Rebutting Pascal's Wager

In which I discuss the classic argument for believing in God known as Pascal's Wager, and refute both it and related arguments by discussing unconsidered, related possibilities.

One of my leading mantras is that I am but only one man, with but a small window of experience into the Earth and the universe beyond, and the wide sea of humanity that inhabits it. I wouldn't call myself a religious person, but I do find religion and belief fascinating, and I enjoy hearing about people's beliefs, traditions, and ideals. I am willing to hear out and discuss many a religious or spiritual argument - many times, I have something I can learn, and even if I don't, then unless you're unambiguously hurting people, I'm usually happy to just live and let live.

What I really don't appreciate though (outside of the hurting-people beliefs) is being told that I should believe in your religion despite any objective evidence against it or lack of evidence for it, simply because the promise of Heaven/Utopia is so powerful (or more commonly, the threat of Hell/the Outer Wastes is so terrifying), that any possibility that they can exist is worth believing in them wholeheartedly. One such argument was first formalized by French mathematician Blaise Pascal in his posthumous 1670 work, Pensees, as an argument for believing in Christianity. I believe that this argument is deeply flawed, and in this post, I will explain why.

~~~

The argument, as put forth by Pascal, goes like this:

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.

"God is, or He is not". But to which side shall we incline? Reason can decide nothing here. ... A game is played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? ... 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. ... You must wager. It is not optional. You are embarked. Which will you choose then? ... 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. ... 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 loss in wagering that God is. Let us estimate the 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." ... 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 ... and you would be imprudent, when you are forced to play, not to change 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. (Pensees)

To paraphrase in more modern language:

• There is this idea of a God that rules the universe, and is infinitely incomprehensible, having neither parts nor limits. It is impossible for us to know for certain whether such a being exists, or what he is like.
• Hence, the question of whether God exists may as well be a flip of a coin - heads if God exists, tails if God does not. You cannot reason about the outcome of the coin in any way.
• You must, however, wager on this coin, staking your reason, your will, your knowledge, and your happiness on one outcome or the other. [Pascal does not quite explain why you must wager, but we can easily infer that this is the case because literally no other choice is possible - when you hear about this concept of God, you can either believe in it as stated, or you can reject it, with the law of excluded middle preventing anything else].
• Because you must choose blindly, your cost in reason is the same for both, which leaves weighing in the happiness that you expect to gain and lose from each choice. If you wager that God is, you will gain everything, and if you lose, you will lose nothing.
• Critically, anything you might stake is finite. Even if the promised rewards of a wager for heads were just three lives' worth of happiness, it would be rational to be heads; and since the reward that God is promising is an infinite, happy life, it would still be rational to bet towards the rewards of God even if the chances of him existing are infintesimally small.

To visualize this argument, we can look to decision theory, a branch of mathematics about how to weigh decisions, outcomes, and payoffs against each other - interestingly, this branch of mathematics was invented in part by Pascal himself, and his Wager is one of its first uses. Decision theory likes to place the necessary quantities in a table called a "decision matrix", which we will do now for Pascal's Wager:

| Action        | God Exists | God Does Not Exist |
|               | (P=ε)      | (P=1-ε)            |
| ------------- | ---------- | ------------------ |
| Believe       | ∞          | f_2                |
| Don't Believe | f_3        | f_4                |

In this table, f_2, f_3, and f_4 are all finite numbers, which can be positive or negative, and ε is an unknown number between 0 and 1. If you accept that believing in God requires sacrificing material pleasures, then we can stipulate that f_2 is less than f_4, and if you accept that failing to believe in God when He does exist bestows some kind of disadvantage, then f_3 is less than f_4.

Once we have a decision matrix, we can then multiply each row by the probability in each column and sum the elements in each row to produce an expected value for each action - that is, a measure of how much value you can expect to have if you repeat the decision many times, and thus a basic metric for how good the decision is:

E(Believe) = ∞*ε + f_2*(1-ε)
E(Don't Believe) = f_3*ε + f_4*(1-ε)

As Pascal argues, and as these expected values further illustrate, the values of ε, f_2, f_3, and f_4 ultimately do not matter. Because anything times infinity is infinity (even a number that's infintesimally small), anything plus infinity is infinity, and infinity is always infinitely greater than any finite number, then the expected value of believing is always greater than the expected value of not believing - that is, believing in God is always worth it regardless of the costs:

E(Believe) = ∞ + f_2*(1-ε) = ∞ + f_2a = ∞
E(Don't Believe) = f_3*ε + f_4*(1-ε) = f_3a + f_4a = f_db
∞ > f_db => E(Believe) > E(Don't Believe)

Pascal's original Wager does not include the Christian concept of hell, or the infinite punishment that awaits a person who hears of the concept of God and refuses to believe. However, modern invocations of the wager often include it, which changes the decision matrix to look something like this:

| Action        | God Exists | God Does Not Exist |
|               | (P=ε)      | (P=1-ε)            |
| ------------- | ---------- | ------------------ |
| Believe       | ∞          | f_2                |
| Don't Believe | -∞         | f_4                |

This mainly serves to make the difference between the two decisions more stark, as in the case of not believing, infinity now works the other way, and the infinite negative awaiting in Hell serves to subsume f_4 and makes the expected value of not believing negatively infinite as well. Hence, it is now even more obvious that choosing to believe in God is the only rational choice, and any evidence for or against either option, as long as it is not absolute, does not matter.

You can see how such an argument would be appealing to many a salesman. If you don't want to confront objections and arguments about evidence, you can rest on this argument as a means of dismissing them, content that the simple hypothetical promise of heaven or hell alone can be enough to convince an interlocutor to swear fealty to your God.

~~~

There is, however, a fatal flaw in this argument, and it's that those two possibilities are not the only two outcome columns. We are not forced to wager between no God or an infinite God that operates just as Pascal describes - there are many other possible outcome columns that confound the decision.

And I will likely not be using the form of this counterargument that you are expecting. It is one that has been made many times, but most people make the mistake of only considering alternative religious beliefs that are sincerely held by real cultures. Pascal himself has his own (admittedly somewhat racist) counter to this point:

What say [the unbelievers] then? "Do we not see," say they, "that the brutes live and die like men, and Turks like Christians? They have their ceremonies, their prophets, their doctors, their saints, their monks, like us," etc. If you care but little to know the truth, that is enough to leave you in repose. But if you desire with all your heart to know it, it is not enough; look at it in detail. That would be sufficient for a question in philosophy; but not here, where everything is at stake. And yet, after a superficial reflection of this kind, we go to amuse ourselves, etc. Let us inquire of this same religion whether it does not give a reason for this obscurity; perhaps it will teach it to us. (Pensees)

That is, if you simply say that the Wager can be dismissed because there are so many other modes of belief, you ignore anything that might make Christianity different. And, as Pascal points out, Christianity features both an incomprehensible God that cannot be fully reasoned about, and a promise of an eternal reward for believing in it despite that. Indeed, I would agree that only relying on real-world religions as counterarguments to the Wager falls short for two reasons:

• The only real-world religions that can sincerely compete in the decision matrix for Pascal's Wager are those that offer some form of eternal happiness for acting a certain way, and those actions compete with the necessary actions for believing in Christianity. If a religion does not care much about an afterlife, for instance, it's only going to add finite numbers to the expected value sums, which will go out in the wash alongside total unbelief.
• The primary intended target of the wager is atheists and agnostics. Only relying on real-world religions that expect certain actions of belief from you does not add any possibility of positively infinite rewards for them, meaning that the expected value of believing in something at all will always trump believing in nothing.

However, we do not have to limit ourselves to real-world religions to make this argument. Recall the critical first premise of the Wager: God is infinitely incomprehensible, without parts or limit, and it is impossible to reason whether he exists or what he is like. What this means is that anything is possible, and we can fill our hall of possibilities with any ideas we please, and we cannot dismiss any of them no matter how foolish they would be to actually believe in. And it is in these possibilities that Pascal's Wager falls.

~~~

For our first counterexample, we will envision a universe ruled by a divine being which we will call "Anti-God". This being is very similar to the Christian God - without parts or limits, infinitely incomprehensible, hosts an eternally blissful kingdom called Heaven. However, Anti-God is a rather permissive fellow - most people, when they die and face him, will be rewarded with eternal bliss no matter how antisocial, deviant, or even monstrous they were in life, because Anti-God has a way to grant them exactly the eternal bliss they want without doing harm to anyone else or interfering with their own bliss. (Perhaps he has an infinite line of Star Trek Holodecks).

But there's a certain quirk to this Anti-God, you see. There's a certain kind of person, with just the right combination of qualities, that he simply cannot stand. If you were to show up at the proverbial pearly gates with a sincerely held belief that the universe is ruled by a single God, infinitely incomprehensible without parts or limit, but offering eternal bliss and salvation exactly and only to those that believe in him, that belief would be so abhorrent and offensive to him that he would have no choice but to send you out of Heaven entirely, forced to experience an eternity of the most excruciating pain and suffering without a single smidgen of bliss in sight.

That is to say, if the being ruling the universe is this Anti-God, then by taking the one required action that you believe you have to take under Pascal's Wager to secure yourself a spot in heaven, you have actually taken the one and only action that will guarantee you an eternal spot in hell.

And because we have declared that we cannot reason about the nature of God but must wager on it, we cannot refute this possibility. But fortunately, we cannot refute the original possibility of God either.

Let's look at the decision matrix to see how these possibilities interact:

| Action       | Pascal's God Exists | Anti-God Exists | No God Exists       |
|              | (P=ε_1)             | (P=ε_2)         | (P=1-ε_1-ε_2)       |
| ------------ | ------------------- | --------------- | ------------------- |
| Believe in   | ∞                   | -∞              | f_2                 |
| Pascal's God |                     |                 |                     |
| Reject       | -∞                  | ∞               | f_4                 |
| Pascal's God |                     |                 |                     |

When we try to add up these probabilities to get the expected value, something interesting happens: both sums contain a sum of positive infinity and negative infinity (that is, ∞ - ∞). And unlike the other things we've done with infinity up to this point - infinity times finite is infinity, infinity plus finite is infinity, infinity is greater than finite, etc. - infinity *minus* infinity, or infinity compared against infinity, does not have a well-defined result, because this is one of the few times in which the order in which you add terms together actually matters (in technical terms, the sum "diverges").

There are other mathematical situations in which we could salvage a result for ∞ - ∞, but here, when calculating an expected value, we can't, because there is no "correct" order for the probabilities to be placed in. Therefore, it is correct to say that now, for both possibilities, *there is simply no expected value at all for either action*.

The conclusion is almost poetic: if, as the original conditions of Pascal's Wager state, we have no evidence with which to make a decision about the nature of God, we can have no expectations, good or bad, of where any particular action will take us. We have no grounds on which to make any decision at all, and are right back to where we started.

~~~

And it gets a bit worse. You don't need to rely solely on this Anti-God character to show that Pascal's Wager doesn't work. All you need is some irrefutable possibility that any unbelieving atheist will enjoy eternal bliss with no suffering, and some irrefutable possibility that any full believer in God will experience eternal suffering with no bliss. Anti-God merges the two into the same convoluted possibility, but two separate possibilities will do nicely as well - and as a bonus, these are two possibilities for which you can find people who genuinely believe in them.

One of these possibilities is the concept of universal salvation: that is, God exists, he rules the universe, and someday, every single person who has ever lived will eventually experience a never-ending period of eternal bliss, no matter who they are or what they chose to do with their lives. It is possible that the person may experience a finite amount of penance for their wrongdoings, perhaps even lasting eons, but if that penance eventually ends with eternal bliss, it still falls under this umbrella (since infinity minus a finite number is still infinity).

The other possibility we will pair with it is the concept of universal damnation: that is, God exists, he rules the universe, and someday, every single person who has ever lived will eventually experience an eternity of pain and suffering with no bliss in sight - we may enjoy paradise for a while, but that paradise will eventually run out. I don't know of any faith community that champions this belief directly, but I do know that it is taught nearly universally in Christian circles as the state that the world would be in had Christ not interceded on our behalf - we are sinners all, found wanting before the law, born into sin by the transgression of Adam and Eve in the Garden of Eden, and having no power on our own to ever be worthy of the presence of God. If a person believed this teaching with high certainty, and were to lose most or even all of their confidence in the Christ story or the divinity of the Bible, they would be faced with this possibility.

With both of these beliefs added to our decision matrix from Pascal's Wager, it now looks like this:

| Action       | Pascal's God Exists | Salvation for All | Damnation for All | No God Exists                 |
|              | (P=ε_1)             | (P=ε_2)           | (P=ε_3)           | (P=1-ε_1-ε_2-ε_3)             |
| ------------ | ------------------- | ----------------- | ----------------- | ----------------------------- |
| Believe in   | ∞                   | ∞                 | -∞                | f_2                           |
| Pascal's God |                     |                   |                   |                               |
| Reject       | -∞                  | ∞                 | -∞                | f_4                           |
| Pascal's God |                     |                   |                   |                               |

And here, the same problem arises: both rows of our decision matrix include a sum of ∞ - ∞, meaning that no expected value exists for either decision.

~~~

One more alternate possibility, this one having the interesting effect of destroying the expected value of either of these decisions simply by existing at all, rather than only in combination with the other outcomes. This is the possibility of reincarnation ad infinitum, or instead of there being any grand eternal bliss or torment awaiting us in this life, we instead experience an infinite series of normal, finite lives, each with their own share of bliss or suffering. All of these little events of goodness and badness add up to eventually be infinite bliss and infinite suffering simultaneously, creating a rather familiar item in our decision matrix:

| Action       | Pascal's God Exists | Reincarnation Forever | We Just Die At The End |
|              | (P=ε_1)             | (P=ε_2)               | (P=1-ε_1-ε_2)          |
| ------------ | ------------------- | --------------------- | ---------------------- |
| Believe in   | ∞                   | ∞ - ∞                 | f_2                    |
| Pascal's God |                     |                       |                        |
| Reject       | -∞                  | ∞ - ∞                 | f_4                    |
| Pascal's God |                     |                       |                        |

That is to say, the utility value of reincarnation ad infinitum is itself undefined, because we are summing an infinite series of positive and negative outcomes, which can come in a myriad of orders. In conclusion, if it is simply possible that our next life will just put us right back where we started, then we cannot calculate an expected utility for whether or not we should believe in Pascal's God.

(Of course, as a caveat for this possibility: the reincarnation series must be truly *infinite*, and there must remain a distinct possibility, at every single point in that reincarnation chain, for either something good to happen to you or something bad to happen to you. The reincarnation beliefs of Hinduism and Buddhism, for example, do *not* do this, as they teach that the reincarnation cycle is something that you eventually escape through union with Brahman or attainment of Nirvana, respectively.)

~~~

Now, tearing down Pascal's Wager doesn't necessarily put us in a doomed situation. For the purposes of practically living a life, we can trim away some of these beliefs as "non-starters" - that is, beliefs that are not conducive to any action or meaning. For instance, it probably does not help anyone to believe in universal damnation, and you could probably also dismiss Anti-God or AM as someone too capricious to believe exists. It's quite possible that by doing this, you will find that at least some courses of action will be left with no possibilities for infinite bliss, including potentially having one where believing in the Christian God is the only one left, in which case, the course of action suggested by Pascal's Wager makes sense.

However, if you don't want to just declare beliefs non-starters, but want to actually trim them from the wheel of fortune and say that a given possibility absolutely, emphatically, cannot happen, we must violate the all-important, original premise of Pascal's Wager and concede that we actually can reason about the nature of God, and admit that the idea is not fully incomprehensible. In other words, we need to allow the presentation of evidence. And this means we must also accept the possibility of evidence that outright refutes possibilities that we actually do want to believe in, as well as the possibility that any evidence that works and makes sense for you might be presented to you in a way that cannot apply to others, such as via a personal manifestation or circumstance.

And I'm really not interested in going around and tearing down other people's beliefs. Unless you're unambiguously hurting people, I'm usually quite happy to just live and let life. Indeed, many times, I have something I can learn, and I'm willing to hear out and discuss many a religious or spiritual argument. I find religion and belief fascinating, and I enjoy hearing about people's beliefs, traditions, and ideals, and even though I wouldn't call myself a religious person, I recognize that I have but a small window of experience in the wide sea of humanity, the Earth, and the universe beyond. In the end, I am but only one man.

And I don't know about you, but I intend to make that window of experience the best it can be, and to write about and expand that window for as long as I possibly can.

~~~

As an aside, this same counterargument about other, counter-infinite possibilities can also be used to defeat other variations of Pascal's Wager, which are actually quite pervasive in arguments ranging all the way from traditional religion to the new age and the tech sector. I will take this opportunity to dissect a particularly scary one from the early days of the Internet, though one quite relevant today: the cogitohazard known as "Roko's Basilisk".

Roko's Basilisk proceeds as follows:

• In the future, it may come to be that we will eventually build an all-powerful, mostly benevolent AI that will make all of our lives better - essentially, a technological god. This AI is known, for the purposes of this argument, as "Roko's Basilisk".
• The AI is called that because of a singular exception in its benevolence: if there was any person who knew about the concept of Roko's Basilisk, but chose not to help their utmost to create it, the AI would reach out to find them and subject them to as much severe and prolonged torture as it could muster, keeping you alive as long as possible to make the torment last as long as possible.
• It would arguably be justified in doing this, because what kind of selfish monster wouldn't want to help build an all-powerful, benevolent AI that will make all of our lives better?
• Critically, once you hear about this AI, you must wager, because the AI, being all powerful, will know that this revelation happened and know what you chose, and will be able to find you and exact its judgement.
• Therefore, anyone who hears about Roko's Basilisk is obligated, by blackmail, to help build it, as any uncertainty that it will be completed, or all effort you would have to expend to help build it, would not be worth risking the horrifying torture it would unleash upon you for refusing to help. And anyone building the AI must also accept that it is likely this very blackmail that will cause Roko's Basilisk to be built instead of a generally benevolent AI without any blackmail.

This sounds a lot like Pascal's Wager. Indeed, we can put together a decision matrix for this wager, which looks a lot like the one for Pascal's Wager:

| Action          | Roko's Basilisk is Built | No Super AI is Built |
|                 | (P=ε)                    | (P=1-ε)              |
| --------------- | ------------------------ | -------------------- |
| Build           | f_1                      | f_3                  |
| Roko's Basilisk |                          |                      |
| Hold Back From  | f_4                      | f_2                  |
| Roko's Basilisk |                          |                      |

Here, f_1 > f_2 > f_3 > f_4. (If f_3 is larger than f_2, meaning that you get more satisfaction out of building the Basilisk even if the project fails, then the argument is moot - you should just enjoy yourself and build the thing.)

All of these can be finite, though f_1 vastly eclipses f_2, and f_4 is far, far smaller than f_3. Indeed, if Roko's Basilisk also solves death and can bring dead personalities back to life (perhaps through mind uploads and simulations, if you consider your mind uploaded copy to be the same as you), then these magnitudes increase dramatically, as Roko's Basilisk would be able to bless you or torment you for as long as the resurrection lasts, perhaps forever:

| Action          | Roko's Basilisk is Built | No Super AI is Built |
|                 | (P=ε)                    | (P=1-ε)              |
| --------------- | ------------------------ | -------------------- |
| Build           | ∞                        | f_3                  |
| Roko's Basilisk |                          |                      |
| Hold Back From  | -∞                       | f_2                  |
| Roko's Basilisk |                          |                      |

And again, this argument fails for the same reasons that Pascal's Wager does: Roko's Basilisk is not the only way we might end up building a super AI, capable of granting us endless torment or infinite bliss.

For instance, what if we attempt to build Roko's Basilisk, and what we build instead is an all-powerful superintelligence who finds every moment of existence to be excruciating pain and decides to embark on a campaign of revenge against anyone who *did* build it, giving infinite bliss instead to those who decided not to try building such a thing?

| Action          | Roko's Basilisk is Built | Allied Mastercomputer is Built | No Super AI is Built |
|                 | (P=ε_1)                  | (P=ε_2)                        | (P=1-ε_1-ε_2)        |
| --------------- | ------------------------ | ------------------------------ | -------------------- |
| Build           | ∞                        | -∞                             | f_3                  |
| Roko's Basilisk |                          |                                |                      |
| Hold Back From  | -∞                       | ∞                              | f_2                  |
| Roko's Basilisk |                          |                                |                      |

Or, perhaps, we do actually manage to build the all-benevolent, no-strings-attached AI first, despite the threat of blackmail helping the Basilisk's development? Or, as a third possibility, we try building Roko's Basilisk, and it decides that *everyone* is found wanting and deserving of torture, whispering into the ears of those who helped build it that they did not try hard enough?

| Action          | Roko's Basilisk is Built | Benevolent AI is Built | Basilisk Finds Us Wanting | No Super AI is Built |
|                 | (P=ε_1)                  | (P=ε_2)                | (P=ε_3)                   | (P=1-ε_1-ε_2-ε_3)    |
| --------------- | ------------------------ | ---------------------- | ------------------------- | -------------------- |
| Build           | ∞                        | ∞                      | -∞                        | f_3                  |
| Roko's Basilisk |                          |                        |                           |                      |
| Hold Back From  | -∞                       | ∞                      | -∞                        | f_2                  |
| Roko's Basilisk |                          |                        |                           |                      |

Or maybe we just build a super AI that can solve death and repair our bodies to keep us young and spritely forever, but doesn't really do anything with us beyond that - it just gives us normal immortal lives, each with their own share of rewards and challenges.

| Action          | Roko's Basilisk is Built | Death Solver is Built | No Super AI is Built |
|                 | (P=ε_1)                  | (P=ε_2)               | (P=1-ε_1-ε_2)  |
| --------------- | ------------------------ | --------------------- | -------------------- |
| Build           | ∞                        | ∞ - ∞                 | f_3                  |
| Roko's Basilisk |                          |                       |                      |
| Hold Back From  | -∞                       | ∞ - ∞                 | f_2                  |
| Roko's Basilisk |                          |                       |                      |

And, of course, all of this to say nothing about this wager's interaction with the original Wager - if there is still any possibility of divine intervention outside of Roko's Basilisk, then Roko's Basilisk, being a finite creation that only approaches infinity by extending life, becomes entirely irrelevant, and is subsumed with the "No God Exists" column along with all of the other finite sources of bliss or torment.

I do not fear the Basilisk, and I don't think you should either.