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Pascal’s Wager Explained

Daniel J. Castellano


1. Introduction
2. Preliminary Argument
3. The Necessity of the Wager
4. The Stakes of the Wager
5. Probability of Wager Outcomes
6. Evaluation of the Wager
7. Making the Wager in Good Conscience
8. Effectiveness of the Argument
Addendum: Alternatives to Monotheism

1. Introduction

I hesitate to include Pascal’s argument of the wager in that domain of philosophy called natural theology. Pascal himself, who thought that all of philosophy was not worth an hour’s pain (Pensées, 79), almost certainly would have objected to such characterization. Pascal’s apologetics does not counter reason with reason, but rather it is the protest of faith against philosophy, arguing that faith occupies a domain that human reason cannot reach even in principle. He begins with the fideistic premise that the existence of God can neither be proven nor disproven by reason, so he instead resorts to a probabilistic argument with an appeal to consequences. The argument of the wager, then, is not an attempt to prove the existence of God, as though it were a demonstrable philosophical thesis, but rather it is a psychological argument, adopting the assumptions of the unbelievers of his time, in order to show that their own epistemological and ethical principles would make it far wiser to believe in God than to deny His existence.

Nonetheless, Pascal’s argument does belong to natural theology insofar as it is an argument that is independent of any determinate body of revelation. In fact, he explicitly appeals to the unknowability of the divine essence, and posits little about God save that He is infinitely incomprehensible. There is nothing in the God of Pascal’s wager that is incompatible with the God described by Jewish and Muslim philosophers of the medieval period. We can even see something of this God in Plato, Aristotle, and the Stoics, so it is a mistake to characterize the God of Pascal’s wager as “the Christian God.” We do know, of course, that Pascal always had in mind “the God of Abraham… not of the philosophers.” (Mémorial) For Pascal, God could only be known as revealed, so it is an uncharacteristic departure for him to speak of God abstracting from revelation. This is because he is now not speaking of knowledge of God, but on the practical wisdom of choosing to believe or disbelieve, based on the premise that reason cannot give us knowledge of God’s existence or non-existence.

Regardless of how we choose to categorize Pascal’s argument, it certainly requires some philosophical knowledge to be interpreted rigorously, just as surely as Pascal shows in its construction much familiarity with the philosophical discipline he scorns. We must be attentive to the fact that this is a classical argumentum ad hominem, taking the assumptions of a defined adversary and showing how they lead to a conclusion contrary to what the adversary would hold. This requires some knowledge of the intellectual culture of the time. We cannot treat Pascal’s wager as an abstract demonstration, since it is in fact an argument of persuasion that presupposes certain assumptions in the listener.

We must take the position of an unbeliever who is willing to recognize that he cannot rationally demonstrate God’s existence or non-existence, nor can his reason grasp anything of God’s essence. Our unbeliever is not an ethical nihilist, but assumes that there is justice and injustice, and that a hypothetical God, being perfect, would certainly be on the side of justice. In the absence of God, he acts as he sees fit, and seeks what is really best for himself, that is, he prefers ethical good (felicity) over apparent good (sensual pleasure). This presumption of basic ethical rectitude is necessary, for if our unbeliever is a sensual hedonist, he should persist in doing what he pleases even if God exists, just as lower animals do. The question of God’s existence only holds importance for those who are concerned with discovering what the highest good is. Only the ethical man, who prefers true felicity over superficial pleasures, has this concern. The ethical man need not be a saint, nor even especially virtuous, but he only needs to desire the good, as understood in terms of the classical ethical virtues. Even those who profess ethical relativism or skepticism may fall into this category, insofar as, in their ordinary lives, they show that they really do believe in the classical ethical virtues. It is a rare intellectual indeed who is so principled as to deny any real distinction between courage and cowardice, justice and injustice, temperance and intemperance, prudence and folly, in the actions of his daily life.

Since the wager argument appeals to a certain type of person in his subjectivity, it is only natural that Pascal uses a subjectivist notion of probability in his argument. This also happened to be the dominant interpretation of probability theory in his time. We will need to scrutinize Pascal’s use of probability in his argument, to verify if his use of a subjectivist interpretation is justified and correctly applied. This will constitute the bulk of our effort.

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2. Preliminary Argument

As noted, Pascal’s argument presumes that God’s existence and essence are unknowable through human reason. He makes an analogy with mathematics, where knowledge of the infinite is necessarily indirect. We know that it is false that there are finitely many numbers, so “there must be an infinity in number.” (Pensées, 233) Yet we do not know what this infinity is. It cannot be even, nor can it be odd, though every finite number is even or odd. “So we may well know that there is a God without knowing what he is. Is there not one substantial truth, seeing there are so many things which are not the truth itself?” Although God’s essence (what He is) is unknowable, this does not make it impossible to know that He exists, which may be inferred from finite things, through the same leap of reasonable faith we make by positing an unknowable infinity in number.

Yet Pascal will not allow that we can know even God’s existence through reason. The analogy with number fails to take us this far. “We know the existence of the infinite, and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.” (Loc. cit.) Recall that in Pascal’s time, due to the enormous influence of Descartes, the entire material world was conceived in terms of geometric extension or corporeality. The very notion of existence, as we ordinarily conceive it, entails the instantiation of some essence. Yet we can only conceive of existents as distinct from each other by imagining them as extended in space. Space is effectively the medium of instantiation, and would seem to be a necessary substratum for an existent. Yet God is said to exist though He is not corporeal and thus has no extension. God’s existence, therefore, is as inconceivable to us as His essence. We may believe and assert that God exists, but we cannot really understand what this means, except by a highly improper analogy with corporeal beings.

Although reason is impotent in bringing us to knowledge of God’s existence or essence, “by faith we know His existence; in glory we shall know his nature.” This fideisic declaration should make clear how wrongheaded it is to depict Pascal’s wager as a proof of the existence of God. The presupposition of his argument is that such a proof is impossible. In this life, God’s existence can only be known through faith, though we do not know what God is. There is no contradiction in knowing that something exists without knowing what it is, as Pascal showed with the example of infinite number. Only in the hereafter may we see God as He really is, in His essence or nature.

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3. The Necessity of the Wager

Given the presuppositions that God’s existence and essence are unknowable through reason, Pascal now argues “according to natural lights,” that is, abstracting from supernatural faith. Since God has neither parts nor limits, He is infinitely incomprehensible to us, so we cannot know what He is or if He is. “This being so, who will dare to undertake the decision of the question?”

Though an unbeliever cannot be expected to try to decide a rationally impenetrable question, the Christians are not to be faulted for proclaiming their answer, Pascal argues. The Christians openly acknowledge that they believe on faith, and that what they believe is indemonstrable, so they should not be faulted for failing to give a rational demonstration of their religion. Pascal supplies the retort: “Yes, but although this excuses those who offer it as such, and takes away the blame from them for putting it forward without reason, it does not excuse those who receive it.” The remainder of Pascal’s argument will be a defense of those who receive the Christian faith without rational proof.

Pascal’s wager argument defends the rationality of one’s choice to become a Christian without a rational demonstration of the existence of God. He is not trying to convince a hypothetical atheist that he too must become a Christian, but rather to explain to that atheist why Pascal’s decision to become or remain a Christian on fidestic grounds is rationally and ethically defensible. This is true apologetics, in the sense of being a defense of what one believes, rather than an attempt to win over or convert others, which is properly called proselytizing or evangelization. It is true that, in many modern discussions of Pascal’s wager, the argument is treated as an attempt to persuade unbelievers to convert, but this is not how Pascal himself conceived the argument.

His defense of his belief was not a mere abstract consideration, for Pascal was in close personal contact with nearly all the leading intellectuals of France, who included unbelievers as well as those who held only those theological theses that could be rationally demonstrated. The latter concerned Pascal no less than the former, as is proved by his repeated references to Descartes throughout his Pensées. Pascal felt compelled to defend not only why he remained a devout Catholic, but also why he did so on faith alone, rejecting Descartes’ demonstration of God’s existence. To be deeply religious was scandalous enough to some intellectuals, but to believe in religious truths while denying that the existence of God was rationally demonstrable may have seemed unconscionable. Pascal’s argument defends his intellectual integrity while maintaining his denial that God’s existence can be proved.

Pascal posits the dichotomy: “God is, or He is not,” where God has been described as an incorporeal being without limit or part. These are surely the only two logical possibilities, however we may choose to subdivide each of them. Pascal has already shown that reason cannot help us decide the question at all. “There is an infinite chaos which separated us.” The determination of God’s existence or non-existence is so remote from human experience and capacities that reason cannot so much as help us make one option or the other more or less probable. For this reason, Pascal depicts it as a coin toss “at the extremity of this infinite distance.” We will examine the justification of this model in more detail later. For now, we need only emphasize that reason is presumed to be completely powerless to render the existence of God more or less probable.

Naturally, this thesis will prove objectionable to most atheists (as well as many theists), who claim that physical science has at least probabilistic bearing on the question of whether God exists. Such claims are philosophically naïve, failing to recognize the impotence of physics before deeper metaphysical questions. Since God, who is hypothesized as a being without limit, could create any possible universe, no actual state of affairs in the physical universe is inconsistent or incongruent with the existence of God. Physical arguments against God’s existence are really arguments against His wisdom or benevolence, based on the determination that the natural order is not as wise or benevolent as it ought to be. This is a purely subjective determination, however, for the question of whether the natural order is wise presumes knowledge of what God would intend to accomplish through the natural order, and the question of benevolence presumes knowledge of what is good on a cosmological scale, as well as the unfounded assumption that creatures are owed a certain degree of comfort by their Creator. These criticisms amount to nothing more than a lowly human saying, “If I were God, I would do things differently,” which has no probative force whatsoever, given the limitations of human understanding.

If there is a rational proof of the necessity or likelihood of God’s existence, it is only to be found in metaphysics. Pascal denies the validity of such philosophical proofs, leaving him with the thesis that reason has nothing to say on the question of God’s existence. Even if we disagree with Pascal’s assessment of natural theology, we may at least acknowledge that, for many people, reason will not help decide this question, to the extent that they are not competent or inclined to evaluate the philosophical arguments regarding the existence of God. The question before us is whether such fideistic believers are intellectually justified in giving their assent to something they hold to be unprovable.

Since reason cannot decide the question of God’s existence, Pascal says that unbelievers should not fault Christians for believing in God, “for you know nothing about it.” He supplies the unbeliever’s retort: “No, but I blame them for having made, not this choice, but a choice… The true course is not to wager at all.” This is a position of agnosticism or negative atheism, which is common even among today’s non-theists. Such people admit that the existence and non-existence of God are unprovable, and conclude that the question is not something with which we should concern ourselves. Despite this seemingly reasonable neutrality, in practice it often resembles de facto atheism, as people live their lives on the assumption that God does not exist.

Pascal takes such false neutrality to task, insisting that “you must wager. It is not optional. You are embarked.” A man may profess with his mouth that the existence or non-existence of God is a matter of indifference to him, but in the way he conducts himself in life he will make a choice. Almost invariably, this choice is to live as though God did not exist. This is only natural, for when we say we are indifferent to whether some person exists, we generally act as though that person did not exist. In the case of God, a cosmologically supreme being, it is impossible to pretend that His existence or non-existence has no bearing on how we conduct our lives, since His scope of action would extend over everything. Those who believe in God, or at least strongly suspect He exists, will find it hard to convince themselves that this fact is unimportant to their lives. They may succeed in doing so only to the extent that they have convinced themselves that God’s existence is unlikely and so not worth worrying over. The necessity of the wager is therefore psychological, not logical. We may hold a purely agnostic stance as a philosophical opinion, but if the theoretical possibility of God’s existence has no effect on how we live, our real wager has been on non-existence.

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4. The Stakes of the Wager

The psychological appeal of the wager is made clear by Pascal’s articulation of the stakes. “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; and your nature has two things to shun, error and misery.” This may be expressed schematically as shown:

StakesReason (Knowledge)Will (Happiness/Felicity)

A man effectively stakes his spiritual soul, which is his reason and his will, on where he stands with respect to the existence of God. It is not that he will lose his faculties of reason and will if he chooses wrongly, but rather he places at stake the objects of these faculties, which are knowledge and happiness respectively. Pascal assumes that a person is rational enough to pursue knowledge and ethical enough to pursue felicity. Indeed, he would say that man’s nature is so constituted that this is necessarily the case. Reason accepts only what appears to be true, and rejects what seems to be error, while the will necessarily desires what it perceives to be good, and rejects what it perceives to be misery. Thus the possible winnings and losses are also defined by human psychology.

Ideally, reason might tell us that we should not wager at all in the absence of evidence, but the psychological necessity of the wager eliminates this option. The remaining options, choosing God’s existence or non-existence, are equally abhorrent to reason, since they both carry the possibility of error, which reason shuns. Reason, then, offers us no basis for preferring one wager over the other.

Pascal’s evaluation of the wager, therefore, focuses only on the stakes of the will’s happiness. He attempts to determine which wager is more likely to maximize the true felicity that every will seeks, but only some actually find. It is no answer to Pascal to say that we should not believe in God to make ourselves happy if He is actually non-existent, for he has already shown at the outset that the truth or falsity of God’s existence cannot be determined by reason, so this question has no bearing on our evaluation of the wager.

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5. Probability of Wager Outcomes

The remainder of Pascal’s argument is probabilistic in nature, and we need to consider two aspects of it. First, we must consider his assessment of the relative probability he assigns to each outcome. Second, we must consider his evaluation of the relative gain or loss in felicity implied by each outcome. The first task requires us to understand classical probability theory, which was a recent discovery in the seventeenth century, and Pascal himself had played an important role in its development.

Modern commentators on the wager argument often accuse Pascal of making some sort of basic error in probability analysis. On its face, it is astonishing that a mathematical genius of Pascal’s caliber could make an elementary error applying a theory he helped develop. It seems far more likely that his less gifted critics are in error. The reason for this incongruity is that modern mathematicians and scientists apply a different interpretation of probability theory than what was common in the seventeenth century.

Classical probability theory was grounded in the fundamental insight that all possible outcomes should be treated as equally likely. Probability could then be computed by enumerating the number of desirable outcomes and taking this as a fraction of all possible outcomes. There was no weighting of individual outcomes; greater or lesser probability of an event was expressed by representing it as a more or less numerous set of outcomes. As philosophically literate men, the mathematicians of the seventeenth century understood that possibility does not admit of degree. If a thing is possible, it has the power to exist, though it does not actually exist. When various things or events are possible, but only one of these possibilities can become actual, we say that the probability of each outcome is 1/N, where N is the total number of possibilities.

The assumption that all possibilities are equal is grounded in the belief that there is no basis for preferring one possibility over the other. This belief was articulated explicitly as an axiom by Pierre-Simon Laplace a century later, and came to be known as the principle of indifference. In the classical conception, probability theory is considered from a subjective perspective. We take a set of outcomes that are possible given our imperfect state of knowledge, and then enumerate how many of these outcomes match the state of affairs we desire or favor.

The intrinsically subjectivist conceptualization of classical probability leads to contradictions if we try to treat probability as a measure of objective reality. For example, suppose I am to reach into a bag and draw a colored marble. What is the probability that the marble will be blue? Since I know nothing about the color distribution of marbles, I have no basis for estimating the relative frequency of blue marbles, and will assign a 50% probability to blue, and 50% to non-blue. Similarly, if I am asked if it is green, I will assign 50% to green and 50% to non-green. If I am then asked it is red, it is clear that my probability assignments cannot be objectively consistent, since I have already assigned 50% to blue and 50% to green.

Such inconsistencies led nineteenth-century mathematicians to posit instead a frequentist model of probability, where the probability of an outcome is instead defined by how often it actually occurs after multiple iterations of a test. Frequentism has the advantage of permitting objectively defined probabilities, but has a practical disadvantage in that probabilities can be determined only after multiple tests are made. Furthermore, such an empirical requirement takes probability theory outside the realm of pure mathematics.

There was never anything wrong with classical probability; rather, it addressed a different set of questions than those answered by frequentist probability. In real life, we often have no way of knowing in advance what the “true” frequency of occurrence is for some phenomena. In such situations, we can do no better than to estimate probability according to classical principles, within the constraints of our knowledge. Such knowledge-contingent probabilistic analysis is used even today in science, especially in the application of Bayesian probability. (It is true that some commentators describe Bayesian probability as “objective,” but this is only an assertion that its rules are grounded in objective logic, not a denial that Bayesian analysis is knowledge-contingent.)

The classical interpretation of probability is appropriate for Pascal’s wager argument not merely because it was the dominant opinion of the time, but because the nature of the scenario demands it. Pascal has already established that we have no way of knowing the true likelihood of God’s existence from reason alone, so only a subjective notion of probability, with the assumption of limited knowledge, is possible. A frequentist notion is obviously inapplicable, since we cannot have successive realities where the eternal God exists and does not exist, and in any event, we can only observe this one universe in our brief lifespan. It is also incoherent to posit an objective anterior probability for an eternal Being that is the First Principle of all that exists.

Furthermore, the notion of God having an objective probability of existence that is fractional is a metaphysical impossibility. God is posited as a metaphysically necessary being, an essence for which non-existence is impossible, since it includes its own act of existence. An existent yet contingent God would be a contradiction, since God is posited as the grounding of all contingent reality. If a god were merely contingent, the necessary Being upon which he ultimately depends should instead be called God in the proper sense of a cosmologically supreme, unlimited Being. There are at most two metaphysically coherent objective probabilities for God’s existence: 1 for an existent God; 0 if God did not exist. In other words, the only way God does not exist is if it is impossible for Him to exist.

Reason cannot give knowledge of the objective probability of God’s existence. If it could, such knowledge would automatically decide the question. Instead, in our ignorant state, we can only assign subjective probabilities to the two subjectively possible outcomes. In the absence of any rational basis for preferring one outcome over the other, Pascal assigns each outcome a possibility of 50%.

Like any estimate of subjective probability, this will lead to contradiction if compared with the probabilities assigned to overlapping questions. For example, suppose we ask instead: “Does the Hindu pantheon exist?” Applying similar principles, we might assign 50% probabilities to the affirmative and negative answers. We might do the same for the ancient Greek pantheon, only to find that we have already assigned 50% probability to the existence of God, and 50% to the Hindu pantheon. This is not due to any flaw in the argument. It is not a false dichotomy to say that God must either exist or not exist. Rather, the contradiction arises only if we treat a succession of subjective probability assessments as though they were objective measures of probability, from which we ought to expect consistency.

Subjective probability is not useful for determining the true, objective probability of an occurrence, but this is not what it pretends to do. We have already accepted at the outset that we cannot determine the likelihood of God’s existence through reason, and Pascal is not trying to get around this impossibility through probability analysis. Subjective probability is designed to maximize our chances of getting a favorable outcome with limited knowledge, which is why it is constantly used by gamblers to great success. If I am asked to wager on whether a marble in a bag is blue or non-blue, I should demand one-to-one odds in order to minimize my losses. It does not matter that this estimate of probability is objectively inconsistent with other hypothetical wagers I might have been asked to make. I must deal with the present wager, and act in a way that maximizes gain or minimizes loss.

Pascal has already established that a wager on whether God exists is a practical psychological necessity for rational, ethical human beings. Since this wager is necessary, we should act in the wisest possible way when placing our bet, irrespective of what we might have done if the wager was something else. If you wish to make a wager regarding the Hindu pantheon as well, you may do so in your own time, but that will not excuse you from the necessity of placing a wager on the existence of the broadly conceived God that Pascal posits. The necessity, recall, is not something that Pascal imposes on his reader as some kind of intellectual or moral imperative. Rather, as a matter of fact, everyone does effectively make such a wager whether he likes it or not. Since everyone is already betting, Pascal argues, the least we can do is try to optimize our bet.

One may object that Pascal has not clearly defined which God he is positing, as there are various mutually inconsistent conceptions of God, even within the monotheistic traditions. If we choose the wrong conception of God, we might be even worse off than if we erroneously wager on non-existence. For Pascal, the question “which God?” would be inapt, since he has already assumed that we cannot know what God is, which is precisely what such theological controversies attempt to decide. All he has defined about God is in the negative: the absence of limit; the absence of corporeality. This is really not so strange, since in the Western tradition, what is essential to God is His transcendence of all limits, as we express with terms such as omnipotence, omniscience, and omnibenevolence.

It is a bit of obfuscation by atheists to affect ignorance of what one means by God, when unbelievers frequently evince a very definite conception in their religious denials. This is why Western atheists will say, “I do not believe in God,” rather than “gods” or “a god.” The Hindu and Greek pantheons barely rate a mention in their act of unbelief, which is emphatically directed against the God conceived by Western monotheism. In the twenty-first century no less than the seventeenth, the proper answer to the atheist who asks, “What do you mean by God?” is, “Why, the very same thing you mean when you deny God.”

Accordingly, Pascal does not address the question of what we should wager regarding the various options offered by pagan religions. Since his is a psychological argument, he assumes what his reader assumes, and no seventeenth-century European considered paganism to be a realistic possibility. The paganism question had been decided unambiguously centuries earlier, and even today it is not seriously considered by atheist intellectuals, however much they may reject other aspects of the legacy of monotheism. It is disingenuous to raise the possibility of such alternatives when one effectively assigns them a probability of zero. Indeed, the atheist raises such alternatives not because he assigns any credence to polytheism or animism, but only to attack the one form of religion he considers a legitimate threat: monotheism. Nonetheless, we will address the question of non-monotheistic religions as an addendum to our discussion, for the sake of completeness. The resolution of such a question has no direct bearing on the present question, however. While it is impossible for both the Abrahamic God and the Hindu pantheon to actually exist, it is no contradiction to say that belief in God is more prudent than disbelief and that belief in the Hindu pantheon is more prudent than disbelief.

Pascal’s assignment of 50% probability to the existence of God seems unintuitive to those accustomed to an objective or frequentist interpretation of probability, but we have seen that such notions are inapplicable here. The objective inconsistency of this probability with other conceivable wagers is not a flaw in the argument, but an unavoidable consequence of the subjectivist conception of probability. As much as we may lament our insufficient knowledge, the necessity of the wager compels us to evaluate the situation according to subjective probabilistic principles. We are not determining the objective probability that God really exists, which is impossible to do, but are judging whether it is more prudent, with respect to maximizing our felicity, to live as though God does or does not exist.

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6. Evaluation of the Wager

The main substance of Pascal’s argument involves his evaluation of the possible outcomes implied from accepting or rejecting the existence of God. The magnitude of potential gain or loss, together with the odds, determines how much we should wager, as in any game of chance. Pascal models the game as a coin toss, for reasons discussed above, but as we shall see, the argument would work just as well if God’s existence is assigned any finite probability.

“If you gain, you gain all; if you lose, you lose nothing.” When the wager is cast in these terms, it is obvious that we should bet on God’s existence, given the necessity of the wager. Still, though we are required to bet, perhaps we should be cautious and wager very little. Pascal replies to this objection with an argument from the newly developed theory of infinite number.

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 (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance 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. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.

In this compact passage, Pascal goes over the range of scenarios that would require us, as a matter of prudence, to bet on the existence of God. In the process, he articulates the stakes and the rewards. What we wager is our very life, that is, our happiness, as discussed earlier. Our potential for gain is an infinite, infinitely happy life. If it turns out we chose wrong by betting on God’s existence, we lose only the life we would have lost anyway, so we lose nothing. Therefore, it is foolish to bet against God’s existence as long as we acknowledge there is a finite probability of this infinite gain.

The force of this argument depends on the opening thesis: “If you gain, you gain all; if you lose, you lose nothing.” Surely, such lopsided outcomes make it a “no-brainer” that we should favor the supposition of God’s existence. It would require a profoundly irrational hostility to the notion of God to disbelieve even when nothing is to be gained by disbelief. This is the stronger portion of the thesis, for everything the atheist thinks he gains by shunning religious duty will be lost. All his sensual pleasure, ethical felicity, grasp of truth, and sense of accomplishment will surely be annihilated by death if God does not exist, and he will not have even the memory of their savor. While eternal life is theoretically conceivable in an atheistic context, no atheist such as Pascal addresses actually believes in an afterlife, so we dismiss this subjective possibility. Surely, then, “if you lose, you lose nothing,” is a correct assessment of affairs, though the folly of the shortsighted world is such that it pretends that we should cling to the happiness of this life even though it will certainly all be taken from us, as though it never happened.

The first portion of the thesis—“if you gain, you gain all”—ought to be given some modification. Even if we accept the supposition that God exists and act accordingly, it is far from necessary that we should gain eternal life even if God does in fact exist. As is said in the New Testament: “You believe that there is one God. You do well: the demons also believe and tremble.” (James 2:19) Recall, however, that Pascal also supposes that the wagerer is an ethically positive person, genuinely seeking his true felicity. Still, even the believer in God’s existence may stumble into ethical error. Further, as Pascal himself was fond of observing, the mere knowledge of what one ought to do does not necessarily translate into the will to actually love and serve God. So, if we are to be fair in our assessment of the wager, we must consider that the probability of gain is considerably less than the 50% posited, since living as though God exists as best we can does not necessarily guarantee eternal life if God exists, though it may at least give us the felicity of contemplating the supreme Good.

As long as we acknowledge that there is a finite probability of the gain of eternal life, it is foolish not to bet everything on God’s existence, since the life we wager will be lost anyway if we are wrong. The infinite magnitude of the reward makes this wager prudentially necessary even for an arbitrarily small but finite probability of gain. In the absence of definite knowledge that an eternal afterlife of bliss is impossible, it is prudent to prefer the supposition that makes such felicity possible (the existence of God) as opposed to the supposition that closes off that possibility (the non-existence of God) without giving us anything in return.

This argument does not imply that it is impossible for an atheist to gain eternal life. Rather, it is impossible to gain eternal life if God does not exist. (This is generally accepted in Western thought, though we will deal with Eastern exceptions in the addendum.) If the atheist does gain eternal life because of his ethical merits, it will be God who gives it to him, which only confirms the greater wisdom of supposing that God exists. Pascal will later address the atheist who cannot believe in God conscientiously, even after recognizing that atheism is an imprudent wager.

Pascal’s wager is in no way dependent on the supposition that unbelievers will receive any sort of eternal punishment. His argument is entirely on the positive side of the ledger, comparing the possible gain of eternal life against the loss of nothing. We have everything to gain, and nothing to lose, so it would be foolish not to be bold enough to place our bet on God’s existence, come what may.

The evaluation of outcomes, in summary, may be presented as follows:

WagerStakesOutcome if correctOutcome if incorrect
God exists1 lifePossible infinite lifeLose 1 life (sacrifice)
God does not exist1 lifeLose 1 life (death)   ?

The worst case scenario for the one who bets that God exists is no worse than the outcome for the one who correctly supposes God does not exist. The believer who freely devotes his life to God is no worse off than the unbeliever who will have his life taken away by death. Pascal does not address what happens to the unbeliever who is incorrect in his guess. If we allow that eternal life is possible for the atheist, we have the counterintuitive result that he is better off being wrong than being right, a scenario that hardly commends the wisdom of atheism. If eternal life is not possible for the atheist, then his best possible outcome is no better than the theist’s worst outcome, making atheism a thoroughly foolish bet.

There remains one aspect of the evaluation in doubt, namely that the theist has a finite probability of eternal life. It is not a logical or metaphysical necessity that an existent God should grant eternal life to anyone at all. We accepted a 50% subjective probability for the existence of God only because reason cannot help us decide the question at all, so the principle of indifference applies. The notion of eternal life, however, is at least partly accessible to reason, as it may be conceived in terms of corporeal existents. This opens the door to philosophical arguments that may enhance or diminish the probability of eternal life. In the austere Jansenist spirituality that Pascal admired (though not uncritically), salvation was far from a foregone conclusion even among the most devoutly observant Catholics. If eternal life is so difficult and uncertain even for those who believe fervently, what hope of gain can there be for one who believes only out of some prudential calculus?

Though we acknowledge that there is no proof strong enough to eliminate a small, finite probability of eternal life for the wagerer, we might still hesitate to see a benefit to betting on God’s existence. After all, if the odds are a trillion to one against eternal life for a given individual, it may seem pointless to sacrifice even the fleeting pleasures of this life for something so unlikely or uncertain. We are being asked to give up palpable realities for the sake of a purely theoretical possibility, which hardly seems prudent.

Pascal addresses this type of concern in quantitative terms, arguing that the contrast between the uncertainty of gain and the certainty of what is staked does not counteract the infinite disparity in possible outcomes.

It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from the fact that there is an infinite distance between them. 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. This is demonstrable; and if men are capable of any truths, this is one.

Pascal reminds us that we are always staking what is certain for a merely hypothetical gain in any ordinary risk or wager. If we are rationally justified in doing so for uncertain finite rewards, we are amply justified in doing so for a potentially infinite reward. The stakes are always certain, so the apt comparison for risk assessment is between the certainty of gain and the certainty of loss. In the case of even odds, the smart strategy is to play even, that is, to place stakes equal to the possible gain. Here, the certain stake is exactly equal to the uncertain gain, refuting the idea that there is an infinite disproportion between the certainty of one and the uncertainty of the other that would counteract the ratio of outcome probabilities. If this is the wise strategy when the reward is finite, it is infinitely more compelling that we should bet similarly when required to place only finite stakes for an infinite reward.

Pascal’s argument should be quite persuasive to an experienced gambler. If someone offers to pay fantastically long odds on a small bet, this is frequently perceived as an excellent opportunity for substantial gain at minimal risk. Bookmakers are well aware of this fact, which is why they usually do not offer odds longer than 50-1 or 100-1, even for wildly improbable events. A great opportunity for a bettor is an unacceptable risk from a bookmaker’s perspective. Casinos and games of chance were quite popular among the aristocracy in Pascal’s time, so they would have understood the appeal of high returns on small bets.

However, note that Pascal is assuming even chances of winning in his game, though we have observed that the probability of reward (as distinct from the probability of God’s existence) may be much less than 50%. This diminishes the “infinite force” of his argument, and we are forced to weigh the uncertainty of gain against the risk of loss differently.

Let us evaluate the probability of eternal life for the believer as p, a small but finite quantity. For the unbeliever, the probability of reward is q, where q may be zero or some positive quantity less than p, on the natural assumption that those who are less concerned with eternal life are less likely to attain it. We denote the infinite value of eternal life as E, while the value of the temporal life we stake is some finite quantity l. The table of payoffs is as follows.

Eternal lifeLoss of life
Wagers“God exists”pE(1 - p)(-l)
“God does not exist”qE(1 - q)(-l)

The payoff expectation for the theistic wager is pE + (1 – p)(-l) = (pE + pll). Since p and l are finite, the last two terms are negligible, leaving a payoff of pE. For the atheist, the payoff is qE if q is greater than zero, or -l if q is zero. When q is zero, the theistic wager is infinitely superior in expectation value; otherwise, it is superior by the ratio p/q. Even if we compare them solely in the “loss of life” column, the theistic wager has an edge, by the ratio (1 – q)/(1 – p). Since p > q, this ratio is greater than one. The theistic wager therefore remains superior, even if we make p much less than 0.5, and even if we allow that atheists may attain eternal life.

Pascal’s wager argument successfully demonstrates that theistic belief is a prudent strategy for maximizing felicity, in the absence of any rationally demonstrative evidence for or against the objective existence of God. As stated at the outset, Pascal’s wagerer is presumed to be concerned with his ethical felicity, which is why he cannot be indifferent to the prospect of eternal beatitude. However, by the same token, a basically ethical person might feel that he cannot bring himself to pretend to believe in God solely out of some calculus of self-interest. Accordingly, the last part of Pascal’s discourse will address such scruples.

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7. Making the Wager in Good Conscience

There may be an ethically well-intentioned person who accepts the validity of this probabilistic argument, yet nonetheless finds he cannot bring himself to believe in God. A person cannot introspectively pretend to believe in God if he does not really believe, even if he admits that it would be advantageous for him to believe. Pascal advises such a person to at least recognize that this inability to believe is a personal deficiency, since it resists following the path of prudential thought. “Endeavour then to convince yourself, not by increase of proofs of God, but by the abatement of your passions.” Here Pascal is drawing upon his own ascetic experience. If a person recognizes that religious faith is prudentially desirable, yet finds he cannot believe, the remedy for his situation is to emulate those who already have faith.

In order to provide such practical guidance, it is now necessary to appeal to a determinate faith tradition, and Pascal naturally refers to the Catholic religion, which he holds to be true. Others who have come to this faith “began by acting as if they believed, taking the holy water, having masses said, etc.” This is not an exhortation to hypocrisy, but rather it is a recognition of how religious education actually worked. External practices often preceded interior contemplation. The external solemnities helped subdue the passions and make a person more disposed to the worship of God. The role of liturgy in developing interior religious life is recognized in the Catholic adage, lex orandi, lex credendi. “Even this will naturally make you believe, and deaden your acuteness.” Speaking from experience, Pascal holds that the natural desires of the soul—from sensual pleasures to worldly vanity—are the obstacles that impede religious faith. It is only when these faculties are deadened, so to speak, that faith can take root.

Pascal urges that the pilgrim soul need not fear the sacrifice of his natural passions, for this will do no ethical harm to him; quite the contrary, he will grow in virtue. “You will be faithful, honest, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others?” As the pilgrim makes his first steps toward faith, he will see that his gain in virtue, at least, is certain, whereas the vicious pleasures he has sacrificed are worth nothing at all, so he has lost nothing. In other words, the proof of the wisdom of his wager is found in the life of faith, the experience of which causes us to evaluate things rightly. As we progress in faith, we more clearly realize how the worldly life we have sacrificed was worth so very little in comparison to what we have gained.

The point of this line of argument is not that moral virtue is impossible without faith, but rather that the denial of the passions, which helps nurture faith, at the same time nurtures virtue. This means that Pascal’s prescription for the one who cannot believe is compatible with good conscience. What is more, if this path is followed, nothing of any worth will be relinquished, and as we progress down this road, we perceive more clearly how the wager of faith did not require us to lose anything at all, making its wisdom all the more manifest.

Although Pascal’s advice to abate the passions is extraneous to his wager argument, it nonetheless contains the core of the attitude needed to appreciate the argument. Pascal presumes his reader is basically ethical, or is at least striving to be. Such a person is concerned with maximizing ethical felicity rather than superficial pleasures, and this attitude enables him to evaluate the wager outcomes properly. If you wish only to maximize vanity and luxury, atheism may serve well to provide such temporary, vulgar “happiness.” Pascal assumes his reader has enough sense to seek a more substantive, lasting happiness if it is available. The glories of heaven will have no appeal to one who seeks only the pleasures that cattle can appreciate. The “happiness” maximized by Pascal’s wager is necessarily ethical, not merely sensual, for our wagerer is presumed to be a rational man, not a brute beast in the form of a man.

This presumption of basic moral philosophy in the wagerer seems to limit the applicability of Pascal’s argument. It will have no effect on the nihilist, who denies there is any such thing as felicity beyond that which he wills or desires. It will have no effect on the hedonist who holds we should maximize the pleasure of the moment without giving thought to the future. This does not mean, however, that the argument is circular, appealing only to those favorably inclined to theism. The basic ethics presumed by Pascal can be found in a variety of pre-Christian cultures and philosophical schools of the East and West. The insight that animal desires often distract man from his greater good is not peculiar to monotheism, but is a bit of wisdom independently attained by countless sages throughout history. Pascal would not be disappointed if told that his argument presumes so much ethics on the part of the wagerer. For him, the question of believing in God was a moral decision, not a mere intellectual problem. It is fitting, then, that only those who seek moral virtue are disposed to persuasion by the wager argument for believing in God. By the same token, Pascal was obligated to persuade his reader that acceptance of the wager is compatible with good conscience.

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8. Effectiveness of the Argument

The perceived effectiveness of Pascal’s wager argument varies according to what readers think Pascal intended to accomplish by it. Some have construed it as a mathematical or logical proof of the existence of God, though Pascal explicitly assumed that such a proof is impossible. Some have tried to construct the argument in terms of objective probabilities, which Pascal declared to be unknowable. Others have cast it in utilitarian game theoretic terms, though Pascal presumed the wagerer is concerned with higher ethics rather than mere pleasure. Some evangelists have used the wager argument to try to persuade non-believers to convert, though Pascal used it primarily to defend the reasonableness of his own act of faith. These faulty constructions of Pascal’s argument are generally convincing only to those already favorably disposed to the conclusion, and therefore have little probative force.

Pascal’s argument accomplishes only the author’s original intent, which was to demonstrate that belief in God on fideistic grounds is consistent with reason and favored by prudence. It does not show that belief in God is required by reason, for that would amount to a rational proof, the possibility of which Pascal denies. The wager argument uses psychological and ethical persuasion rather than rational demonstration. When Pascal says the wager is necessary, he is speaking not of logical necessity but of a psychological necessity, grounded in our need to resolve questions of truth and happiness. When defining the stakes of the wager, he assumes we have a rational intellect that seeks truth and an ethical will that desires genuine, lasting happiness. Since reason can teach us nothing about the existence of an incomprehensible, infinite God, we must turn to a prudential evaluation of ethical outcomes.

Evaluation of the wager outcomes requires a subjectivist interpretation of probability, since reason cannot teach us the objective probability of God’s existence, which at any rate cannot be fractional for a metaphysically necessary Being. Although Pascal believed that the unbeliever would certainly be condemned (Pensées, 239), his evaluation of the wager looks only at the positive side of the ledger, namely the prospect of eternal life. The strategic superiority of the theistic wager holds as long as we allow that the theist’s probability of eternal life is greater than that of the atheist who neither expects nor attempts to attain it. This supposition is no more fantastic than the belief that the man aiming for the mark is more likely to hit it.

In my view, the most instructive part of argument is: “if you lose, you lose nothing.” Most secular thinking is mired in the folly that the happiness of this life amounts to anything in the face of death. Countless wise men throughout history, East and West, have exposed the shortsightedness of this view. Once you contemplate the eternal, this world becomes as nothing. Yet the intellectuals today frequently devote their higher faculties and learning to the vulgar endeavor of making this life a bit more physically comfortable. For a common, coarse person to be concerned with such animal pleasures is understandable, but we should expect better from those who spend time immersed in the world beyond sensation. The hedonist is a fool even by secular standards of utility, for the physical pleasure he prizes will taken away by death, and not even a memory of such pleasure will remain. The corpse will not know if it lived well or poorly, happily or unhappily; all is for nought. The atheist’s highest expectation is no better than what the theist gets if he is wrong. Prudence unilaterally favors theism.

Yet most secular unbelievers are not disposed to contemplate the eternal or to regard lower pleasures with contempt, so the argument is less likely to persuade an atheist to become a theist, but better suited to persuade the theist to remain so. Pascal teaches the theist: there is no advantage to atheism, so why throw away the prospect of eternal life? For what? For the pleasures of this world, which are nothing? Those who are already educated in faith or in classical ethical wisdom cannot easily unlearn how worthless worldly pleasures are. Those who are ignorant, on the other hand, might not ever come to experience the realization of how foolish it is to cling to fleeting pleasures.

Pascal’s argument is insightful for its emphasis on the ethical grounds of theistic and atheistic belief; after all, most people are theists or atheists for ethical reasons, not because we are robots who coldly pursue what is rigorously demonstrated. Most people are incapable of properly evaluating philosophical arguments about the existence of God, and that includes most modern intellectuals, who have an impoverished understanding of rational demonstration, due to their mistaking of symbolic logic for real logic. The fact that belief or disbelief in God is primarily an ethical decision comports well with the Christian doctrine that faith or its absence should have moral consequences. Although atheists prefer to cast their unbelief as a purely intellectual decision, they can scarcely resist the urge to make moral condemnations of religion, betraying their own ethical concerns, in particular the desire not to be subject to any external moral authority. The fact that atheists are ethical does not diminish Pascal’s argument; on the contrary, he is counting on it, in order for them to evaluate the wager properly and see that their intellectual belief is in tension with their higher moral aspirations.

The continuing prevalence of religious belief from remote antiquity into the modern era may be construed as an anthropological confirmation that it is indeed a successful strategy for maximizing felicity. Even atheists must devise consolations that are surrogates for religion, such as the vain hope that the progress of the human race will continue in perpetuity, substituting a collective immortality for individual immortality. Yet these consolations, unlike the hopes of the faithful, are demonstrably falsified by physics, and are at odds with the modern liberal belief in the immeasurable value of the irreplaceable individual, which arose historically from the Christian tradition.

Nonreligious liberalism is doomed to be ethically disjointed, at once declaring the supreme moral worth of the individual and that only the collective endures. The Communists, at least, were consistent in making the collective supreme, while the secular West finds itself in a love-hate relationship with liberal statism, as is proved by recent political history. If we were to cleave only to our nobler inclinations, we would realize that only the prospect of personal immortality is consonant with the transcendent dignity that we recognize in each man. If there is no God, it is foolish to treat any ephemeral collection of matter as though it were a son of God, born with inviolable rights and other divine prerogatives. Although unbelievers like to say bleak things about the insignificance of humanity, they mostly think highly of themselves, of their work, and of those they love. Their intellectual life is completely at odds with their ethical life.

Pascal attempts to bridge this gap by pointing out the powerlessness of reason before divine questions, and urging all men of good will to look to their nobler ethical nature for the more prudent response to such questions. Such a prudential calculation, it turns out, will not force us to become hypocrites, but will actually facilitate the development of the classical virtues. In fact, our reason and our will shall be in greater harmony than they would be if we pursued virtue only as a futile exercise in the face of annihilation. It is the atheist who has to merely pretend that life is objectively meaningful in the moral order; the theist truly believes that this is so, so only in him is there a consistency between metaphysics and higher ethics.

Addendum: Alternatives to Monotheism

© 2011 Daniel J. Castellano. All rights reserved.