• Arguing about Risk


    According to Wikipedia, this is how Blaise Pascal originally worded his famous wager:

    Pascal’s Wager
    1. God is, or He is not.
    2. A Game is being played… where heads or tails will turn up.
    3. According to reason, you can defend neither of the propositions.
    4. You must wager. (It’s not optional.)
    5. 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.
    6. Wager, then, without hesitation that He is. (…) 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. 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.

    I have never heard a Christian articulate it in quite this way, and most Christians would say that simply “believing in God” is not what it takes to be saved (cf James 2:19).  Most Christians I know would probably also object to Premise 3, but the point of the wager is not whether or not there is good evidence that God exists; it explicitly avoids that.  Instead, it argues that even if you are not convinced of God’s existence, it is still better to believe in God than not.  It attempts to construct a win-win situation, where the theist risks a finite loss for an infinite gain, but the atheist risks an infinite loss for a finite gain.

    I have been presented with the wager on a number of occasions (and I am ashamed to admit that I sometimes presented it myself in my Christian years).  But it is rarely the first argument used.  In my experience, the evangelist usually deploys the wager when their primary arguments have not been successful.  As a last resort, they will often say something like:

    If a non-Christian is wrong, then he suffers an infinite loss (eternity in Hell1), but if a Christian is wrong, he suffers only a finite loss (a finite life lived according to a delusion).  So you should be a Christian.

    Sometimes it is tempting to say everything one can on a certain topic, and there is a lot one could say about Pascal’s Wager.  But here I will limit myself to what I see as its most fatal flaw.

    Nothing to lose?

    Many Christians find the logic of the wager compelling, including some pastors I have spoken with.  To such a Christian, it is difficult to see that Pascal is guilty of committing the logical fallacy known as the False Dichotomy, where two alternatives are wrongly presented as the only alternatives.  It is easier to see this when considering how an adherent to a different faith might present their own version of the wager.  Imagine a Muslim is trying to convert a Christian to Islam.  After everything else has failed, the Muslim says:

    If a non-Muslim is wrong, then he suffers an infinite loss (eternity in Muslim Hell), but if a Muslim is wrong, he suffers only a finite loss (a finite life lived according to a delusion).  So you should be a Muslim.

    Clearly the logic is exactly the same.  But I would hazard a guess that no Christian reading this would be tempted to jump ship and become a Muslim!  Not only does the Christian think the Muslim is wrong; the Christian believes the Muslim will suffer infinite loss, by virtue of ending up in the Christian hell.

    If you are not already committed to the truth of Islam, the flaw with the argument should be easy to see.  If a Muslim is wrong, he could be wrong for several reasons.  He could be wrong because all religions are wrong, in which case, as claimed, there is no infinite loss to worry about.  Or he could be wrong because another religion (such as Christianity, Mormonism, Hinduism, Judaism, or even some as-yet-undiscovered religion) is true, in which case there may indeed be an infinite loss to worry about.

    Of course, the same criticism applies to the original Christian version of the wager.  It too relies on a false dichotomy:

    Either Christianity is true, or no religion is true.

    Since this dichotomy is simply not true, the wager loses all its force.  No matter what you wager – whether on Christianity or another religion, or no religion at all – you are risking an infinite loss.

    Nothing to gain?

    At this point, a Christian (or Muslim, Mormon, etc) might be prepared to concede that all people are indeed risking an infinite loss.  But they might counter that any kind of theism is to be preferred to atheism for the simple reason that a theist at least has a chance of an infinite gain, whereas an atheist does not.  But why should an atheist accept that he has no chance of infinite gain?

    This is immediately contradicted by Christian Universalism, for example, which teaches that all people will receive eternal life.  (This also provides a scenario in which being “rewarded” has nothing to do with your beliefs being “right”.)  Unless such possibilities can all be eliminated (and remember, the point of the wager is that all options are treated as legitimate possibilities), such a counter-argument will not work.  There are also plausible scenarios in which non-believers are rewarded but believers punished.  For example, the universe may have been created by a powerful being who, for whatever reason, decides not to reveal himself to his creation.  He also decides to punish those who believed in a God on insufficient evidence, and reward those who proportioned their belief to the evidence and therefore withheld belief in the supernatural.  Or the universe could have been created by a powerful but malevolent being who will flip a coin to see who gets rewarded or punished.  Or perhaps people with blue eyes will be rewarded, and all others punished.  Maybe God will punish those who defend atrocious deeds ascribed to him in ancient texts, but will reward those whose moral convictions will not allow them to do so……

    It seems there are endless scenarios in which an atheist stands to receive an infinite gain.  So this line of defence will not work.

    What are the chances?

    To motivate the idea behind the theist’s final counter, consider a simple lottery in which a single number from 1 to 1,000,000 is selected at random.  If you have the matching ticket, you gain an infinite reward; if you don’t have it, you suffer an infinite loss.  There are two people in the draw; you have one ticket, and I have all the rest.  We both have a chance at an infinite gain, and a chance at an infinite loss.  But clearly my situation is more desirable than yours, due to my increased chances of winning.

    Perhaps a Christian (or Muslim, Mormon, etc), who, by now, might be prepared to concede that all people are indeed risking an infinite loss for an infinite gain, will counter that a theist has more chances at an infinite gain.  Analogously to the above lottery example, if there are more scenarios in which a given theist has an infinite gain than does an atheist, then (that specific kind of) theism is to be preferred.2

    To formalise the theist’s counter-argument, let N be the number of possible scenarios, and let T and A stand for the number of such scenarios in which the theist and atheist, respectively, have an infinite gain.  Essentially, the theist is here claiming that the fraction T/N (which is equal to the probability that the theist will gain an infinite reward) is greater than the fraction A/N (which represents the corresponding probability for the atheist).

    But how could a theist make such a claim?  Without giving reasonable estimates of the numbers NTA, the theist would just be making a baseless assertion.  But how might one attempt to estimate the value of these numbers?  This seems to me to be difficult in the extreme, but I am willing to listen to any theist who wants to give me their estimates.

    It is also worth emphasising that if T and A both happen to be infinite (which seems plausible to me), then any talk of “more chances” is irrelevant.  So, again, the theist is stuck.


    At the end of the day, there is no win-win situation.  Any wager risks an infinite loss for an infinite reward.  So the best way to make your decision is to determine what you honestly think is true, not just gamble on the most appealing option.  But if you are tempted to simply wager on some religion, I suggest you follow the advice of the late Christopher Hitchens, and play it safe by wagering on the religion with the worst hell 😉



    1.  Or at least missing out on eternal life.  (Some Christians do not believe in a literal hell.)  This seems to reduce the force of the argument, since, once you have been permanently erased, you won’t exactly be unhappy about that.  The annihilationist may still attempt to use the wager and argue that, although an atheist and Christian both risk finite loss, only a Christian has the possibility of infinite gain.  But this adaptation is still defeated by the argument I present.

    2.  Note that the reference to a specific kind of theism here is warranted.  For example, suppose there were 1,000 possible scenarios, 10 of which were favourable for the atheist.  Suppose further that there were 198 kinds of theism, each with 5 favourable scenarios.  Here there is only a 10/1,000=1% chance of an atheist receiving an infinite gain.  Thus, there is a 99% chance that Heaven will be populated by theists.  However, any given theist only has a 5/1,000=0.5% chance, so atheism is the best wager in this situation.  (Obviously these numbers are contrived, but the point is that atheism must be compared to one specific kind of theism at a time, not theism as a general whole.)

    Category: AtheismPascal's WagerTheism


    Article by: Reasonably Faithless

    Mathematician and former Christian