and B, and choosing A dominates choosing B, it is rational to choose A.


Let’s say that if we are choosing between actions A and B,

A dominates B if and only if there is at least one scenario in which A leads to a better outcome than B, and no scenario in which A leads to a worse outcome than B.

One way to read Pascal here is as saying that believing in God dominates nonbelief. If this were true, this would be a very strong argument that it is rational to believe in God. Here’s how he might be thinking about the choice whether to believe:

3.1 The argument from superdominance

One version of Pascal’s argument is that the decision to believe in God superdominatesthe decision not to believe in God, in the above sense. He seems to have this in mindwhen he writes,

“. . . if you win, you win everything, if you lose you lose nothing.”

This indicates that, at least at this point in the next, he sees the decision to believe ornot believe in God as follows:

Courses of action Possibility 1:God exists

Possibility 2:God does not exist

Believe in God Infinite reward Lose nothing, gain nothingDo not believe in God Infinite loss Lose nothing, gain nothing

How should you respond to the choice of either believing, or not believing, in God? Itseems easy: just as in the above case you should choose ‘heads’, so in this case you shouldchoose belief in God. After all, belief in God superdominates non-belief: the worst casescenario of believing in God is as good as the best case scenario of non-belief, and thebest case scenario of believing in God is better than the best case scenario of non-belief.

Pascal, however, seems to recognize that there is an objection to this way of representingthe choice of whether or not one should believe in God: one might think that if one decidesto believe in God, and it turns out that God does not exist, there has been some loss:you are then worse o! than if you had not believed all along. Why might this be?

If this is right, then it looks like the following is a better representation of our choice:



Believe in God Infinite reward LossDo not believe in God Infinite loss Gain

If this is a better representation of the choice, then it is not true that believing in Godsuperdominates non-belief.

3.2 The argument from expected utility

How, then, should we decide what to do? One method was already suggested earlier: youshould see which of the two courses of action has the higher expected utility.

But, to figure this out, we have to know what probabilities we should assign to thepossibilities that God exists, and that God does not exist. Pascal suggests when settingup the argument that there is “an equal chance of gain and loss”, which would put theprobabilities of each at 1/2.

We also need to figure out how to measure the utility of each of the two outcomes, giveneither of the two choices. Below is one way to do that:

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If this is the right way to think about the choice between belief and non-belief, then believing seems to dominate not believing.

Is this the right way to think about the choice?

3.1 The argument from superdominance

One version of Pascal’s argument is that the decision to believe in God superdominatesthe decision not to believe in God, in the above sense. He seems to have this in mindwhen he writes,

“. . . if you win, you win everything, if you lose you lose nothing.”

This indicates that, at least at this point in the next, he sees the decision to believe ornot believe in God as follows:



Believe in God Infinite reward Lose nothing, gain nothingDo not believe in God Infinite loss Lose nothing, gain nothing

How should you respond to the choice of either believing, or not believing, in God? Itseems easy: just as in the above case you should choose ‘heads’, so in this case you shouldchoose belief in God. After all, belief in God superdominates non-belief: the worst casescenario of believing in God is as good as the best case scenario of non-belief, and thebest case scenario of believing in God is better than the best case scenario of non-belief.

Pascal, however, seems to recognize that there is an objection to this way of representingthe choice of whether or not one should believe in God: one might think that if one decidesto believe in God, and it turns out that God does not exist, there has been some loss:you are then worse o! than if you had not believed all along. Why might this be?

If this is right, then it looks like the following is a better representation of our choice:



Believe in God Infinite reward LossDo not believe in God Infinite loss Gain

If this is a better representation of the choice, then it is not true that believing in Godsuperdominates non-belief.

3.2 The argument from expected utility

How, then, should we decide what to do? One method was already suggested earlier: youshould see which of the two courses of action has the higher expected utility.

But, to figure this out, we have to know what probabilities we should assign to thepossibilities that God exists, and that God does not exist. Pascal suggests when settingup the argument that there is “an equal chance of gain and loss”, which would put theprobabilities of each at 1/2.

We also need to figure out how to measure the utility of each of the two outcomes, giveneither of the two choices. Below is one way to do that:

5

If this is the right way to think about the choice between belief and non-belief, then believing seems to dominate not believing.

Is this the right way to think about the choice?

However, it is not obvious that belief does dominate nonbelief, since it is not obvious that one really loses nothing if one believes in God in the scenario in which God does not exist. Wouldn’t one be undertaking religious obligations which one might have avoided? And isn’t having a false belief something bad in itself?

If we think about a scenario in which one believes in God but God does not exist as involving some loss -- either because one would not do things which one might like to do, or because having a false belief is in itself a loss -- then believing does not dominate not believing.

There is, however, another way to think about Pascal’s argument, which does not involve dominance reasoning. This uses another concept from decision theory, namely expected utility.

There is, however, another way to think about Pascal’s argument, which does not involve dominance reasoning. This uses another concept from decision theory, namely expected utility.

Recall the example of the coin toss above, in which, if you bet heads and the coin comes up heads, you win $10, and if it comes up tails, you win $5. Suppose I offer you the following deal: I will give you those payoffs on a fair coin flip in exchange for you paying me $7 for the right to play. Should you take the bet?

Here is one way to argue that you should take the bet. There is a 1/2 probability that the coin will come up heads, and a 1/2 probability that it will come up tails. In the first case I win $10, and in the second case I win $5. So, in the long run, I’ll win $10 about half the time, and $5 about half the time. So, in the long run, I should expect the amount that I win per coin flip to be the average of these two amounts -- $7.50. So the expected utility of my betting heads is $7.50. So it is rational for me to pay any amount less than the expected utility to play (supposing for simplicity, of course, that my only interest is in maximizing my money, and I have no other way of doing so).

To calculate the expected utility of an action, we assign each outcome of the action a certain probability, thought of as a number between 0 and 1, and a certain value (in the above case, the relevant value is just the money won). In the case of each possible outcome, we then multiply its probability by its value; the expected utility of the action will then be the sum of these results.

In the above case, we had (1/2 * 10) + (1/2 * 5) = 7.5.

The notion of expected utility seems to lead to a simple rule for deciding what to do:

The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.



Our question is then: how might Pascal argue that believing in God has higher expected utility than nonbelief? Our answer seems to be given in the following passage:

Let us assess the two cases: if you win you win everything, if you lose you losenothing. Do not hesitate then; wager that he does exist.”

A di!erence in kind between this argument and the arguments for the existence of Godwe have considered. Pascal does not provide us any evidence for thinking that God exists.He gives us prudential rather than theoretical reasons for forming a belief that God exists.The distinction between these two kinds of reasons.

Pascal goes on to spell out more explicitly his reasoning for thinking that it is rational tobelieve in God, using an analogy with gambling:

“. . . since there is an equal chance of gain and loss, if you stood to win onlytwo lives for one you could still wager, but supposing you stood to win three?. . . it would be unwise of you, since you are obliged to play, not to risk yourlife in order to win three lives at a game in which there is an equal chance ofwinning and losing. . . . But here there is an infinity of happy life to be won,one chance of winning against a finite number of chances of losing, and whatyou are staking is finite. That leaves no choice; wherever there is infinity, andwhere there are not infinite chances of losing against that of winning, thereis no room for hesitation, you must give everything. And thus, since you areobliged to play, you must be renouncing reason if you hoard your life ratherthan risk it for an infinite gain, just as likely to occur as a loss amounting tonothing.”

Clearly Pascal thinks that there is some analogy between believing in God and making aneven-odds bet in which you stand to win three times as much as you stand to lose; to bemore precise about what this analogy is supposed to be, we can introduce some conceptsfrom decision theory, the study of the principles which govern rational decision-making.

2 The wager and decision theory

Pascal was one of the first thinkers to systematically investigate what we now call ‘decisiontheory’, and elements of his thought on this topic clearly guide his presentation of thewager.

Suppose that we have two courses of action between which we must choose, and the con-sequences of each choice depend on some unknown fact. E.g., it might be the case thatwe have to bet on whether a coin comes up heads or tails, and what the result of ourbet is depends on whether the coin actually does come up heads or tails. Imagine first asimple bet in which if you guess correctly, you win $1, and if you guess incorrectly, youlose $1. We could represent the choice like this:



Choose ‘heads’ Win $1 Lose $1Chose ‘tails’ Lose $1 Win $1

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What is Pascal’s argument here? How should we perform the relevant expected utility calculations?



Let us assess the two cases: if you win you win everything, if you lose you losenothing. Do not hesitate then; wager that he does exist.”

A di!erence in kind between this argument and the arguments for the existence of Godwe have considered. Pascal does not provide us any evidence for thinking that God exists.He gives us prudential rather than theoretical reasons for forming a belief that God exists.The distinction between these two kinds of reasons.

Pascal goes on to spell out more explicitly his reasoning for thinking that it is rational tobelieve in God, using an analogy with gambling:

“. . . since there is an equal chance of gain and loss, if you stood to win onlytwo lives for one you could still wager, but supposing you stood to win three?. . . it would be unwise of you, since you are obliged to play, not to risk yourlife in order to win three lives at a game in which there is an equal chance ofwinning and losing. . . . But here there is an infinity of happy life to be won,one chance of winning against a finite number of chances of losing, and whatyou are staking is finite. That leaves no choice; wherever there is infinity, andwhere there are not infinite chances of losing against that of winning, thereis no room for hesitation, you must give everything. And thus, since you areobliged to play, you must be renouncing reason if you hoard your life ratherthan risk it for an infinite gain, just as likely to occur as a loss amounting tonothing.”

Clearly Pascal thinks that there is some analogy between believing in God and making aneven-odds bet in which you stand to win three times as much as you stand to lose; to bemore precise about what this analogy is supposed to be, we can introduce some conceptsfrom decision theory, the study of the principles which govern rational decision-making.

2 The wager and decision theory

Pascal was one of the first thinkers to systematically investigate what we now call ‘decisiontheory’, and elements of his thought on this topic clearly guide his presentation of thewager.

Suppose that we have two courses of action between which we must choose, and the con-sequences of each choice depend on some unknown fact. E.g., it might be the case thatwe have to bet on whether a coin comes up heads or tails, and what the result of ourbet is depends on whether the coin actually does come up heads or tails. Imagine first asimple bet in which if you guess correctly, you win $1, and if you guess incorrectly, youlose $1. We could represent the choice like this:



Choose ‘heads’ Win $1 Lose $1Chose ‘tails’ Lose $1 Win $1

2

Pascal says two things which help us here. First, he emphasizes that “there is an equal chance of gain and loss” -- an equal chance that God exists, and that God does not exist. This means that we should assign each a probability of 1/2.

Second, he says that in this case the amount to be won is infinite. We can represent this by saying that the utility of belief in God if God exists is ∞.



Pascal says two things which help us here. First, he emphasizes that “there is an equal chance of gain and loss” -- an equal chance that God exists, and that God does not exist. This means that we should assign each a probability of 1/2.

Second, he says that in this case the amount to be won is infinite. We can represent this by saying that the utility of belief in God if God exists is ∞.

Let’s concede the objection made above: if we believe in God, and God does not exist, this involves some loss of utility. This loss will be finite -- let’s symbolize it by word “LOSS”.

We can then think of the decision as follows:

Courses of actionPossibility 1: God exists (Prob. = 0.5)

Possibility 2: God does not exist (Prob. = 0.5)

Believe in God ∞ LOSS

Don’t believe 0 0

Supposing that this is the correct assignment of values, what is the expected utility of each action?



Supposing that this is the correct assignment of values, what is the expected utility of each action?

Expected utilities

.5* ∞ + .5-LOSS= ∞

.5*0 +.5*0 = 0

So it looks as though the expected utility of believing in God is infinite, whereas the expected utility of nonbelief is 0. If the rule of expected utility is correct, it follows that it is rational to believe in God -- and it is not a very close call.

We will consider two sorts of objections to this expected utility argument for the rationality of belief in God. One sort is based on the values in the above table: one might dispute Pascal’s claims about the values or probabilities of different outcomes. A second sort of objection is based on the rule of expected utility itself.




Don’t believe 0 0



Let’s consider a few objections of the first sort. First, suppose that the probability that God exists is not 1/2, but some much smaller number -- say, 1/100. Would that affect the argument above?

Expected utilities

.5* ∞ + .5-LOSS= ∞

.5*0 +.5*0 = 0




Don’t believe 0 0

It seems not. Let’s suppose that the probability that God exists is some nonzero finite number m, which can be as low as one likes. Then it seems that we should think of the choice between belief and nonbelief as follows:

Expected utilities

m* ∞ + .(1-m)-LOSS= ∞

m*0 + (1-m)*0 = 0

Courses of actionPossibility 1: God exists (Prob. = m)

Possibility 2: God does not exist (Prob. = 1-m)


Don’t believe 0 0



It seems not. Let’s suppose that the probability that God exists is some nonzero finite number m, which can be as low as one likes. Then it seems that we should think of the choice between belief and nonbelief as follows:

Expected utilities

m* ∞ + .(1-m)-LOSS= ∞

m*0 + (1-m)*0 = 0

Courses of actionPossibility 1: God exists (Prob. = m)

Possibility 2: God does not exist (Prob. = 1-m)


Don’t believe 0 0

This is a real strength of Pascal’s argument: it does not depend on any assumptions about the probability that God exists other than the assumption that it is nonzero. In other words, he is only assuming that we don’t know for sure that God does not exist, which seems to many people -- including many atheists -- to be a reasonable assumption.

But another objection might seem more challenging. It seems that Pascal is assuming that, if God exists, there is a 100% chance that believers will get infinite reward. But why assume that? Why not think that if God exists there is some chance that believers will get infinite reward, and some chance that they won’t? How would that affect the above chart?



Expected utilities

m* ∞ + n*0 + (1-(m

+n))*LOSS= ∞

m*0 + n*0 + (1-(m+n))*0 = 0

Courses of action

Possibility 1: Rewarding God exists (Prob. = m)

Possibility 2: No reward God exists (Prob. = n)

Possibility 3: God does not exist (Prob. = 1-(m+n))

Believe in God ∞ 0 LOSS

Don’t believe 0 0 0

But another objection might seem more challenging. It seems that Pascal is assuming that, if God exists, there is a 100% chance that believers will get infinite reward. But why assume that? Why not think that if God exists there is some chance that believers will get infinite reward, and some chance that they won’t? How would that affect the above chart?

To accommodate this possibility, we would have to add another column to our chart, to represent the two possibilities imagined. Let’s call these possibilities “Rewarding God” and “No reward God”, and let’s suppose that each has a nonzero probability of being true -- respectively, m and n. The resulting chart looks like this:

As this chart makes clear, adding this complication has no effect on the result. Pascal needn’t assume that God will certainly reward all believers; he need only assume that there is a nonzero chance that God will reward all believers.



Expected utilities

m* ∞ + n* ∞ + (1-(m

+n))*LOSS= ∞

m*0 + n* ∞ + (1-(m+n))*0

= ∞

Courses of action


Possibility 2: Generous God exists (Prob. = n)


Believe in God ∞ ∞ LOSS

Don’t believe 0 ∞ 0

As this chart makes clear, adding this complication has no effect on the result. Pascal needn’t assume that God will certainly reward all believers; he need only assume that there is a nonzero chance that God will reward all believers.

However, one might think that there is yet another relevant possibility that we are overlooking. After all, isn’t there some chance that God might give eternal reward to believers and nonbelievers alike? This is surely possible. Let’s call this the possibility of “Generous God.” Setting aside the possibility of No reward God, which we have seen to be irrelevant, taking account of the possibility of Generous God has a striking effect on the expected utilities of belief and nonbelief:

Now, it appears, belief and nonbelief have the same infinite expected utility, which undercuts Pascal’s argument for the rationality of belief in God.



Expected utilities

m* ∞ + n* ∞ + (1-(m

+n))*LOSS= ∞

m*0 + n* ∞ + (1-(m+n))*0

= ∞

Courses of action


Possibility 2: Generous God exists (Prob. = n)


Believe in God ∞ ∞ LOSS

Don’t believe 0 ∞ 0

Now, it appears, belief and nonbelief have the same infinite expected utility, which undercuts Pascal’s argument for the rationality of belief in God.

However, Pascal seems to have a reasonable reply to this objection. It seems that the objection turns on the fact that any probability times an infinite utility will yield an infinite expected value. And that means that any two actions which have some chance of bring about an infinite reward will have the same expected utility.

But this is extremely counterintuitive. Suppose we think of a pair of lotteries, EASY and HARD. Each lottery has an infinite payoff, but EASY has a 1/3 chance of winning, whereas HARD has a 1/1,000,000 chance of winning. What is the expected utility of EASY vs. HARD? Which would you be more rational to buy a ticket for?

How might we modify our rule of expected utility to explain this case? Would this help Pascal respond to the case of Generous God?



Even if we can solve the case of Generous God in this way, it raises a troubling sort of objection to Pascal’s argument. That argument turns essentially on the possibility of decisions with infinite expected utility. But it is plausible that, in at least some cases which involve infinite expected utility, the rule of expected utility gives us incorrect results. Consider the following bet:

Would you pay $2 to take this bet? How about $4?

The St. Petersburg

I am going to flip a fair coin until it comes up heads. If the first time it comes up heads is on the 1st toss, I will give you $2. If the first time it comes up heads is on the second toss, I will give you $4. If the first time it comes up heads is on the 3rd toss, I will give you $8. And in general, if the first time the coin comes up heads is on the nth toss, I will give you $2n.

Suppose now I raise the price to $10,000. Should you be willing to pay that amount to play the game?

What is the expected utility of playing the game?



The St. Petersburg


What is the expected utility of playing the game?

We can think about this using the following table:

OutcomeFirst heads

is on toss #1First heads




is on toss #5.....

Probability $2 $4 $8 $16 $32 .....

Payoff 1/2 1/4 1/8 1/16 1/32 .....

The expected utility of playing = the sum of probability * payoff for each of the infinitely many possible outcomes. So, the expected utility of playing equals the sum of the infinite series

1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+......



The St. Petersburg


OutcomeFirst heads





is on toss #5.....

Probability $2 $4 $8 $16 $32 .....

Payoff 1/2 1/4 1/8 1/16 1/32 .....

The expected utility of playing = the sum of probability * payoff for each of the infinitely many possible outcomes. So, the expected utility of playing equals the sum of the infinite series

1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+......

But it follows from this result, plus the rule of expected utility, that you would be rational to pay any finite amount of money to have the chance to play this game once. But this seems clearly mistaken. What is going on here?

Does this show that the rule of expected utility can lead us astray? If so, in what sorts of cases does this happen? Does this result depend essentially on their being infinitely many possible outcomes?

Pascal seems to consider this reply to his argument when he imagines someone replyingas follows:

“. . . is there really no way of seeing what the cards are? . . . I am being forcedto wager and I am not free; I am begin held fast and I am so made that Icannot believe. What do you want me to do then?”

Pascal’s reply:

“That is true, but at least get it into your head that, if you are unable tobelieve, it is because of your passions, since reason impels you to believeand yet you cannot do so. Concentrate then not on convincing yourself bymultiplying proofs of God’s existence, but by diminishing your passions. . . . ”

4.2 Rationality does not require maximizing expected utility

The St. Petersburg paradox; the counterintuitive consequences which result from (i) therequirement that we should act so as to maximize expected utility, and (ii) the possibilityof infinite expected utilities.

Why the result that we should sometimes fail to maximize expected utility is puzzling.

4.3 We should assign 0 probability to God’s existence

How this blocks the argument.

The case against assignment of 0 probability to the possibility that God exists.

4.4 The ‘many gods’ objection

(For more detail, and a list of relevant further readings, see the excellent entry “Pascal’sWager” in the Stanford Encyclopedia of Philosophy by Alan Hajek, from which much ofthe above is drawn.)

7

Pascal seems to consider this reply to his argument when he imagines someone replyingas follows:

“. . . is there really no way of seeing what the cards are? . . . I am being forcedto wager and I am not free; I am begin held fast and I am so made that Icannot believe. What do you want me to do then?”

Pascal’s reply:

“That is true, but at least get it into your head that, if you are unable tobelieve, it is because of your passions, since reason impels you to believeand yet you cannot do so. Concentrate then not on convincing yourself bymultiplying proofs of God’s existence, but by diminishing your passions. . . . ”

4.2 Rationality does not require maximizing expected utility

The St. Petersburg paradox; the counterintuitive consequences which result from (i) therequirement that we should act so as to maximize expected utility, and (ii) the possibilityof infinite expected utilities.

Why the result that we should sometimes fail to maximize expected utility is puzzling.

4.3 We should assign 0 probability to God’s existence

How this blocks the argument.

The case against assignment of 0 probability to the possibility that God exists.

4.4 The ‘many gods’ objection

(For more detail, and a list of relevant further readings, see the excellent entry “Pascal’sWager” in the Stanford Encyclopedia of Philosophy by Alan Hajek, from which much ofthe above is drawn.)

7

If I offer you $5 to raise your arm, you can do it. But, to illustrate objection (1), suppose I offered you $5 to believe that you are not now sitting down. Can you do that (without standing up)?

Let’s now turn to a pair of quite different objections, which focus on the fact that Pascal is asking us to make decisions about what to believe on the basis of expected utility calculations. The objections are that this is (1) impossible and (2) the wrong way to form beliefs.

Pascal considered this objection, and gave the following response:

What does he have in mind here?

Let’s now turn to a pair of quite different objections, which focus on the fact that Pascal is asking us to make decisions about what to believe on the basis of expected utility calculations. The objections are that this is (1) impossible and (2) the wrong way to form beliefs.

To see how objection (2) might be developed, consider the following claim about how we ought to form beliefs:

The rule of responsible belief formation

It is irrational to form a belief in a claim unless you have more reason to think that it true than that it is false.

It is plausible that, in at least many cases, we hold each other to this sort of rule. If your friend often forms beliefs despite having no reason for thinking that the belief is more likely to be true than false, you would likely take him to be irrational -- as well as untrustworthy.

But now recall the rule used in Pascal’s wager:



This also seemed plausible; and it led, without the assumption that we have any more reason to think that God exists than not, to the conclusion that it is rational to believe in God. This shows that, if we think of the rule of expected utility as applying to beliefs, these two rules sometimes come into conflict. This means that at least one must be, as it stands, false.

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