Refutation of Pascals Wager by Massimo Pigliucci

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A refutation of Pascal’s wager and why skeptics should be non-theists

by Massimo Pigliucci

“Man is obviously made for thinking. Therein lies all his dignity and his merit; and his all duty is to think as he ought” (Blaise Pascal, 1623-1662, Pensees)

One of the most popular arguments of people who believe in a God and would like to make an “irrefutable” argument for their beliefs is the (in)famous “Pascal wager”. Within the skeptic and humanist communities, one of the most delicate, politically wide-ranging, and emotionally charged debates concerns the question: should a coherent skeptic also be an atheist (or at least agnostic)? This essay attempts to discuss a fundamental link between the two issues, while making use of an essential component of the scientific method to solve both.

Blaise Pascal was a mathematical genius of the 17th century, and by all means the father of modern probability theory. He was also a deeply religious man, follower of the Jansenist movement, a French Christian sect which emphasized personal holiness and especially predestination. In his Pensee (Thoughts), published posthumously in 1670, Pascal mounted a philosophical defense of faith as a fundamental mean of understanding the universe. The so-called Pascal’s wager is a unique blend of theism and probability theory, that is - in my opinion - an archetypal example of that contradiction between mysticism and rational thought that was so characteristic of the origin of modern science. The same mix was typical of all the path-breaking thinkers in equilibrium between the old medieval set of assumptions about the universe and the forthcoming Enlightenment, from Descartes to Newton, from Copernicus to Galilei.

The wager consists of the following simple argument. We do not know if God exists. Therefore, we are faced with an arbitrary choice of believing in him or not. If we do not believe, we can live whatever life we please on Earth, but we are faced with potential eternal damnation afterwards. If we believe, we will forgo some earthly pleasures on the short run, but we may gain an everlasting reward after we die. Pascal’s conclusion, based on his budding probability theory, is that we should obviously go for the latter choice. The potential reward is infinite, while the sure loss is meager indeed (at least, so it seemed to a 17th century conservative Christian). What is wrong with Pascal’s wager? If you think about it, his argument is the same that modern lottery agents use to convince you to buy their tickets (scaled down, obviously, since the reward is very high, but not infinite, and the material loss is much smaller than your entire life). The reason Pascal’s wager does not work is the same reason why you should never plan your retirement on winning the lotto. [i] See, for the wager to work, Pascal has to make a fundamental (not explicitly stated) assumption: that the a priori probabilities of God’s existence or inexistence are the same, a Solomonic 50-50. Well, but if the probability of winning the lotto were indeed 50%, you would be a fool not to plan your retirement on it! The reason you do not is exactly because you know that while the reward may be very high, the likelihood of actually getting it is incredibly small (minuscule would be a more appropriate adjective).

Now, how do we attach an actual probability to the existence (or lack thereof) of God’s existence? Clearly, we cannot do it in the way a statistician would like to do it, because we have no data from repeated or controlled experiments (i.e., we do not have the luxury of observing several universes, some with and some without Gods, and then calculate their relative frequencies. Neither do we have a theoretical model predicting the likelihood of godless universes - although Stephen Hawking may have gotten something very close to the latter: Hawking 1993) . It is equally clear that we can easily find human beings willing to attach either extreme value to such probability: staunch atheists will estimate it at zero, while equally stubborn theists will assure you that it is 100%. So, isn’t Pascal’s assumption of 50-50 a reasonable alternative after all?

French philosopher Blaise Pascal (1623-1662), a mathematical prodigy who – together with Pierre de Fermat - formulated some of the basic principles of modern probability theory. Among other things, he invented the first mechanical calculator and was able to relate the level of mercury in a column to the external barometric pressure. Religiously, he was a Jansenist, and therefore a believer in absolute predestination. On the other hand, he believed in human progress through science and empirical investigation. Go figure. Source: Microsoft Bookshelf 1998, Encarta 98 Desk Encyclopedia. Of course not. All of the above simply means that humans disagree profoundly, and that we have no clear experimental evidence one way or the other. However, if we cannot think in quantitative terms, perhaps we can achieve a qualitative estimate. After all, we do not need to know exactly what our odds of winning the lotto are, all we require to make our decision is to know that they are very, very small. Well, there are two orders of reasoning that would lead me to conclude that the probability of the inexistence of God is in fact exceedingly small. First, every time we consider a God with physical attributes, that is one that actually does something in the universe (as opposed to the unfalsifiable but sterile deistic position), science invariably tells us that that God does not exist. We thought that God caused lightning, now we know better; we attributed to him a worldwide flood that modern geology says never occurred; and so on and so forth. Second, even a much reduced version of God (either the deistic one, or the so-called “God of the gaps” [ii]), assumes the existence of the supernatural, that is of something we have absolutely no evidence of, which is not necessary to explain the world, and quite plainly is the result of wishful thinking on the part of a pathologically insecure humanity. Either way, we cannot prove the inexistence of God (or of anything else, for that matter), but we are certainly warranted - at this stage of our quest - to conclude that its likelihood is astoundingly small. If that is correct, than Pascal’s wager reduces to the general case of lottery tickets: you shouldn’t buy either.

Two fundamental types of mistakes

What does that have to do with the relationship between theistic belief and a reasonable intellectual commitment to skepticism? To get to the second part of my argument we have first to briefly discuss a fundamental concept of modern statistics (and therefore, ironically, an intellectual grandchild of Pascal): the difference between type I and type II errors in hypothesis testing.

The relationships among type I error (the likelihood of rejecting a truth), type II (the likelihood of embracing a falsehood), and the logistic problems imposed by larger and larger sample sizes. There is an inverse proportionality both between the two types of errors and between these and the sample size of an experiment. Alas, these relationships make it impossible to reach certain (or even very highly probable) conclusions on anything. It is a basic tenet of modern quantitative science (such as most of biology, for example), that there is no way to get to prove or disprove any particular hypothesis with absolute certainty. All that statistically-based hypothesis testing can offer is “a very likely maybe”. That is, of course, a source of comfort, not despair for scientists. After all, actually knowing the probability of being correct (or wrong) while making a decision is much better than basing your choice on the flip of a coin. It also follows from elementary statistical theory that every time we accept a hypothesis we may be committing what is know as type II error: we may in fact being embracing the wrong conclusion. On the other hand, any time we reject a hypothesis on the basis of a statistical test, we may be committing a type I error: we may be closing the gates to a correct answer.

In an ideal world, we would like to reduce both kinds of errors to zero, of course. Alas, only religion promises (and cannot deliver) as much. In science, you can arbitrarily reduce the likelihood of type I errors, but in so doing you will have to face an equally dramatic increase of type II errors, and vice versa. There is a third option, in that it is indeed possible to minimize (but never annihilate) both kinds of error by correspondingly increasing your sample size, that is the number of observations or data points you rely upon when making whatever decision. In other words, the only way to make more informed decisions is, well, to increase the amount of information at your disposal. This, of course, is only apparently a way out: increasing the amount of available information implies logistical problems, since the cost of an experiment (or whatever other way to accumulate knowledge) is usually proportional to the amount of information one wishes to gather. Especially in this era of budget-minded everything, it seems unlikely that even decisions associated with substantial human costs (such as the efficacy of a new AIDS drug) are going to be funded at an adequate level to minimize the errors associated with hypothesis testing (see figure).

Now, type I or II errors can (and are) regularly committed by any person who makes decisions, not just by scientists. Which is why this discussion is vital for everybody. Not only you have a certain likelihood to fall prey to either kind of error while you make more or less important decisions in your life (like which car to get, or whom to marry), but these problems are very much at the forefront of social policy and, therefore, of politics. For example, let us consider the debate on capital punishment. The liberal position that this should be abolished corresponds to an attempt to minimize type II error: liberals don’t want to risk putting to death innocent people; of course, by so doing they will inevitably free some real criminals (and the likelihood of both outcomes will decrease only if we spend more money researching each case). The conservative choice of retaining the death penalty, of course, is in accordance to a desire to reduce type I errors (freeing criminals); but this too comes with a price, namely the probability of frying an innocent on the electric chair. (Incidentally, one cannot think of liberals as always reducing type II errors and conservative preoccupied with type I mistakes: on fiscal policy, the liberal spending mentality corresponds to funding all that is worth, and in so doing giving money to useless projects – a minimization of type I errors. On the other hand, the fiscally conservative approach does not waste money, but it also ends up not funding worthy causes – a type II error)

So, the real question is: which kind of error should we attempt to minimize within the limits posed by the logistics of the situation? Wisdom (1997) schematized the situation by suggesting the existence of two fundamental types of personality among humans, the “gullible” and the “skeptic”. Gullible people essentially put a premium onto minimizing type I errors: they are terrorized by the idea of rejecting an important truth; therefore they lower their standards of acceptance in order to reduce such possibility. Of course, in so doing they also end up believing a lot of baloney. Skeptics, on the other hand, are folks who really would like to avoid embracing false truths, that is they concentrate on minimizing type II errors. Contemporary society (at least in theory) shuns the gullible and rewards the skeptic. Yet, as Wisdom points out in his article, this was not always the case. In fact, for most of human history, “gullible” people have made up the respectable part of society (together with that other apparently necessary component, the oppressed), while skeptics where burnt at the stakes.

The question that faces every skeptic and scientist (and which indeed should face every human being) is then: what is it better, to reduce type I or type II errors? If the choice were only a matter of personal preferences, than I’m afraid that the current social status of scientists is usurped, and that skeptics should not feel too good about themselves because they think they got a better philosophy than gullible people.

But there is one powerful argument out of this conundrum, an argument which - I am sure this comes as no surprise to the reader - favors the skeptic position by a long shot. A rather negativistic way of putting it would be that there are a lot more falsehoods than truths out there. A more accurate statement would be that since there is only one reality, there are many more wrong hypotheses about that reality. In other words, if you accept that reality is not just a figment of your imagination, and that in fact you can “know” something about it, you also concur with the conclusion that only one among a series of hypotheses to explain that reality is correct. Ergo, the majority of hypotheses out there are false. As in the case of Pascal’s wager, it is much safer to reduce type II error because the actual probability of mistake is not 50-50, but very much skewed toward falsehood.

Where does that leave the debate about theism within the skeptic and humanist community? Let me first clarify that the latter is not, and should not be, a debate about rights. Everybody has the right to believe whatever mix of ideas she feels comfortable with, no matter how contradictory these ideas are (furthermore, just as an exercise in humbleness, one would be better off to remember that regardless of how careful we are in reducing type II errors, we still probably believe a lot of nonsense, unless we embrace the nihilist and intellectually sterile position of the original skeptics such as Socrates, who knew he knew nothing Stevenson 1998) . This said, one can still argue that some beliefs are mutually contradictory, or at least logically inconsistent. I think that a theist skeptic falls just into this kind of situation.

Theism is a form of belief in the supernatural, that is a form of mysticism. By the same exact argument I used in the beginning against Pascal’s wager, we can conclude that - while a supernatural realm may exist - it certainly is very, very unlikely. Therefore, in believing in the supernatural, one behaves like a gullible person (in the benign sense of someone who reduces type I errors), not as a skeptic. Hence, a logically consistent skeptic should be either an atheist or, at most, an agnostic. The usual objection to this conclusion is the so-called “empirical” evidence that many skeptics are also theists [iii]. However, this is no objection at all, in that it only shows that humans have an uncanny ability to simultaneously hold to beliefs which are mutually incompatible. But we didn’t really need another demonstration of that, did we?


Hawking, S. (1993) Black Holes and Baby Universes. Bantam Doubleday, New York, NY.

Stevenson, J. (1998) The complete idiot's guide to philosophy. Alpha Books, New York.

Wisdom, B. (1997) Skepticism and credulity. Skeptic 5(2):96-100.

[i] Although, at the time of this writing, in November 1999, newspapers around the United States have published the astounding statistics that 25% of the American public thinks that playing the lotto is a better retirement strategy than tax-deferred investment plans. Even Pascal would have been horrified by this widespread lack of understanding of probability theory.

[ii] Deists, such as another French philosopher, Voltaire, basically say that God created the universe and then retired. The God of the gaps is invoked by some moderate creationists, which point to any piece of incomplete modern scientific knowledge to curve some possible space for a supernatural intervention. The first God, even if he existed, would not give you the kind of reward Pascal was after; the second one is simply another word for “I don’t know”, and it is bound to shrink farther and farther as long as science will progress in its understanding of the natural world.

[iii] Anthropologist Eugenie Scott used this argument in a private conversation with me, and Stephen Gould published an entire book partially based on this premise. I therefore call this the “Scott-Gould fallacy”. Scott is a good sport, and I am sure she will not get offended. I cannot vouch for Gould.