Pascal’s wager

Argument for adopting a divinely favoured way of life -named after French philosopher, mathematician, physicist and pious gambler Blaise Pascal (1623-1662) who stated it in his Pensees (§233) – but apparently stemming from Islam.

One statement of it (not Pascal’s) is this. Let the utility of a policy be the gain it promises multiplied by the probability of getting it. Let the gain of a life of pleasure be some finite quantity X, certainly achievable. But if God exists he may punish our sins with infinite suffering in Hell. Let the probability that God both exists and will do this be as small as you like, but finite (above zero).

The product of an infinite loss and a finite probability is still infinite. Therefore the utility of choosing pleasure over virtue is X minus infinity; that is, is infinity. But the utility of choosing virtue is at worst (that is, if God will not reward us, and death is the end) minus X (that is, the certain loss of X units of pleasure), which is finite.

Therefore, it must be rational to choose virtue; that is one should act as if God exists because any positive probability of going to Hell and suffering infinite punishment cannot be balanced by the finite advantages gained by sin.

Unfortunately, if the argument proves anything it proves too much, since it could be reiterated for infinitely many different hypotheses about God’s preferences and intentions.

Source:
P T Landsberg, ‘Gambling on God’, Mind (1971)

The wager uses the following logic (excerpts from Pensées, part III, §233):

  • God is, or God is not. Reason cannot decide between the two alternatives
  • A Game is being played… where heads or tails will turn up
  • You must wager (it is not optional)
  • Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing
  • Wager, then, without hesitation that He is. (…) There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.
  • But some cannot believe. They should then ‘at least learn your inability to believe…’ and ‘Endeavour then to convince’ themselves.

Pascal asks the reader to analyze humankind’s position, where our actions can be enormously consequential, but our understanding of those consequences is flawed. While we can discern a great deal through reason, we are ultimately forced to gamble. Pascal cites a number of distinct areas of uncertainty in human life:

Category Quotation(s)
Uncertainty in all This is what I see, and what troubles me. I look on all sides, and everywhere I see nothing but obscurity. Nature offers me nothing that is not a matter of doubt and disquiet.[6]
Uncertainty in man’s purpose For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either.[7]
Uncertainty in reason There is nothing so conformable to reason as this disavowal of reason.[8]
Uncertainty in science There is no doubt that natural laws exist, but once this fine reason of ours was corrupted, it corrupted everything.[9]
Uncertainty in religion If I saw no signs of a divinity, I would fix myself in denial. If I saw everywhere the marks of a Creator, I would repose peacefully in faith. But seeing too much to deny Him, and too little to assure me, I am in a pitiful state, and I would wish a hundred times that if a god sustains nature it would reveal Him without ambiguity.[6]We understand nothing of the works of God unless we take it as a principle that He wishes to blind some and to enlighten others.[10]
Uncertainty in skepticism It is not certain that everything is uncertain.[11]

Pascal describes humanity as a finite being trapped within an incomprehensible infinity, briefly thrust into being from non-being, with no explanation of “Why?” or “What?” or “How?” On Pascal’s view, human finitude constrains our ability to achieve truth reliably.

Given that reason alone cannot determine whether God exists, Pascal concludes that this question functions as a coin toss. However, even if we do not know the outcome of this coin toss, we must base our actions on some expectation about the consequence. We must decide whether to live as though God exists, or whether to live as though God does not exist, even though we may be mistaken in either case.

In Pascal’s assessment, participation in this wager is not optional. Merely by existing in a state of uncertainty, we are forced to choose between the available courses of action for practical purposes.

Explanation

The Pensées passage on Pascal’s wager is as follows:

If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is….

…”God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not, then, reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. 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. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

“That is very fine. Yes, I must wager; but I may perhaps wager too much.” Let us see. 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.[8]

Pascal begins by painting a situation where both the existence and non-existence of God are impossible to prove by human reason. So, supposing that reason cannot determine the truth between the two options, one must “wager” by weighing the possible consequences. Pascal’s assumption is that, when it comes to making the decision, no one can refuse to participate; withholding assent is impossible because we are already “embarked”, effectively living out the choice.

We only have two things to stake, our “reason” and our “happiness”. Pascal considers that if there is “equal risk of loss and gain” (i.e. a coin toss), then human reason is powerless to address the question of whether God exists. That being the case, then human reason can only decide the question according to possible resulting happiness of the decision, weighing the gain and loss in believing that God exists and likewise in believing that God does not exist.

He points out that if a wager were between the equal chance of gaining two lifetimes of happiness and gaining nothing, then a person would be a fool to bet on the latter. The same would go if it were three lifetimes of happiness versus nothing. He then argues that it is simply unconscionable by comparison to betting against an eternal life of happiness for the possibility of gaining nothing. The wise decision is to wager that God exists, since “If you gain, you gain all; if you lose, you lose nothing”, meaning one can gain eternal life if God exists, but if not, one will be no worse off in death than if one had not believed. On the other hand, if you bet against God, win or lose, you either gain nothing or lose everything. You are either unavoidably annihilated (in which case, nothing matters one way or the other) or miss the opportunity of eternal happiness. In note 194, speaking about those who live apathetically betting against God, he sums up by remarking, “It is to the glory of religion to have for enemies men so unreasonable…”

Inability to believe

Pascal addressed the difficulty that ‘reason’ and ‘rationality’ pose to genuine belief by proposing that “acting as if [one] believed” could “cure [one] of unbelief”:

But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavour then to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith, and do not know the way; you would like to cure yourself of unbelief and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness.

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