I’m sure you’ve all heard of Pascal’s Wager. The gist of it is that if you bet on there being a God (meaning, to Pascal, the Medieval Catholic God), you have an infinite expected return on investment, because at worst it costs you nothing (or at least, relatively little, compared to an infinite return) and at best you get to go to an eternal heaven, whereas if you bet against there being a God, you have an infinite expected negative return on investment, because at worst it costs you an infinite torment in an eternal hell, and at best it earns you nothing (or at least, relatively little, compared to an infinite cost). And by the ordinary principles of expected utility, these then average to “infinite reward” and “infinite cost” regardless of the probability that God actually exists. So you should always bet on there being a (Medieval Catholic) God.

There are several problems with this argument. Here you’ll find a survey of some of them, illustrating why this argument is either invalid or unsound no matter how you try to revise it. There is a somewhat fawning summary of the argument and its history and logic in the Stanford Encyclopedia of Philosophy. But this is surpassed by a concise critique in the Internet Encyclopedia of Philosophy (the IEP).

Its Validity Would Warrant Belief

It is not a problem with Pascal’s Wager, however, that “we can’t choose to believe what we do.” Despite that being often the leading objection to it voiced, it’s misguided. Yes, real belief must follow from the formation of a genuine conviction. But this can arise from recognizing the force of a bet. The IEP rightly adds further reasons to dismiss this objection to the Wager. But more salient still is that recognizing better bets can produce beliefs.

For example, suppose we were speaking instead of the following scenario: a tsunami is rising behind you on the road, and you see no traffic, but the road is heavily fogged in so you have no real way to know; you can surrender and be drowned, or attempt to drive faster than the wave in the mere hope there is no one ahead of you on the road to collide with. A simple utility equation will reveal you should attempt to outrun the wave. That option has a less than 100% probability of resulting in your death, whereas the surrendering option effectively has a 100% probability of your death. In other words, the outcome of that decision is “always death”; of the other, “sometimes survive.”

Once you fully, rationally recognized that, the very fact of doing so will indeed cause you to genuinely believe survival is more likely when you assume without evidence the road is clear. This might not be what you would normally call “a belief the road is clear,” since you would also still correctly believe that it yet might not be. But it is nevertheless a belief that “the road is more likely clear than that the tsunami will not overtake me if I remain still.” Analogously, if Pascal’s Wager is as sound as this reasoning would be about the road and the tsunami, then your recognizing that will likewise cause in you a genuine belief that “God more likely exists than that I will end up in an eternal paradise if he doesn’t or that I will avoid an eternal hell if he does.” You will therefore believe in God to some significant degree, even as you admit you are not certain that God exists. Just as you would believe to some significant degree the road is clear—which means, not wholly, but enough to motivate your every decision. Just as Pascal was arguing for Catholicism.

Pascal’s Wager is often presented in a formally invalid way, such that the conclusion really doesn’t follow from the premises. But it nevertheless could be written in a form that would be formally valid (many versions exist in the literature; see the bibliographies in the encyclopedias cited above). Formal validity means the conclusion must be true if the premises are. But to get that outcome, those premises must be more numerous than those usually stated by Pascal or anyone defending or using the argument, and it is the fallacies inherent in those premises that cause it to fail. Logicians must then say the argument is unsound. And it is. Its conclusion does not follow because its premises lack sufficient probability to render its conclusion probable. So to that we now turn.

The Hidden Ad Baculum

The first problem is that Pascal’s Wager is a covert ad baculum fallacy, an “appeal to force” or “argument by intimidation.” It isn’t presented that way, of course. It’s presented as a seemingly sensible argument from expected utility. But even if it were formally valid as such, it still rests on a premise that is, actually, a veiled threat.

All critiques of Pascal’s Wager implicitly reveal this as its defining flaw. The most salient example being the “many-gods objection” that I’ll get to in the end. Since there are many different gods one could bet on, and all those bets cancel each other out, we are faced with asking why we are to prefer Pascal’s singular premise of Catholicism. The obvious answer is the very thing that purports to make the bet work: its implicit threat of death or, worse, hell. In Pascal’s version, you are being threatened with eternal damnation, the “infinite cost” to atheism.

If you remove that threat, what remains is only death: if God exists and you bet not, you die; if God doesn’t exist and you bet not, you die. The outcome is the same either way. So the only motivator remaining is the promise of eternal reward: if you bet God exists, you go to heaven; ergo, if God exists and you don’t bet on it, you lose out on an infinite opportunity. Which is called an opportunity cost. It therefore is a hidden threat. In this version of the bet, contrary to Pascal’s own belief, there is no hell and the unsaved simply stay dead (called annihilationism, some evidence suggests Saint Paul held to this view). But that means, now, you are simply being threatened with death: if you don’t believe, God won’t save you.

This is like saying, “If you don’t swear allegiance to Kim Jong-un, hospitals will refuse to treat you if you come to harm.” You are being refused a benefit. That’s coercion. Only worse, since hospitals can only restore you to life for so long; God, in this scenario, is able to do so indefinitely. Thus the opportunity cost to remaining an atheist is (by Pascal’s reasoning) infinite. It’s thus no accident that Pascal’s Wager only works for threat-bearing gods, whether threatening not to save you if you come to harm, or (horrifically worse) threatening to torture you eternally. It fails as an argument as soon as you adopt a God who wouldn’t threaten you to coerce belief: a God who would morally, responsibly, always save everyone, never allowing them to come to eternal harm, much less torture them.

Hence Pascal’s Wager depends on the ad baculum fallacy. And it does so at the step of selecting which God is to be bet upon. Only threatening gods are proposed. Universal saviors are ignored or hand waved out of existence (“Oh, don’t mind the American legal system, trust us, Kim Jong-un decides who gets into hospitals in the U.S.”). This reveals the immorality of the gods the Wager depends upon…and the moral depravity of anyone who would attempt to persuade you with it. But whether or not you deem coercing people into belief to be moral, it remains fallacious: things are not true merely because you threaten us lest we believe them. Merely because you can claim Kim Jong-un controls hospital access in America, it does not follow that I should believe he does. Likewise merely because you can claim there is a God who is coercing me to believe in him, it does not follow that I should believe there is. The rest of Pascal’s Wager as an argument is designed to distract you from noticing this singular defect in its design. Pascal was trying to hide the cudgel. How the Wager attempts to do that, is where its other flaws lie. To those we now turn.

An Incoherent Utility Function

Pascal’s Wager falters on the problem of infinite returns. It depends on your blithely accepting the dubious proposition that once we merely propose infinite costs or rewards, no probability of being wrong matters anymore. This is fishy. And it’s fishy for a reason. It remains fishy even if you drop its appeal to infinities.

Of course the costs of betting on God are not literally zero, as adhering to the requisite norms and behaviors and terrifying beliefs required of Medieval Catholicism are actually quite arduous and deprive a human of a great deal of accessible happiness, even causing society considerable losses in this respect as well (see: the entire history of Catholicism). But that is always finite—just one human lifetime of it—which will of course be infinitely dwarfed by an infinite return on investment. Even a totally miserable mortal life is as nothing to pay for eternal bliss. And yes, the return on investment when betting against God is likewise not zero, but is, nevertheless, still finite. Whatever differential benefit you and society get from your not being an oppressed, guilt-ridden Catholic, it only lasts one lifetime. That is thus also infinitely dwarfed by assigning an infinite cost to the bet. Even a totally blissful mortal life is as nothing to the cost of an eternal torment in exchange for it; or even, in the case of annihilationism, an infinite opportunity cost.

That sounds superficially compelling. But it’s deeply incoherent. It’s not just that it conflates finite with transfinite arithmetic—one could reconstruct the argument without infinities; it’s that the costs or gains proposed are arbitrary. Of course we can say the expected return on an investment equals simply the difference between A, the cost of a bet times the probability of losing, and B, the payout of the bet times the probability of winning. But we don’t get to just invent costs and benefits. To allow that to change our epistemic status would be irrational.

For example, consider a straightforward raffle example: we can buy a raffle ticket for $2, there’s a 1 in 1000 chance of winning the raffle, and the prize is $1000. If we disregard any fractions of a cent, then the cost of playing times the probability of losing is $2 x 0.999 = $1.99, and the payout if you win times the probability of winning is $998 (the $1000 earned minus the $2 paid) x 0.001 = 99 cents. The expected utility of buying a raffle ticket is minus one dollar ($0.99 – $1.99 = $1). In other words, you can expect to lose money buying these raffle tickets. Though the average loss is small (just a dollar), so maybe you won’t care (that’s a different equation).

But now imagine a shady carnival barker tells you the raffle’s payout is a billion dollars, or $1,000,000,000. Wow! Then everything changes. The expected utility becomes an expected gain of $1,000,000,000 x 0.001 = $1,000,000, minus only a $1.99 expected loss, or $999,998 or so. We should totally buy a ticket! But…wait a minute. Why should I buy this ticket? The only thing that’s changed is that the barker changed what they told me the prize is. All they had to do was make up a larger prize. That should in no way increase the epistemic probability that I actually will win a billion dollars. On this thinking, all I have to do to get you to believe literally anything is to just tell a bigger lie about how much it will benefit you to believe it. As that’s clearly irrational, so is Pascal’s Wager.

It is true that infinite quantities are always by definition infinitely larger than finite quantities, and all nonzero probabilities, no matter how absurdly small, always produce an infinite quantity when multiplied by an infinity. But Pascal is just making all these values up. He’s like the carnival barker, just inventing a larger prize, and expecting you to just “believe him” and thus update your estimate of the expected utility. But you shouldn’t believe him. In fact, the larger the prize claimed, the lower the probability that that’s really the prize. Because the more exaggerated a claim, the more likely it’s a lie—or in any other respect false. Maybe Pascal is just a fool, and thus simply in error, and not actually a liar; but the outcome is the same: the more outrageous the claim, the less likely it is to be true. And that means infinitely outrageous claims are infinitely unlikely to be true. So by the same logic that Pascal is expecting us to apply to “infinite payoffs” when multiplied against any probability of a payoff, the epistemic probability of infinite payoffs is infinitesimal, and an infinity times an infinitesimal in such a case is an undefined quantity. Not an infinite one.

It’s not just that Pascal simply got the math wrong. He did. But the problem here is more fundamental than that. Suppose we do away with the vexing complexities of dealing with transfinite arithmetic and just say, instead, that heaven will consist of 1000 years of bliss, or 1,000,000 years of bliss, or 1 x 10^100 years of bliss. These are finite numbers, so we are back doing completely straightforward arithmetic, no weirdness. Why are we assigning equal probability to these bets? Pascal wants us to assume not merely that there is a cosmic raffle—without, unlike an actual raffle, any evidence there is one or even plausibly could be one—but that its prize is absurd.

That’s maximally unlikely to be true. That we will live a million years is always going to be less likely no matter what Pascal says than that we will live a thousand. And so on up the line. “1,000,000 years of bliss” and “1 x 10^100 years of bliss” are not equally likely. And indeed, why should we believe any god will keep things going an infinitely long time? Since there is always a nonzero probability he’ll give up at some point and hang the whole business, and all probabilities approach 100% as eventualities approach infinity, no matter how unlikely it is that God will chuck it in, your utility function is always going to tell you it’s all but 100% certain he eventually will. Likewise if we adopt a diminishing returns theory of lifespan, such that the longer you live, the less relatively valuable adding more years becomes, such that an infinite lifespan actually tops out at a finite value. Either way, that value is still an arbitrary invention of whatever shady carnival barker like Pascal is promising it.

We can call this the Liar’s Wager. Imagine I tell you you must give me all your wealth within twenty four hours or I will cast a spell causing your soul to be irrevocably defiled and thus trapped in a hell dimension for all eternity after you die—but if you obey me, I will cast a different spell that ensures your soul will go to a heavenly dimension. Of course you would not believe me. Infinite utility functions aren’t going to change your mind, in this case any more than Pascal’s. And they shouldn’t. That’s the problem. And it’s a problem fatal to Pascal’s Wager. Because yes, there is always a nonzero chance I am not lying—indeed, even as I am here telling you I am lying, there is still a nonzero probability I am lying about that, and thus actually telling the truth! And any probability, no matter how small, multiplied by an infinite gain, remains an infinite gain. So you should either believe me, or else abandon this clearly defective utility calculus. The latter is obviously what a rational person should do.

If the utility function we have to adopt as a premise in Pascal’s Wager to make that Wager work entails that all I have to do is tell a bigger lie (about “infinite” costs and rewards, or even just “absurdly large” ones, like merely “a thousand years” of bliss or hell), and you should believe me, when you wouldn’t believe me if I told a smaller lie (like that you will receive just “ten years” of riches or misery “while still living”), the Wager is clearly depending on an irrational utility function. Telling a bigger lie should not increase the probability of its truth. If you wouldn’t believe the smaller claim, you should even less believe the bigger claim. And this is because, indeed, the epistemic probability of a payoff (or cost) always declines as the claimed payoff (or cost) increases. And when you increase it infinitely, the epistemic probability is likewise infinitely decreased.

Imagine plotting every possible wager on a graph, from “you’ll live an extra ten years if you eat this magic bean” to “a hundred years” to “a thousand” to “a million” and so on, for every possible finite lifespan I could claim. If we walked the wager up from small to big that way, we would see that as the claimed benefit rises, the probability of my being able to ensure it correspondingly drops, such that merely increasing the claim does not increase the epistemic probability of it being true. Nor can that increase the utility function, because that function is dependent not merely on the odds of a payout if the scheme is real, but also on the odds the scheme is even real. So as that goes down, so does the utility function. Bigger is always downward. Not the other way around. The probability that a church raffle will pay a billion dollars is much less than that it pays a million, which is much less than that it pays a thousand, and so on. It therefore cannot be that as we approach infinity, the probability of the payout thus continually drops, but for some weird reason at infinity, it suddenly jumps all the way to 100%! Picture this on the graph; it doesn’t make sense. Obviously the trend is down, not up. So at infinity, the probability of the claim being true should be infinitely close to zero, not infinitely close to 100%. That’s what any rational utility function would generate. Pascal’s Wager thus relies on an irrational utility function. A false premise.

One might pause here and ask whether any evidence can ever persuade us of an infinite afterlife. To answer that, re-phrase the question as “infinite reward,” rather than infinite “time,” and the answer is no. We could never have enough information to be certain we would ever really achieve an infinite benefit. Such is probably even beyond human comprehension. But even if intelligible, such a marvel would require infinite evidence. However, we should not confuse time with value, which I think the Wager does, though it needn’t. I don’t actually think eternal life has infinite value; I think the more you live, diminishing returns will reduce the value of more life—not to zero or negative value, but always to some finite value. At some point the difference between living 1 x 10^100 and 2 x 10^100 years is so inconsequential as to hardly be measurable in relative terms. The difference will still be nonzero (it will always still be to some degree better to live longer). But that positive amount will continue to be less and less as the spans involved increase, and terminate at some finite limit at infinity. As such, it’s possible to have sufficient finite evidence to believe you will live an infinite time. But it would require some really extraordinary evidence. Not just some dude assuring us it’s today’s raffle prize.

The Bayes’ Function

For the same reason, Pascal’s Wager also falters on the rule of total evidence. And this many critics have long pointed out already, via the “many-gods objection.” The probability of some claim G can only ever be the converse of the sum of the probabilities of all other competing claims. In other words, P(G) must necessarily equal 1 – P(~G), and P(~G) must necessarily equal P(G1) + P(G3) + (PG4) + … P(Gn) for every n. In short, the Wager cannot be taken without taking into account all possible outcomes of the Wager—in other words, all possible Wagers. Thus to be at all rational, or indeed at all logically coherent, Pascal’s Wager must divide the total utility space among all possible Wagers. The result is no differential recommendation as to choice. There is no bet to make.

Since there are infinitely many possible diverse outcomes (infinitely many possible gods, systems of gods, and systems lacking gods), they all negate each other. For example, for any “believe in God(x) or else” Wager, there is a contrary Wager that is equally valid, such as “God(y) will punish you for believing in God(x), therefore you should believe in God(y).” These two Wagers have the exact same utility value (even if it’s infinite)—and therefore negate each other. Therefore all such Wagers are invalid. That wouldn’t necessarily be the case, but it is the case for Pascal’s Wager because of that invalid premise of an irrational utility function we just examined. If we grant that the utility function for a God is infinite, then it is infinite for every contrary God; and the mathematical result is zero net utility (as the IEP puts it, “as soon as we allow infinite utilities, decision theory tells us that any course of action is as good as any other”). Of course, the utility function can’t be infinite (owing to the Liar’s Wager, as we just saw). But if it were, then the Wager still collapses on this other premise, of there being only one God to Wager on—and that, conveniently, only a God who just happens to be a Coercer God (hence reducing the whole Wager to an ad baculum fallacy).

Even if we tried to fix this by getting rid of the infinite utility function, whereby every God produces infinite utility no matter how differentially unlikely that God’s existence is, we still don’t get a differential recommendation of what to do—which bet to take. Because there still isn’t sufficient evidence to decide. That “most people” believe in either the Christian or Muslim god, is an accident of history that is itself not evidence that either’s God is more likely to exist than some yet other god no one has even thought of or that just happens to be less popular—including secular gods (yes, there are even secular versions of Pascal’s Wager: see Roko’s Basilisk). But worse, there is no such thing as “the Christian” or “the Muslim” God: there are thousands of such Gods, because every sect disagrees as to what gets you saved, and thus what “bet” you should be taking. Which gets us right back to the evidence for gods in general (see Bayesian Counter-Apologetics). So Pascal’s Wager is of no help.

And there is no escaping this. As the IEP elegantly puts it:

According to the many-gods objection, Pascal’s wager begs the question and hence is irrational. It assumes that if God exists then God must take a rather specific form, which few open-minded agnostics would accept. Pascalians reply by invoking the notion of a genuine option (which is not defined), by devising run-off decision theory (which is not justified), by claiming that Pascal was understandably unaware of other cultures (which is not true), and by appealing to generic theism (which does not solve the problem).

Carrier’s Wager

There is an equally valid “Pascal’s” Wager for atheism even on the presumption that God exists. Let’s call it Carrier’s Wager. I lay out the case in The End of Pascal’s Wager. In short, there is strong evidence to believe that if a moral God exists, that God wants us to be atheists, and will reward us for it. Therefore we should bet on atheism even if God exists. It can’t be objected to this that atheists should never bet against God if he exists, since anyone trusting Pascal’s Wager is already conceding we should be theists even if God does not exist. It’s doing the same thing either way. Thus Pascal’s Wager becomes self-refuting. Which exposes its vacuity. It was all along asking us to believe things we already know to be false (that its defective utility function is rational, that there is only one God to bet on, and that that God just conveniently happens to be a Coercive one). That those same false things can be used to prove exactly the contrary conclusion only cooks the goose further.

In fact, we can generalize this reasoning. If we should take any Wager, and the probability of a morally good god is higher than a god who is not morally good (as anyone trusting Pascal’s Wager is committed to believing), then we should wager on a morally good god. But a morally good god would not reward or punish anyone for failing to adopt an epistemically doubtful belief. Just try to think of any scenario in which any moral person you know would do that to someone; you will be unable, because acting like that would be immoral in every moral system anyone deems credible; you would always condemn it. Yet for whatever reasons, if God exists, they have chosen to leave their existence epistemically doubtful. And that is a state of affairs they can only have chosen, as establishing abundant epistemic evidence of their existence is extremely easy—as easy as for the existence of any other person. Therefore, insofar as a morally good god would reward or punish us for anything, it would be on the merits of our moral character alone, and not on our belief in their existence. Again, try to think of any scenario in which any moral person you know would judge differently; you will be unable, because acting like that would be immoral in every moral system anyone deems credible; you would always condemn it. Therefore, if God exists, it solely follows that we should be a moral person, the sort of person any decent being would keep from needless harm—it does not follow that we should believe in God, or adopt any system of religious beliefs whatever (much less Pascal’s Catholicism).

Which conclusion already follows if God does not exist. Atheists should be moral people, for their own conscience’s benefit as well as the reciprocal benefit it accrues from society. So if we bet on atheism, we should be moral, and will be rewarded for it; if we bet on theism, we should be moral, and will be rewarded for it. No difference in outcome is produced by Pascal’s Wager. Moreover, a morally good god would only judge moral character developed in accordance with what available evidence alone warrants believing is a moral person. In other words, moralities that rely on hidden or inaccessible or dubious evidence could not be the basis for such a god’s judgment, but only moralities for which there can be little or no epistemic doubt. Anything else would be, again, grossly unfair and thus immoral. And since God has chosen to keep even their existence in epistemic doubt, the moralities by which a god would judge us cannot require or involve or follow from any premise that there is a god, but only from premises epistemically evident.

Which means, basically, a morally good god will only judge us according to our adherence to ethical naturalism. Which the natural evidence, wholly without God, already prescribes we do. Therefore, again, if a moral god is more likely than an amoral one, and Pascal’s Wager can produce any true conclusion at all, then the conclusion of Pascal’s Wager is simply, “You ought to live your life in accordance with ethical naturalism,” and in particular, “whichever ethical naturalism the evidence most clearly indicates is most conducive to our enjoying a satisfying inner life and external community.” Only an immoral God would not rescue those who live moral lives by that standard, be they atheists or believers; and since an immoral God is less likely than a moral one, Pascal’s Wager produces no sound argument that we shouldn’t be atheists. We should simply believe what the evidence evinces; and live good lives.

Conclusion

Pascal’s Wager follows only from three invalid premises—premises that are not only dubious, but certainly false, and yet essential to producing the conclusion Pascal wanted. It relies on an ad baculum fallacy (by privileging only one Wager, the one that just happens to imagine God threatening us to compel compliance), an irrational utility function (whereby the more absurdly we exaggerate a threat or benefit, the more likely we should deem that threat or benefit to be real), and the disregarding of competing Wagers (failing to divide the total utility space among all possibilities, rather than only two). Pascal’s Wager is therefore not a sound argument for any conclusion—other than, at best, being a moral person in whatever way secular evidence already evokes, regardless of one’s belief or disbelief in any gods. But we already knew that. So Pascal is of no use here.

Pascal would have been better off heeding his own Bible: as the Book of Matthew claims Pascal’s God said, “For in the same way you judge others, you will be judged, and with the measure you use, it will be measured to you,” Matthew 7:2. I would never judge others merely according to whether they believed in a god; therefore neither shall I be so judged. Unless Pascal thinks his God is a liar. What’s his wager on that?