Pascal's wager: a decision argument, not a proof

Pascal's wager does not try to prove that God exists. It argues that, since reason cannot settle the question, betting on God is the rational choice: if God exists the gain is infinite, and if not, little is lost. The argument has three premises, and each fails on its own. The strongest objection, that there are many possible gods, is not the end of it. What survives is the real contribution: a way of reasoning about stakes that have no upper limit.

What kind of argument it is

The wager comes from a single note in Pascal's Pensées, headed "Infinite—nothing". The Stanford Encyclopedia stresses how much is packed into it: the justification of theism, probability theory and decision theory used for almost the first time in history, pragmatism, and voluntarism, the thesis that belief is a matter of the will.

It is easy to confuse it with proofs of God's existence, and Pascal's text is plain that it is not one. He begins by saying reason can decide nothing here: we cannot know whether God exists, yet we must wager one way or the other. What reason can do, he says, is weigh the outcomes: "If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is."

So the wager is about what to do, not what is true. That one change of target explains most of what follows, including why an atheist and a theist can both think it a clever piece of reasoning with a weak foundation.

The argument in three premises

Put as the Stanford Encyclopedia does, it is a decision under risk. There is a table of outcomes: wager for God or against, and God exists or does not. Salvation is infinite in value. The rest are finite. With any positive, finite probability that God exists, the expected utility of wagering for God is infinite, as Pascal says, because infinite value times any positive probability stays infinite. The expected utility of wagering against is finite. Rationality, the third premise runs, requires you to take the action with the highest expected utility. So you should wager for God.

Premises, then:

The table. The outcomes and values are as stated, and the same for everyone.
The probability. You should give God's existence some positive, finite probability.
The rule. Rationality requires maximising expected utility.

Where each premise breaks

Premise	The objection	The reply
1. The table	The argument assumes that the same decision table applies to everybody; perhaps there is a predestined infinite reward for the Chosen, whatever they do. Above all, the many-gods objection: other possible gods would reward other things	Some defenders argue the other gods can be weighted by how choiceworthy each seems
2. The probability	Perhaps you could rationally fail to assign a probability at all	Pascal's own text says reason can decide nothing, which supports leaving it undefined
3. The rule	The Allais and Ellsberg paradoxes show maximising expectation can mislead; the St. Petersburg paradox makes it absurd to pay any finite amount to play a game with infinite expectation	Refinements of expected utility theory have been proposed

The St. Petersburg paradox is the most telling for the wager. A rule that tells you to pay any price for an infinite-expectation game is the same rule that tells you to wager for God, and most people reject the first conclusion. If the rule fails there, it is not obvious it holds here.

The many-gods objection is the one the Stanford Encyclopedia treats as the most important. The Internet Encyclopedia states its sting: the wager assumes that if God exists then God must take a rather specific form, which few open-minded agnostics would accept. Once other possible gods are admitted, the decision table has more rows than Pascal gave it. The Internet Encyclopedia lists four classic challenges in all: that belief cannot be chosen, that rival gods undermine each other, that prudential reasoning about belief is irresponsible, and that reference to infinite values is decision-theoretic nonsense.

The part about belief

The wager's second half is often forgotten. Pascal expects the objection that he cannot simply choose to believe. His answer is practical: learn from those who have been bound like you, and follow the way by which they began, by acting as if they believed, taking the holy water, having masses said. The Stanford Encyclopedia reads this as two pieces of advice to the non-believer: act like a believer, and suppress those passions that are obstacles to becoming a believer. These are actions a person can perform at will, so the wager is, on this reading, a plan for becoming a believer and not a demand to believe on the spot.

The reading has its own critic. As the entry notes, critics may question the psychology of belief formation that Pascal presupposes, since one could strive to believe, following the prescription exactly, and fail.

What philosophers think of the conclusion

The conclusion is a minority position among professionals. In the 2020 PhilPapers Survey, 18.93% accepted or leaned towards theism and 66.95% towards atheism. Disliking a conclusion is not an objection, and the survey does not ask about the wager. It shows only that the conclusion is a minority position among those who study it.

On our own map, Pascal sits at the end of the meaning row that holds meaning to be given by God, with the one-line reason we wrote beside him. Theism is the one philosophy on our map for which that question is close to decisive, together with the others at its two ends. The placements are our own, not survey results.

Our view

We think the wager's staying power comes from a good argument form attached to the wrong target. Take the form out and ask of any argument that offers unbounded stakes: who else is offering stakes of the same size, can the stakes really be unbounded, and can the payoff be obtained by the action recommended? Pascal's own wager fails the first test, since the many-gods objection supplies rival offers. It fails the second in the St. Petersburg way, because a rule that follows any infinite payoff leads to absurd conclusions elsewhere. And it strains the third, since belief is not fully under the will.

We would add one observation that the standard objections obscure. Premise 2 asks only for some positive, finite probability, yet that is enough to carry the whole argument, because any positive number makes the expectation infinite. That means the wager is not decided by how likely God is. It is decided by whether infinity belongs in the calculation at all.

The strongest objection to our reading is that the many-gods objection is not conclusive: some formal responses weigh each possible god by its plausibility and still favour Pascal. We think that is the right place for the argument to be fought, and we do not claim the question closed.

What would change our mind is a treatment of infinite utilities that tells you which gods to weigh, how, and does not depend on picking the answer first.

Where to read next

The Stanford Encyclopedia's entry follows the text of Pascal's note step by step and is the fullest guide to the objections. The note itself is short. On this site, Pascal and theism have pages, and faith and evidence takes up the question of what belief should rest on. To see where your own answers place you on the map, take the walk.

Spotted a mistake? Tell us at the foot of the page and it will be fixed and listed here.