Even if the chance of God existing is small, as long as it is greater than zero, the expected value of believing is infinite. While the Wager has its advocates, there are many objections. There are many religions, and believing in the God of one religion might prevent gaining the infinite rewards of another religion. Assuming the probabilities of Christianity, Islam, and atheism are greater than zero, we get confusing expected values.
You might think the decision matrix tells us that believing either religion is a better bet than believing atheism. Thus, all options seem to have the same expected value. A common response to the many-gods objection can be summarized in two words: probability matters. It matters even when dealing with infinite values. Christianity and Islam actually do not have the same expected value—wagering on the more probable religion gives you a higher chance at an infinite good, and so has a higher expected value.
Most philosophers reject doxastic voluntarism , the view that we can directly control our beliefs. This might be indirect control, like the control you could exercise over your political beliefs by changing the news sources you read. A second response—which Pascal himself favored—frames the wager in terms of action , rather than belief.
The wager gives you a reason to commit to God—by going to church, praying, and immersing yourself in a religious community—rather than trying to directly believe in God.
It seems like forming a belief on the basis of a wager would violate evidentialism , the view that we should proportion our beliefs to the evidence. We should believe because of evidence, not because a belief is beneficial. This is because more than one belief-attitude fits your evidence. Also, we can again make the wager about a commitment to God, rather than about belief. Since evidentialism applies to belief and not action, you could then take the wager without violating evidentialism.
The wager is unique because it leads us to consider many kinds of reasons for belief, including evidence, arguments, risks, and rewards. Instead of focusing on whether it is true or false that God exists, the wager concerns whether belief in God is beneficial, or pragmatic, for the believer.
The Allais and Ellsberg paradoxes, for example, are said to show that maximizing expectation can lead one to perform intuitively sub-optimal actions. So too the St. Petersburg paradox, in which it is supposedly absurd that one should be prepared to pay any finite amount to play a game with infinite expectation.
That paradox is particularly apposite here. Various refinements of expected utility theory have been suggested as a result of such problems. Or we might consider suitably defined utility ratios , and prefer one option to another if and only if the utility ratio of the former relative to the latter is greater than 1—see Bartha If we either admit refinements of traditional expected utility theory, or are pluralistic about our decision rules, then premise 3 is apparently false as it stands.
Indeed, Bartha argues that his ratio-based reformulation answers some of the most pressing objections to the Wager that turn on its invocation of infinite utility. Finally, one might distinguish between practical rationality and theoretical rationality. One could then concede that practical rationality requires you to maximize expected utility, while insisting that theoretical rationality might require something else of you—say, proportioning belief to the amount of evidence available.
But when these two sides of rationality pull in opposite directions, as they apparently can here, it is not obvious that practical rationality should take precedence.
For a discussion of pragmatic, as opposed to theoretical, reasons for belief, see Foley A number of authors who have been otherwise critical of the Wager have explicitly conceded that the Wager is valid—e. Mackie , Rescher , Mougin and Sober , and most emphatically, Hacking Their point is that there are strategies besides wagering for God that also have infinite expectation—namely, mixed strategies, whereby you do not wager for or against God outright, but rather choose which of these actions to perform on the basis of the outcome of some chance device.
The expectation of the entire strategy is:. It can be argued that the problem is still worse than this, though, for there is a sense in which anything that you do might be regarded as a mixed strategy between wagering for God, and wagering against God, with suitable probability weights given to each. Suppose that you choose to ignore the Wager, and to go and have a hamburger instead. Still, you may well assign positive and finite probability to your winding up wagering for God nonetheless; and this probability multiplied by infinity again gives infinity.
So ignoring the Wager and having a hamburger has the same expectation as outright wagering for God. Even worse, suppose that you focus all your energy into avoiding belief in God.
Still, you may well assign positive and finite probability to your efforts failing, with the result that you wager for God nonetheless. In that case again, your expectation is infinite again. Rather, there is a many-way tie for first place, as it were.
All hell breaks loose: anything you might do is maximally good by expected utility lights! He argues that an atheist or agnostic has more than one opportunity to follow a mixed strategy. Returning to the first example of one, suppose that the fair coin lands tails. You are back to where you started. But since it was rational for you to follow the mixed strategy the first time, it is rational for you to follow it again now—that is, to toss the coin again.
And if it lands tails again, it is rational for you to toss the coin again … With probability 1, the coin will land heads eventually, and from that point on you will wager for God. Similar reasoning applies to wagering for God just in case an n-sided die lands 1 say : with probability 1 the die will eventually land 1, so if you repeatedly base your mixed strategy on the die, with probability 1 you will wind up wagering for God after a finite number of rolls.
Robertson replies that not all such mixed strategies are probabilistically guaranteed to lead to your wagering for God in the long run: not ones in which the probability of wagering for God decreases sufficiently fast on successive trials. However, Easwaran and Monton counter-reply that with a continuum of times at which the dice can be rolled, the sequence of rolls that Robertson proposes can be completed in an arbitrarily short period of time. In that case, what should you do next?
Easwaran and Monton prove that if there are uncountably many times at which one implements a mixed strategy with non-zero probability of wagering for God, then with probability 1, one ends up wagering for God at one of these times. And they assume, as is standard, that once one wagers for God there is no going back.
They concede that imagining uncountably rolls of a die, say, involves an idealization that is surely not physically realizable.
But they maintain that you should act in the way that an idealized version of yourself would eventually act, one who can realize the rolls as described—that is, wager for God outright. There is a further twist on the mixed strategies objection. But we have seen numerous reasons not to grant all his premises. But if it is, according to the mixed strategies objection, all hell breaks loose. Hence, it seems that each action that gets infinite expected utility according to Pascal similarly gets infinite expected utility according to you ; but by the previous reasoning, that is anything you might do.
The full force of the objection that hit Pascal now hits you too. But that is just another way for all hell to break loose for you: in that case, you cannot evaluate the choiceworthiness of your possible actions at all. Either way, you face decision-theoretic paralysis. It still does not obviously follow that you should wager for God. All that we have granted is that one norm—the norm of rationality—prescribes wagering for God.
For all that has been said, some other norm might prescribe wagering against God. And unless we can show that the rationality norm trumps the others, we have not settled what you should do, all things considered. There are several arguments to the effect that morality requires you to wager against God.
Pascal himself appears to be aware of one such argument. One way of putting the argument is that wagering for God may require you to corrupt yourself, thus violating a Kantian duty to yourself. Penelhum contends that the putative divine plan is itself immoral, condemning as it does honest non-believers to loss of eternal happiness, when such unbelief is in no way culpable; and that to adopt the relevant belief is to be complicit to this immoral plan.
See Quinn for replies to these arguments. For example, against Penelhum he argues that as long as God treats non-believers justly, there is nothing immoral about him bestowing special favor on believers, more perhaps than they deserve. Finally, Voltaire protests that there is something unseemly about the whole Wager. Let us now grant Pascal that, all things considered rationality and morality included , you should wager for God. What exactly does this involve? A number of authors read Pascal as arguing that you should believe in God—see e.
Quinn , and Jordan a. But perhaps one cannot simply believe in God at will; and rationality cannot require the impossible. What, then, would you have me do? However, he contends that one can take steps to cultivate such belief:. But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.
We find two main pieces of advice to the non-believer here: act like a believer, and suppress those passions that are obstacles to becoming a believer. And these are actions that one can perform at will. Believing in God is presumably one way to wager for God. This passage suggests that even the non-believer can wager for God, by striving to become a believer. To this, a follower of Pascal might reply that the act of genuine striving already displays a pureness of heart that God would fully reward; or even that genuine striving in this case is itself a form of believing.
It is not optional. But of course Pascal does not think that you would be infinitely rewarded for wagering for God momentarily, then wagering against God thereafter; nor that you would be infinitely rewarded for wagering for God sporadically—only on the last Thursday of each month, for example. Indeed, the Wager arguably has greater influence nowadays than any other such argument—not just in the service of Christian apologetics, but also in its impact on various lines of thought associated with infinity, decision theory, probability, epistemology, psychology, and even moral philosophy.
It has provided a case study for attempts to develop infinite decision theories. In it, Pascal countenanced the notion of infinitesimal probability long before philosophers such as Lewis and Skyrms gave it prominence.
It continues to put into sharp relief the question of whether there can be pragmatic reasons for belief, and the putative difference between theoretical and practical rationality.
Kenny suggests that nuclear Armageddon has negative infinite utility, and some might say the same for the loss of even a single human life.
This is plausibly read, then, as an assignment of negative infinite utility to the Andromeda scenario. Colyvan, Justus and Regan canvas difficulties associated with assigning infinite value to the natural environment.
Bartha and DesRoches respond, with an appeal to relative utility theory. As we have seen, it is also a great deal more besides. Background 2. The Argument from Superdominance 3. The Argument From Expectation 4. Reason can decide nothing here. A game is being played at the extremity of this infinite distance where heads or tails will turn up… 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… But your happiness? Let us weigh the gain and the loss in wagering that God is… If you gain, you gain all; if you lose, you lose nothing.
Wager, then, without hesitation that He is. Yes, I must wager; but I may perhaps wager too much. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play 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. Wagering for God brings infinite reward if God exists. 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… Again this passage is difficult to understand completely. Specifically: Either God exists or God does not exist, and you can either wager for God or wager against God.
Rationality requires you to perform the act of maximum expected utility when there is one. In short, this form of the objection is just another version of the many-gods objection. Moore — for us to base any belief on decision-theoretic self-interest Clifford , Nicholls Since utilitarians would tend to favor Pascalian reasoning while Kantians and virtue ethicists would not, the issue at stake belongs to a much larger debate in moral philosophy.
If you regularly brush your teeth, there is some chance you will go to heaven and enjoy infinite bliss. On the other hand, there is some chance you will enjoy infinite heavenly bliss even if you do not brush your teeth.
In fact, as soon as we allow infinite utilities, decision theory tells us that any course of action is as good as any other Duff In reply to such difficulties, Jordan proposes a run-off decision theory as described above.
Imagine tossing a coin until it lands heads-up, and suppose that the payoff grows exponentially according to the number of tosses you make. It follows you should be willing to pay any finite amount for the privilege of playing this game. Yet it clearly seems irrational to pay very much at all. The conclusion is that decision theory is a bad guide when infinite values are involved for discussion of this very old paradox, see Sorensen Byl points out that instead of referring to infinite payoffs we can speak of arbitrarily high ones.
No matter how improbable be the existence of God, it is still decision-theoretically rational to believe in God if the reward for doing so is sufficiently, yet only finitely, high.
However, this does not address the heart of the problem, for the St. Petersburg paradox too may be cast in terms of an arbitrarily high limit. Intuitively, one would not be willing to pay a million dollars, say, for the privilege of playing a game capped at one-million-and-one coin tosses, and it is not just because of the diminishing value of money.
There is something unsettling about decision theory, at least as applied to extreme cases, and so we might be skeptical about using it as a basis for religious commitment. A good sourcebook is Jordan a. Paul Saka Email: paul-saka live. The Equi-utility Paradox The St. Petersburg Paradox References and Further Reading 1. A Reason for Believing in God There are two kinds of argument for theism. The Super-Dominance Argument Pascal begins with a two-by-two matrix: either God exists or does not, and either you believe or do not.
The Expectations Argument What if the atheist is a happy hedonist, or if the theist is a miserable puritan? Run-off Decision Theory Some Pascalians propose combining pragmatic and epistemic factors in a two-stage process. The Equi-utility Paradox If you regularly brush your teeth, there is some chance you will go to heaven and enjoy infinite bliss.
The St. Petersburg Paradox Imagine tossing a coin until it lands heads-up, and suppose that the payoff grows exponentially according to the number of tosses you make.
Also in the 8 th , 9 th , 10 th editions; in Philosophy and the Human Condition , 2 d edition ed. Tom Beauchamp et al. See also Schlesinger Mackie, J.
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