“Pascal’s Wager” is the name given to an argumentdue to Blaise Pascal for believing, or for at least taking steps tobelieve, in God. The name is somewhat misleading, for in a singlesection of hisPensées, Pascal apparentlypresents four such arguments, each of which mightbe called a ‘wager’—it is only the third ofthese that is traditionally referred to as “Pascal’sWager”. We find in it the extraordinary confluence of severalimportant strands of thought: the justification of theism; probabilitytheory and decision theory, used here for almost the first time inhistory; pragmatism; voluntarism (the thesis that belief is a matterof the will); and the use of the concept of infinity.

We will begin with some brief stage-setting: some historicalbackground, some of the basics of decision theory, and some of theexegetical problems that thePensées pose. Then wewill follow the text to extract three main arguments. The bulk of theliterature addresses the third of these arguments, as will the bulk ofour discussion here. Some of the more technical and scholarly aspectsof our discussion will be relegated to lengthy footnotes, to whichthere are links for the interested reader. All quotations are from§233 ofPensées (1910, Trotter translation), the‘thought’ whose heading is“Infinite—nothing”.

• 1. Background
• 2. The Argument from Superdominance
• 3. The Argument From Expectation
• 4. The Argument From Generalized Expectations: “Pascal’s Wager”
• 5. Objections to Pascal’s Wager
  • 5.1 Premise 1: The Decision Table
  • 5.2 Premise 2: The Probability Assigned to God’s Existence
  • 5.3 Premise 3: Rationality Requires Maximizing Expected Utility
  • 5.4 Is the Argument Valid?
  • 5.5 Moral Objections to Wagering for God
• 6. A Fourth “Wager”?
• 7. What Does It Mean to “Wager for God”?
• 8. The Continuing Influence of Pascal’s Wager
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Background

It is important to contrast Pascal’s argument with variousputative ‘proofs’ of the existence of God that had comebefore it. Anselm’s ontological argument, Aquinas’‘five ways’, Descartes’ ontological and cosmologicalarguments, and so on, purport to prove that God exists. Pascal isapparently unimpressed by such attempted justifications of theism:“Endeavour … to convince yourself, not by increase ofproofs of God…” Indeed, he insists that “we do notknow if He is …”. Pascal’s project, then, isradically different. He aims to show thatwe ought tobelieve in God, rather than thatGodexists. And he seeks to provideprudentialreasons rather thanevidential reasons for believing inGod. To put it simply, we should wager that God exists because it isthebest bet. Ryan 1994 finds precursors to this line ofreasoning in the writings of Plato, Arnobius, Lactantius, and others;we might add Ghazali to his list—see Palacios 1920. Franklin2018 presents striking parallels to Pascal’s Wager by Sirmondand Chillingworth from 1637 and 1638 respectively, thus predatingPascal by a few years. But what is distinctive is Pascal’sexplicitly decision-theoretic formulation of the reasoning. In fact,Hacking 1975 describes the Wager as “the first well-understoodcontribution to decision theory” (viii). Thus, we should pausebriefly to review some of the basics of that theory.

In any decision problem, the way the world is, and what an agent does,together determine an outcome for the agent. We may assignutilities to such outcomes, numbers that represent the degreeto which the agent values them. It is typical to present these numbersin a decision table, with the columns corresponding to the variousrelevant states of the world, and the rows corresponding to thevarious possible actions that the agent can perform.

Indecisions under uncertainty, nothing more isgiven—in particular, the agent does not assign subjectiveprobabilities to the states of the world. Still, sometimes rationalitydictates a unique decision nonetheless. Consider, for example, a casethat will be particularly relevant here. Suppose that you have twopossible actions, \(A_1\) and \(A_2\), and the worst outcomeassociated with \(A_1\) is at least as good as the best outcomeassociated with \(A_2\); suppose also that in at least one state ofthe world, \(A_1\)’s outcome is strictly better than\(A_2\)’s. Let's say in that case that \(A_1\)superdominates \(A_2\). Then rationality seems to require youtoperform \(A_1\).^[1]

Example. A magician will toss a coin. You knownothing about the coin—it might be a normal coin, it might betwo-headed, it might be two-tailed, and it might be biased to anydegree. Suppose that in this state of complete ignorance about thecoin, you do not assign any probability whatsoever to its landingheads. Suppose that you can either bet on heads or on tails; it costsnothing to bet, and you will win $1 if you bet correctly. But I willpay you an extra $1 if you bet on heads (I especially like bets onheads). Your possible total pay-offs are given by this decisiontable:

Betting on heads superdominates betting on tails. The worst outcomeassociated with betting on heads (which pays $1)  is at least asgood as the best outcome associated with betting on tails (which pays$1); and if the coin lands heads, the outcome associated with bettingon heads paysmore than that associated with tails ($2 >$0). Moreover, it seems clear that you should bet on heads.

Indecisions under risk, the agent assigns subjectiveprobabilities to the various states of the world. Assume that thestates of the world are independent of what the agent does. A figureof merit called theexpected utility, or theexpectation of a given action can be calculated by a simpleformula: for each state, multiply the utility that the action producesin that state by the state’s probability; then, add thesenumbers. According to decision theory, rationality requires you toperform the action of maximum expected utility (if there is one).

Example. Suppose that the utility of money is linearin number of dollars: you value money at exactly its face value.Suppose now that you know that afair coin will betossed, and so you assign probability 1/2 to heads and 1/2 to tails.It costs a dollar to play the following game. If the coin lands heads,you will win $3; if it lands tails, you will get nothing. (Including the initial cost of playing, the total possible payoffs are$1 less than these respective amounts.) Should you play? Here is thedecision table:

The expectation of playing is (2 x 1/2)  + (–1 x 1/2) =1/2. This exceeds the expectation of not playing—namely0—so you should play.

And now suppose that the payoff if the coin lands heads is reducedby $1, so that the decision table becomes:

Then consistent with decision theory, you could either play or not,for either way your expectation would be 0.

Considerations such as these will play a crucial role inPascal’s arguments. It should be admitted that there are certainexegetical problems in presenting these arguments. Pascal neverfinished thePensées, but rather left them in the formof notes of various sizes pinned together. Hacking 1972 describes the“Infinite—nothing” as consisting of “twopieces of paper covered on both sides by handwriting going in alldirections, full of erasures, corrections, insertions, andafterthoughts” (24).^[2] This may explain why certain passages are notoriously difficult tointerpret, as we will see. Furthermore, our formulation of thearguments in the parlance of modern Bayesian decision theory mightappear somewhat anachronistic. For example, Pascal did not distinguishbetween what we would now callobjective andsubjective probability, although it is clear that it is thelatter that is relevant to his arguments. To some extent,“Pascal’s Wager” now has a life of its own, and ourpresentation of it here is perfectly standard. Still, we will closelyfollow Pascal’s text, supporting our reading of his arguments asmuch as possible. (See also Golding 1994 for another detailed analysisof Pascal’s reasoning, broken down into more steps than thepresentation here.)

There is the further problem of dividing theInfinite-nothinginto separate arguments. We will locate three arguments that eachconclude that rationality requires you to wager for God, although theyinterleave in the text.^[3] Finally, there is some disagreement over just what “wageringfor God” involves—is itbelieving in God, ormerelyengendering belief? We will conclude with a discussionof what Pascal meant by this.

2. The Argument from Superdominance

Pascal maintains that we are incapable of knowing whether God existsor not, yet we must “wager” one way or the other. Reasoncannot settle which way we should incline, but a consideration of therelevant outcomes supposedly can. Here is the first key passage:

“God is, or He is not.” But to which side shall weincline? Reason can decide nothing here. There is an infinite chaoswhich separated us. A game is being played at the extremity of thisinfinite distance where heads or tails will turn up… Which willyou choose then? Let us see. Since you must choose, let us see whichinterests you least. You have two things to lose, the true and thegood; and two things to stake, your reason and your will, yourknowledge and your happiness; and your nature has two things to shun,error and misery. Your reason is no more shocked in choosing onerather than the other, since you must of necessity choose… Butyour happiness? Let us weigh the gain and the loss in wagering thatGod is… If you gain, you gain all; if you lose, you losenothing. Wager, then, without hesitation that He is.

There are exegetical problems already here, partly because Pascalappears to contradict himself. He speaks of “the true” assomething that you can “lose”, and “error” assomething “to shun”. Yet he goes on to claim that if youlose the wager that God is, then “you lose nothing”.Surely in that case you “lose the true”, which is just tosay that you have made an error, and since this is something “toshun”, it is presumably a cost. Pascal believes, of course,that the existence of God is “the true”—butthat is not something that he can appeal to in this argument.Moreover, it is not because “you must of necessity choose”that “your reason is no more shocked in choosing one rather thanthe other”. Rather, by Pascal’s own account, it is because“[r]eason can decide nothing here”. (If it could, then itmight well be shocked—namely, if you chose in a way contrary toit.)

Following McClennen 1994, Pascal’s argument seems to be bestcaptured as presenting the following decision table:

Wagering for God superdominates wagering against God: the worstoutcome associated with wagering for God (status quo) is at least asgood as the best outcome associated with wagering against God (statusquo); and if God exists, the result of wagering for God is strictlybetter than the result of wagering against God. (The fact that theresult ismuch better does not matter yet.) Pascal draws theconclusion at this point that you should wager for God.

Without any assumption about your probability assignment toGod’s existence, the argument is invalid. Rationality doesnot require you to wager for God if you assign probability 0to God existing, as a strict atheist might. And Pascal does notexplicitly rule this possibility out until a later passage, when heassumes that you assign positive probability to God’s existence;yet this argument is presented as if it is self-contained. His claimthat “[r]eason can decide nothing here” may suggest thatPascal regards this as a decision under uncertainty, which is toassume that you donot assign probability at all toGod’s existence. If that is a further premise, then the argumentis apparently valid; but that premise contradicts his subsequentassumption that you assign positive probability. See McClennen for areading of this argument as a decision under uncertainty.

Pascal appears to be aware of a further objection to this argument,for he immediately imagines an opponent replying:

“That is very fine. Yes, I must wager; but I may perhaps wagertoo much.”

The thought seems to be that if I wager for God, and God does notexist, then I really do lose something. In fact, Pascal himself speaksofstakingsomething when one wagers for God, whichpresumably one loses if God does not exist. (We have already mentioned‘the true’ as one such thing; Pascal also seems to regardone’s worldly life as another.) In that case, the table ismistaken in presenting the two outcomes under ‘God does notexist’ as if they were the same, and we do not have a case ofsuperdominance after all.

Pascal addresses this at once in his second argument, which we willdiscuss only briefly, as it can be thought of as just a prelude to themain argument.

3. The Argument From Expectation

He continues:

Let us see. Since there is an equal risk of gain and of loss, if youhad only to gain two lives, instead of one, you might still wager. Butif there were three lives to gain, you would have to play (since youare under the necessity of playing), and you would be imprudent, whenyou are forced to play, not to chance your life to gain three at agame where there is an equal risk of loss and gain. But there is aneternity of life and happiness.

Understanding “equal risk” here as “equalprobability”, the probability of gain (winning the wager) and ofloss (losing the wager) must each be 1/2. His hypotheticallyspeaking of “two lives” and “three lives” maystrike one as odd. It is helpful to bear in mind Pascal’sinterest in gambling (which after all provided the initial motivationfor his study of probability) and to take the gambling model quiteseriously here. Indeed, the Wager is permeated with gamblingmetaphors: “game”, “stake”, “heads ortails”, “cards” and, of course, “wager”.Now, recall our calculation of the expectations of the two dollar andthree dollar gambles. Pascal apparently assumes now that utility islinear in number oflives, that wagering for God costs“one life”, and then reasons analogously to the way we didin our expectation calculations above! This is, as it were, a warm-up.Since wagering for God is rationally required even in the hypotheticalcase in which one of the prizes is three lives, then all the more itis rationally required in the actual case, in which one of the prizesis aneternity of life (salvation).

So Pascal has now made two striking assumptions:

     (1). The probability of God’s existence is1/2.

     (2). Wagering for God bringsinfinitereward if God exists.

Morris 1994 is sympathetic to (1), while Hacking 1972 finds it“a monstrous premiss” (189). One way to defend it is viathe classical interpretation of probability, according to which allpossibilities are given equal weight. The interpretation seemsattractive for various gambling games, which by design involve anevidential symmetry with respect to their outcomes; and Pascal evenlikens God’s existence to a coin toss, evidentially speaking.However, unless more is said, the interpretation yields implausible,and even contradictory results. (You have a one-in-a-million chance ofwinning the lottery; but either you win the lottery or youdon’t, so each of these possibilities has probability 1/2?!)Pascal’s argument for (1) is presumably that “[r]eason candecide nothing here”. (In the lottery ticket case, reason candecidesomething.) But it is not clear that completeignorance should be modeled as equiprobability. Morris imagines,rather, an agent who does have evidence for and against the existenceof God, but it is equally balanced. In any case, it \(is\) clear thatthere are people in Pascal’s audience who do not assignprobability 1/2 to God’s existence. This argument, then, doesnot speak to them.

However, Pascal realizes that the value of 1/2 actually plays no realrole in the argument, thanks to (2). This brings us to the third, andby far the most important, of his arguments.

4. The Argument From Generalized Expectations: “Pascal’s Wager”

We continue the quotation.

But there is an eternity of life and happiness. And this being so, ifthere 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 actstupidly, being obliged to play, by refusing to stake one life againstthree at a game in which out of an infinity of chances there is onefor you, if there were an infinity of an infinitely happy life togain. But there is here an infinity of an infinitely happy life togain, a chance of gain against a finite number of chances of loss, andwhat you stake is finite. It is all divided; wherever the infinite isand 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.Pascal’s talk of winning two, or three, lives is a littlemisleading. By his own decision theoretic lights, you wouldnot act stupidly “by refusing to stake one life againstthree at a game in which out of an infinity of chances there is onefor you”—in fact, you should not stake more than aninfinitesimal amount in that case (an amount that is bigger than 0,but smaller than every positive real number). The point, rather, isthat the prospective prize is “an infinity of an infinitelyhappy life”. In short, if God exists, then wagering for Godresults in infinite utility.

What about the utilities for the other possible outcomes? There issome dispute over the utility of “misery”. Hackinginterprets this as “damnation” (188), and Pascal doeslater speak of “hell” as the outcome in this case. Martin1983 among others assigns this a value ofnegative infinity.Sobel 1996, on the other hand, is one author who takes this value tobe finite. There is some textual support for this reading: “Thejustice of God must be vast like His compassion. Now justice to theoutcast is less vast … than mercy towards the elect”. Asfor the utilities of the outcomes associated with God’snon-existence, Pascal tells us that “what you stake isfinite”. This suggests that whatever these values are, they arefinite.

Pascal’s guiding insight is that the argument from expectationgoes through equally wellwhatever your probability forGod’s existence is, provided that it is non-zero and finite(non-infinitesimal)—“a chance of gain against a finitenumber of chances of loss”.^[4]

Pascal’s assumptions about utilities and probabilities are nowin place. In another landmark moment in this passage, he next presentsa formulation of expected utility theory. When gambling, “everyplayer stakes a certainty to gain an uncertainty, and yet he stakes afinite certainty to gain a finite uncertainty, without transgressingagainst reason”. How much, then, should a player be prepared tostake without transgressing against reason? Here is Pascal’sanswer: “… the uncertainty of the gain is proportioned tothe certainty of the stake according to the proportion of the chancesof gain and loss …” It takes some work to show that thisyields expected utility theory’s answer exactly, but it is workwell worth doing for its historical importance and is included in a footnote.^[5].)

Let us now gather together all of these points into a single argument.We can think of Pascal’s Wager as having three premises: thefirst concerns the decision table of rewards, the second concernsthe probability that you should give to God’s existence, and thethird is a maxim about rational decision-making. Specifically:

1. Either God exists or God does not exist, and you can either wager forGod or wager against God. The utilities of the relevant possibleoutcomes are as follows, where \(f_1, f_2\), and \(f_3\) are numberswhose values are not specified beyond the requirement that they befinite:

   	God exists	God does not exist
   Wager for God	\(\infty\)	\(f_1\)
   Wager against God	\(f_2\)	\(f_3\)

2. Rationality requires the probability that you assign to Godexisting to be positive (and finite).
3. Rationality requires you to perform the act of maximum expectedutility (when there is one).
4. Conclusion 1. Rationality requires you to wager forGod.
5. Conclusion 2. You should wager for God.

We have a decision under risk, with probabilities assigned to the waysthe world could be, and utilities assigned to the outcomes. Inparticular, we represent the infinite utility associated withsalvation as ‘\(\infty\)’. We assume that the real line isextended to include the element ‘\(\infty\)’, and that thebasic arithmetical operations are extended as follows:

For all real numbers \(r\): \(\infty + r = \infty\).
For all real numbers \(r\): \(\infty \times r = \infty\) if \(r \gt0\).

The first conclusion seems to follow from the usual calculations ofexpected utility (where \(p\) is your positive (and finite)probability for God’s existence):

That is, your expected utility of belief in God is infinite—asPascal puts it, “our proposition is of infinite force”. Onthe other hand, your expected utility of wagering against God is

This is finite.^[6] By premise 3, rationality requires you to perform the act of maximumexpected utility. Therefore, rationality requires you to wager forGod.

We now survey some of the main objections to the argument.

5. Objections to Pascal’s Wager

5.1 Premise 1: The Decision Table

Here the objections are manifold. Most of them can be stated quickly,but we will give special attention to what has generally been regardedas the most important of them, ‘the many Gods objection’(see also the link to footnote 7).

1. Different decision tables for different people. The argumentassumes that the same decision table applies to everybody.However, perhaps the relevant rewards are different for differentpeople. Perhaps, for example, there is a predestined infinite rewardfor the Chosen, whatever they do, and finite utility for the rest, asMackie 1982 suggests. Or maybe the prospect of salvation appeals moreto some people than to others, as Swinburne 1969 has noted.

Even granting that a single \(2 \times 2\) table applies toeverybody, one might dispute the values that enter into it. Thisbrings us to the next two objections.

2. The utility of salvation could not be infinite. One mightargue that the very notion of infinite utility is suspect—seefor example Jeffrey 1983 and McClennen 1994.^[7] Hence, the objection continues, whatever the utility of salvationmight be, it must be finite. Strict finitists, who are suspicious ofthe notion of infinity in general, will agree—see Dummett 1978and Wright 1987. Or perhaps the notion of infinite utility makessense, but an infinite reward could only be finitely appreciated by ahuman being.

3. There should be more than one infinity in the table. Thereare also critics of the Wager who, far from objecting to infiniteutilities, want to seemore of them in the table. Forexample, it might be thought that a forgiving God would bestowinfinite utility upon wagerers-for and wagerers-againstalike—Rescher 1985 is one author who entertains thispossibility. Or it might be thought that, on the contrary, wageringagainst an existent God results innegative infinite utility.(As we have noted, some authors read Pascal himself as saying asmuch.) Either way, \(f_2\) is not really finite at all, but \(\infty\)or \(-\infty\) as the case may be. And perhaps \(f_1\) and \(f_3\)could be \(\infty\) or \(-\infty\). Suppose, for instance, that Goddoes not exist, but that we are reincarnatedad infinitum,and that the total utility we receive is an infinite sum that divergesto infinity or to negative infinity.

4. The table should have more rows. Perhaps there is morethan one way to wager for God, and the rewards that God bestows varyaccordingly. For instance, God might not reward infinitely those whostrive to believe in Him only for the very mercenary reasons thatPascal gives, as James 1956 has observed. One could also imaginedistinguishing belief based on faith from belief based on evidentialreasons, and posit different rewards in each case.

5. The table should have more columns: the many Godsobjection. If Pascal is really right that reason can decidenothing here, then it would seem that various other theistichypotheses are also live options. Pascal presumably had in mind theCatholic conception of God—let us suppose that this is the Godwho either ‘exists’ or ‘does not exist’. Byexcluded middle, this is a partition. The objection, then, is that thepartition is not sufficiently fine-grained, and the ‘(Catholic)God does not exist’ column really subdivides into variousother theistic hypotheses. The objection could equally runthat Pascal’s argument ‘proves too much’: byparallel reasoning we can ‘show’ that rationality requiresbelieving in various incompatible theistic hypotheses. As Diderot(1746) puts the point: “An Imam could reason just as well this way”.^[8]

Since then, the point has been presented again and refined in variousways. Mackie 1982 writes, “the church within which alonesalvation is to be found is not necessarily the Church of Rome, butperhaps that of the Anabaptists or the Mormons or the Muslim Sunnis orthe worshippers of Kali or of Odin” (203). Cargile 1966 showsjust how easy it is to multiply theistic hypotheses: for each realnumber \(x\), consider the God who prefers contemplating \(x\) morethan any other activity. It seems, then, that such ‘alternativegods’ are a dime a dozen—or \(\aleph_1\), for thatmatter.

In response, some authors argue that in such a competition amongvarious possible deities for one’s belief, some are moreprobable than others. Although there may be ties among the expectedutilities—all infinite—for believing in various ones amongthem, their respective probabilities can be used as tie-breakers.Schlesinger (1994, 90) offers this principle: “In cases wherethe mathematical expectations are infinite, the criterion for choosingthe outcome to bet on is its probability”. (Note that thisprinciple is not found in the Wager itself, although it might beregarded as a friendly addition. Askell 2018 proposes a similarprinciple.) Are there reasons, then, for assigning higher probabilityto some Gods than others? Jordan (1994a, 107) suggests that someoutlandish theistic hypotheses may be dismissed for having “nobacking of tradition”. Similarly, Schlesinger maintains thatPascal is addressing readers who “have a notion of what genuinereligion is about” (88), and we might take that to suggest thatCargile’s imagined Gods, for example, may be correspondinglyassigned lower probability than Pascal’s God. Franklin (2018,41) writes that “[Pascal's] rhetoric is addressed to realagents, namely ”men of the world“ in the Paris of 1660” forwhom “the spectrum of religious theories to which they attachedgrounded subjective non-zero probability consisted of just Catholicismand atheism”. Saka (2018, 190–191) replies to various authorswho relativise the Wager to Pascal's intended audience that“Pascal's peers knew of Greco-Roman paganism, Judaism, Islam,new-world paganism, and multiple brands of Protestantism; they knew ofalleged Satanism … and they knew, from their acquaintance with theforegoing, that still other religions could readily behypothesized.” But still there is the issue of whatprobabilities should be assigned to alternative deities. Lycanand Schlesinger 1989 give more theoretical reasons for favoringPascal’s God over others in one’s probability assignments.They begin by noting the familiar problem in science ofunderdetermination of theory by evidence. Faced with a multiplicity oftheories that all fit the observed data equally well, we favor thesimplest such theory. They go on to argue that simplicityconsiderations similarly favor a conception of God as“absolutely perfect”, “which is theologically uniquein that it implies all the other predicates traditionally ascribed toGod” (104), and we may add that this conception isPascal’s. Conceptions of rival Gods, by contrast, leave openvarious questions about their nature, the answering of which woulddetract from their simplicity, and thus their probability.

Finally, Bartha 2012 models one’s probability assignments tovarious theistic hypotheses as evolving over time according to a‘deliberational dynamics’ somewhat analogous to thedynamics of evolution by natural selection. So understood,Pascal’s Wager is not a single decision, but rather a sequenceof decisions in which one’s probabilities update sequentially inproportion to how choiceworthy each God appeared to be in the previousround. (This relies on a sophisticated handling of infinite utilitiesin terms of utility ratios given in his 2007; see below.) He arguesthat a given probability assignment is choiceworthy only if it is anequilibrium of this deliberational dynamics. He shows that certainassignments are choiceworthy by this criterion, thus providing a kindof vindication of Pascal against the many Gods objection.

5.2 Premise 2: The Probability Assigned to God’s Existence

There are four sorts of problem for this premise. The first two arestraightforward; the second two are more technical, and can be foundby following the link to footnote 9.

1. Undefined probability for God’s existence. Premise 1presupposes that you shouldhave a probability forGod’s existence in the first place. However, perhaps you couldrationallyfail to assign it a probability—yourprobability that God exists could remainundefined. We cannotenter here into the thorny issues concerning the attribution ofprobabilities to agents. But there is some support for this responseeven in Pascal’s own text, again at the pivotal claim that“[r]eason can decide nothing here. There is an infinite chaoswhich separated us. A game is being played at the extremity of thisinfinite distance where heads or tails will turn up…” Thethought could be that any probability assignment is inconsistent witha state of “epistemic nullity” (in Morris’ 1986phrase): to assign a probability at all—even 1/2—toGod’s existence is to feign having evidence that one in facttotally lacks. For unlike a coin that we know to be fair, thismetaphorical ‘coin’ is ‘infinitely far’ fromus, hence apparently completely unknown to us. Perhaps, then,rationality actually requires us torefrain from assigning aprobability to God’s existence (in which case at least theArgument from Superdominance would apparently be valid). Or perhapsrationality does not require it, but at leastpermits it.Either way, the Wager would not even get off the ground.

2. Zero probability for God’s existence. Strict atheistsmay insist on the rationality of a probability assignment of 0, asOppy 1990 among others points out. For example, they may contend thatreason alonecan settle that God does not exist, perhaps byarguing that the very notion of an omniscient, omnipotent,omnibenevolent being is contradictory. Or a Bayesian might hold thatrationality places no constraint on probabilistic judgments beyondcoherence (or conformity to the probability calculus). Then as long asthe strict atheist assigns probability 1 to God’s non-existencealongside his or her assignment of 0 to God’s existence, no normof rationality has been violated.

Furthermore, an assignment of \(p = 0\) would clearly block the routeto Pascal’s conclusion, under the usual assumption that

For then the expectation calculations become:

And nothing in the argument implies that \(f_1 \gt f_3\). (Indeed,this inequality is questionable, as even Pascal seems to allow.) Inshort, Pascal’s wager has no pull on strict atheists.^[9]

5.3 Premise 3: Rationality Requires Maximizing Expected Utility

Finally, one could question Pascal’s decision theoreticassumption that rationality requires one to perform the act of maximumexpected utility (when there is one). Nowperhaps this is ananalytic truth, in which case we could grant it to Pascal withoutfurther discussion—perhaps it isconstitutive ofrationality to maximize expectation, as some might say. But thispremise has met serious objections. The Allais 1953 and Ellsberg 1961paradoxes, for example, are said to show that maximizing expectationcan lead one to perform intuitively sub-optimal actions. So too theSt. Petersburg paradox, in which it is supposedly absurd that oneshould be prepared to pay any finite amount to play a game withinfinite expectation. (That paradox is particularly apposite here.)^[10]

Various refinements of expected utility theory have been suggested asa result of such problems. For example, we might consider expecteddifferences between the pay-offs of options, and prefer oneoption to another if and only if the expected difference of the formerrelative to the latter is positive—see Hájek and Nover2006, Hájek 2006, Colyvan 2008, and Colyvan & Hájek2016. Or we might consider suitably defined utilityratios,and prefer one option to another if and only if the utility ratio ofthe former relative to the latter is greater than 1—see Bartha2007. If we either admit refinements of traditional expected utilitytheory, or are pluralistic about our decision rules, then premise 3 isapparently false as it stands. Nonetheless, the door is opened to somesuitable reformulation of it that might serve Pascal’s purposes.Indeed, Bartha argues that his ratio-based reformulation answers someof the most pressing objections to the Wager that turn on itsinvocation of infinite utility.

Finally, one might distinguish betweenpractical rationalityandtheoretical rationality. One could then concede thatpractical rationality requires you to maximize expected utility, whileinsisting that theoretical rationality might require something else ofyou—say, proportioning belief to the amount of evidenceavailable. This objection is especially relevant, since Pascal admitsthat perhaps you “must renounce reason” in order to followhis advice. But when these two sides of rationality pull in oppositedirections, as they apparently can here, it is not obvious thatpractical rationality should take precedence. (For a discussion ofpragmatic, as opposed to theoretical, reasons for belief, see Foley1994.)

5.4 Is the Argument Valid?

A number of authors who have been otherwise critical of the Wager haveexplicitly conceded that the Wager is valid—e.g. Mackie 1982,Rescher 1985, Mougin and Sober 1994, and most emphatically, Hacking1972. That is, these authors agree with Pascal that wagering for Godreally is rationally mandated by Pascal’s decision table intandem with positive probability for God’s existence, and thedecision theoretic account of rational action.

However, Duff 1986 and Hájek 2003 argue that the argument is infact invalid. Their point is that there are strategies besideswagering for God that also have infinite expectation—namely,mixed strategies, whereby you do not wager for or against Godoutright, but rather choose which of these actions to perform on thebasis of the outcome of some chance device. Consider the mixedstrategy: “Toss a fair coin: heads, you wager for God; tails,you wager against God”. By Pascal’s lights, withprobability 1/2 your expectation will be infinite, and withprobability 1/2 it will be finite. The expectation of the entirestrategy is:

That is, the ‘coin toss’ strategy has the same expectationas outright wagering for God. But the probability 1/2 was incidentalto the result. Any mixed strategy that gives positive and finiteprobability to wagering for God will likewise have infiniteexpectation: “wager for God iff a fair die lands 6”,“wager for God iff your lottery ticket wins”, “wagerfor God iff a meteor quantum tunnels its way through the side of yourhouse”, and so on.

It can be argued that the problem is still worse than this, though,for there is a sense in whichanything that you do might beregarded as a mixed strategy between wagering for God, and wageringagainst God, with suitable probability weights given to each. Supposethat you choose to ignore the Wager, and to go and have a hamburgerinstead. Still, you may well assign positive and finite probability toyour winding up wagering for God nonetheless; and this probabilitymultiplied by infinity again gives infinity. So ignoring the Wager andhaving a hamburger has the same expectation as outright wagering forGod. Even worse, suppose that you focus all your energy intoavoiding belief in God. Still, you may well assign positiveand finite probability to your efforts failing, with the result thatyou wager for God nonetheless. In that case again, your expectation isinfinite again. So even if rationality requires you to perform the actof maximum expected utility when there is one, here there isn’tone. Rather, there is a many-way tie for first place, as it were. Allhell breaks loose: anything you might do is maximally good by expectedutility lights!^[11]

Monton 2011 defends Pascal’s Wager against this line ofobjection. He argues that an atheist or agnostic has more than oneopportunity to follow a mixed strategy. Returning to the first exampleof one, suppose that the fair coin lands tails. Monton’s thoughtis that your expected utility now changes; it is no longer infinite,but rather that of an atheist or agnostic who has no prospect of theinfinite reward for wagering for God. You are back to where youstarted. But since it was rational for you to follow the mixedstrategy the first time, it is rational for you to follow it againnow—that is, to toss the coin again. And if it lands tailsagain, it is rational for you to toss the coin again … Withprobability 1, the coin will land heads eventually, and from thatpoint on you will wager for God. Similar reasoning applies to wageringfor God just in case an n-sided die lands 1 (say): with probability 1the die will eventually land 1, so if you repeatedly base your mixedstrategy on the die, with probability 1 you will wind up wagering forGod after a finite number of rolls. Robertson 2012 replies that notall such mixed strategies are (probabilistically) guaranteed to leadto your wagering for God in the long run: not ones in which theprobability of wagering for God decreases sufficiently fast onsuccessive trials. Think, for example, of rolling a 4-sided die, thena 9-sided die, and in general an \((n+1)^2\)-sided die on the\(n\)^th trial …, a strategy for which theprobability that you will eventually wager for God is only 1/2, asRobertson shows. However, Easwaran and Monton 2012 counter-reply thatwith a continuum of times at which the dice can be rolled, thesequence of rolls that Robertson proposes can be completed in anarbitrarily short period of time. In that case, what should you donext? By Monton’s argument, it seems you should roll a dieagain. Easwaran and Monton prove that if there are uncountably manytimes at which one implements a mixed strategy with non-zeroprobability of wagering for God, then with probability 1, one ends upwagering for God at one of these times. (And they assume, as isstandard, that once one wagers for God there is no going back.) Theyconcede that imagining uncountably rolls of a die, say, involves anidealization that is surely not physically realizable. But theymaintain that you should act in the way that an idealized version ofyourself would eventually act, one whocan realize the rollsas described—that is, wager for God outright.

There is a further twist on the mixed strategies objection. To repeat,the objection’s upshot is that even granting Pascal all hispremises, still wagering for God is not rationally required. But wehave seen numerous reasonsnot to grant all his premises.Very well then; let’s not. Indeed, let’s suppose that yougivetiny probability p to them all being true, where \(p\)is positive and finite. So you assign probability \(p\) to yourdecision problem being exactly as Pascal claims it to be. But if itis, according to the mixed strategies objection, all hell breaksloose. Yet again, \(p\) multiplied by infinity gives infinity. Hence,it seems that each action that gets infinite expected utilityaccording to Pascal similarly gets infinite expected utility accordingtoyou; but by the previous reasoning, that is anything youmight do. The full force of the objection that hit Pascal now hits youtoo. There are some subtleties that we have elided over; for example,if you also assign positive and finite probability to a source ofnegative infinite utility, then the expected utilitiesinstead become \(\infty\) – \(\infty\), which is undefined. Butthat is just another way for all hell to break loose for you: in thatcase, you cannot evaluate the choiceworthiness of your possibleactions at all. Either way, you face decision-theoretic paralysis. Wemight call thisPascal’s Revenge. See Hájek(2015) for more discussion.

Jackson and Rogers (2019), developing points in Jackson (2016), arguethat the mixed-strategies objection is a “structural, but notsubstantive” (61) objection to Pascal's Wager. They providecases in which it is clearly rational to prefer one infinite good toanother. They suggest a reformulation of how prospects of infiniterewards should be compared. (This also provides a response to the manyGods objection.) Hájek (2003 and especially 2018) offers manyvalid reformulations of the Wager with more nuanced representations ofthe utility of salvation, such that the lower the probability ofwagering for God, the lower the expected utility. Seefootnote 11 forfurther discussion.

5.5 Moral Objections to Wagering for God

Let us grant Pascal’s first conclusion for the sake of theargument: rationality requires you to wager for God. The secondconclusion, that youshould wager for God, does not obviouslyfollow. All that we have granted is that one norm—the norm ofrationality—prescribes wagering for God. For all that has beensaid, someother norm might prescribe wagering against God.And unless we can show that the rationality norm trumps the others, wehave not settled what you should do, all things considered.

There are several arguments to the effect thatmoralityrequires you to wageragainst God. Pascal himself appears tobe aware of one such argument. He admits that if you do not believe inGod, his recommended course of action “will deaden youracuteness” (This is Trotter’s translation. Pascal’soriginal French wording is “vous abêtira”, whoseliteral translation is even more startling: “will make you abeast”.) One way of putting the argument is that wagering forGod may require you to corrupt yourself, thus violating a Kantian dutyto yourself. Clifford 1877 argues that an individual’s believingsomething on insufficient evidence harms society by promotingcredulity. Penelhum 1971 contends that the putative divine plan isitself immoral, condemning as it does honest non-believers to loss ofeternal happiness, when such unbelief is in no way culpable; and thatto adopt the relevant belief is to be complicit to this immoral plan.See Quinn 1994 for replies to these arguments. For example, againstPenelhum he argues that as long as God treats non-believers justly,there is nothing immoral about him bestowing special favor onbelievers, more perhaps than they deserve. (Note, however, that Pascalleaves open in the Wager whether the payoff for non-believers \(is\)just; indeed, as far as his argument goes, it may be extremelyunjust.)

Finally, Voltaire protests that there is something unseemly about thewhole Wager: “That article seems a bit indecent and childish;that notion of gambling, of losses and winnings, does not suit thegravity of the subject” (Voltaire 1778 [1961, 123]). This doesnot so much support wagering against God, as dismissing all talk of‘wagerings’ altogether. Schlesinger (1994, 84) canvasses asharpened formulation of this objection: an appeal to greedy,self-interested motivations is incompatible with “the quest forpiety” that is essential to religion. He replies that thepleasure of salvation that Pascal’s Wager countenances is“of the most exalted kind”, and that if seeking it countsas greed at all, then it is “the manifestation of a noble greedthat is to be acclaimed” (85). Franklin (1998, 2018) regardsVoltaire as caricaturing Pascal's Wager and missing his key point that“you must wager”. Franklin argues that given that thechoice Pascal presents is unavoidable, it should be made on the basisof a rational calculation, and that his conclusion concernsaction, not the truth of theism.

6. A Fourth “Wager”?

We have concluded our discussion of what is traditionally known as“Pascal’s Wager”. But Pascal has one last twist instore for us. In his “End of this address” he writesregarding wagering for God:

Now, what harm will befall you in taking this side? You will befaithful, humble, grateful, generous, a sincere friend, truthful.Certainly you will not have those poisonous pleasures, glory andluxury; but will you not have others? I will tell you that you willthereby gain in this life, and that, at each step you take on thisroad, you will see so great certainty of gain, so much nothingness inwhat you risk, that you will at last recognise that you have wageredfor something certain and infinite, for which you have givennothing.

This passage makes two further striking claims regarding wagering forGod: you will “gain in this life”, and “you havewagered for something certain”. The decision table can bepresented as follows (with the outcomes ranked):

This is again a decision under uncertainty (in our technicalsense)—it involves no considerations of probability. Indeed,“the wager” dissolves, twice over: utilities alonedefinitively settle that you should wager for God, and in any case itis not really a gamble at all, since your gain is certain! (That is myunderstanding of “you have wagered for something certain”;it is not God's existence itself thatis “certain”.) The worst outcome associated withwagering for God (gain in earthly life) is strictlybetter than the best outcome associated with wageringagainst God (earthly life). It follows immediately that you shouldwager for God. Hájek (2018) calls this an argumentfrom superduperdominance. This is a valid argument forwagering for God, even if we allow that God's existence might beimpossible.

Pascal has come full circle back to the first wager, and he now goeseven beyond it. The solution to this decisionproblem is trivial. The status of this “wager”—thesoundness or otherwise of this argument—turns on whether one'sprospects are as this decision table portrays them. See Jordan (2006)and (2018, 108–109) for further discusion.

7. What Does It Mean to “Wager for God”?

Let us now grant Pascal that, all things considered (rationality andmorality included), you should wager for God. What exactly does thisinvolve?

A number of authors read Pascal as arguing that you shouldbelieve in God—see e.g. Quinn 1994, and Jordan 1994a.But perhaps one cannot simply believe in God at will; and rationalitycannot require the impossible. Pascal is well aware of this objection:“[I] am so made that I cannot believe. What, then, would youhave me do?”, says his imaginary interlocutor. However, hecontends that one can take steps to cultivate such belief:

You would like to attain faith, and do not know the way; you wouldlike to cure yourself of unbelief, and ask the remedy for it. Learn ofthose who have been bound like you, and who now stake all theirpossessions. 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 theway by which they began; by acting as if they believed, taking theholy water, having masses said, etc. …

But to show you that this leads you there, it is this which willlessen the passions, which are your stumbling-blocks.

We find two main pieces of advice to the non-believer here: act like abeliever, and suppress those passions that are obstacles to becoming abeliever. And these are actions that onecan perform atwill.

Believing in God is presumably one way to wager for God. This passagesuggests that even the non-believer can wager for God, by striving tobecome a believer. Critics may question the psychology of beliefformation that Pascal presupposes, pointing out that one could striveto believe (perhaps by following exactly Pascal’s prescription),yet fail. To this, a follower of Pascal might reply that the act ofgenuine striving already displays a pureness of heart that God wouldfully reward.

According to Pascal, ‘wagering for God’ and‘wagering against God’ are contradictories, as there is noavoiding wagering one way or another: “you must wager. It is notoptional.” The decision to wager for or against God is one thatyou make at a time—at \(t\), say. But of course Pascal does notthink that you would be infinitely rewarded for wagering for Godmomentarily, then wagering against God thereafter; nor that you wouldbe infinitely rewarded for wagering for God sporadically—only onthe last Thursday of each month, for example. What Pascal intends by‘wagering for God’ is an ongoing action—indeed, onethat continues until your death—that involves your adopting acertain set of practices and living the kind of life that fostersbelief in God. The decision problem for you at \(t\), then, is whetheryou should embark on this course of action; to fail to do so is towager against God at \(t\).

8. The Continuing Influence of Pascal’s Wager

Pascal’s Wager vies with Anselm’s Ontological Argument forbeing the most famous argument in the philosophy of religion. Indeed,the Wager arguably has greater influence nowadays than any other suchargument—not just in the service of Christian apologetics, butalso in its impact on various lines of thought associated withinfinity, decision theory, probability, epistemology, psychology, andeven moral philosophy. It has provided a case study for attempts todevelop infinite decision theories. In it, Pascal countenanced thenotion of infinitesimal probability long before philosophers such asLewis 1980 and Skyrms 1980 gave it prominence. It continues to putinto sharp relief the question of whether there can be pragmaticreasons for belief, and the putative difference between theoreticaland practical rationality. It raises subtle issues about the extent towhich one’s beliefs can be a matter of the will, and the ethicsof belief.

Reasoning reminiscent of Pascal’s Wager, often with an explicitacknowledgment of it, also informs a number of debates in moralphilosophy, both theoretical and applied. Kenny 1985 suggests thatnuclear Armageddon has negative infinite utility, and some might saythe same for the loss of even a single human life. Stich 1978criticizes an argument that he attributes to Mazzocchi, that thereshould be a total ban on recombinant DNA research, since such researchcould lead to the “Andromeda scenario” of creating akiller strain of bacterial culture against which humans are helpless;the ban, moreover, should be enforced if the “Andromeda scenariohas even the smallest possibility of occurring” (191), inMazzocchi’s words. This is plausibly read, then, as anassignment of negative infinite utility to the Andromeda scenario.More recently, Colyvan, Cox, and Steele 2010 discuss Pascal’sWager-like problems for certain deontological moral theories, in whichviolations of duties are assigned negative infinite utility. Colyvan,Justus and Regan 2011 canvas difficulties associated with assigninginfinite value to the natural environment. Bartha and DesRoches 2017respond, with an appeal to relative utility theory. Stone 2007 arguesthat a version of Pascal’s Wager applies to sustaining patientswho are in a persistent vegetative state; see Varelius 2013 for adissenting view. Pascal’s Wager has even been appealed to in themedical debate over whether antibiotics should be used to prevent acertain kind of inflammation in the heart (Shaw and Conway 2010).

Pascal’s Wager is a watershed in the philosophy of religion. Aswe have seen, it is also a great deal more besides.

Bibliography

• Allais, Maurice, 1953. “Le Comportment de l’HommeRationnel Devant la Risque: Critique des Postulats et Axiomes del’École Américaine”,Econometrica,21: 503–546.
• Askell, Amanda, 2019. “Prudential Objections toAtheism”, in Graham Oppy (ed.),A Companion to Atheism andPhilosophy, Oxford: Wiley-Blackwell.
• Bartha, Paul, 2007. “Taking Stock of Infinite Value:Pascal’s Wager and Relative Utilities,”Synthese,154: 5–52.
• –––, 2012. “Many Gods, Many Wagers:Pascal’s Wager Meets the Replicator Dynamics”, in JakeChandler and Victoria S. Harrison (eds.),Probability in thePhilosophy of Religion, Oxford: Oxford University Press,187–206.
• –––, 2016. “Probability and the Philosophyof Religion”, in Alan Hájek and Christopher Hitchcock(eds.),The Oxford Handbook of Probability and Philosophy,Oxford: Oxford University Press, 738–771.
• –––, 2018. “Pascal’s Wager and theDynamics of Rational Deliberation”, in Bartha and Pasternack (eds.)2018, 236–259.
• Bartha, Paul and C. Tyler DesRoches, 2016. “The RelativelyInfinite Value of the Environment”,Australasian Journal ofPhilosophy, 95(2): 328–353.
• Bartha, Paul and Lawrence Pasternack (eds.), 2018.Pascal'sWager (Classic Philosophical Arguments), Cambridge: CambridgeUniversity Press.
• Brown, Geoffrey, 1984. “A Defence of Pascal’sWager”,Religious Studies, 20: 465–79.
• Cain, James, 1995. “Infinite Utility”,Australasian Journal of Philosophy, 73(3):401–404.
• Cargile, James, 1966. “Pascal’s Wager”,Philosophy, 35: 250–7.
• Chen, Eddy, and Daniel Rubio, 2020. “SurrealDecisions”,Philosophy and Phenomenological Research,100(1): 54–74.
• Clifford, William K., 1877. “The Ethics of Belief”, inR. Madigan (ed.),The Ethics of Belief and Other Essays,Amherst, MA: Prometheus, 70–96.
• Colyvan, Mark, 2008. “Relative Expectation Theory”,Journal of Philosophy, 105(1): 37–54.
• Colyvan, Mark, Damian Cox, and Katie Steele, 2010.“Modelling the Moral Dimension of Decisions”,Noûs, 44(3): 503–529.
• Colyvan, M., J. Justus, and H.M. Regan, 2010. “The NaturalEnvironment is Valuable but Not Infinitely Valuable”,Conservation Letters, 3(4): 224–8.
• Colyvan, Mark and Alan Hájek, 2016. “Making AdoWithout Expectations”,Mind, 125(499):829–857.
• Conway, John, 1976.On Numbers and Games, London:Academic Press.
• Cutland, Nigel (ed.), 1988.Nonstandard Analysis and itsApplications, London: London Mathematical Society, Student Texts10.
• Diderot, Denis, 1746.Pensées Philosophiques,reprinted Whitefish, MN: Kessinger Publishing, 2009.
• Duff, Antony, 1986. “Pascal’s Wager and InfiniteUtilities”,Analysis, 46: 107–9.
• Dummett, Michael. 1978. “Wang’s Paradox”, inTruth and Other Enigmas, Cambridge, MA: Harvard UniversityPress.
• Duncan, Craig, 2018. “The Many Gods Objection toPascal’s Wager: A Defeat, Then a Resurrection”, in Barthaand Pasternack (eds.) 2018, 148–167.
• Easwaran, Kenny and Bradley Monton, 2012. “Mixed Strategies,Uncountable Times, and Pascal’s Wager: A Reply toRobertson”,Analysis, 72(4): 681–685.
• Ellsberg, D., 1961. “Risk, Ambiguity and the SavageAxioms”,Quarterly Journal of Economics, 25:643–669.
• Feller, William, 1971.An Introduction to Probability Theoryand its Applications (Volume II), 2nd edition, London: Wiley.
• Flew, Anthony, 1960. “Is Pascal’s Wager the Only SafeBet?”,The Rationalist Annual, 76: 21–25.
• Foley, Richard, 1994. “Pragmatic Reasons for Belief”,in Jordan 1994b, 27–44.
• Franklin, James, 1998. “Two Caricatures, I: Pascal’sWager”,International Journal for Philosophy ofReligion, 44: 109–14.
• –––, 2018. “Pascal’s Wager and theOrigins of Decision Theory: Decision-Making by RealDecision-Makers”, in Bartha and Pasternack (eds.) 2018,31–46.
• Golding, Joshua, 1994. “Pascal’s Wager”,TheModern Schoolman, 71(2): 115–143.
• Hacking, Ian, 1972. “The Logic of Pascal’sWager”,American Philosophical Quarterly, 9(2):186–92; reprinted in Jordan (ed.) 1994b, 21–29. (Pagereferences are to the 1972 original.)
• –––, 1975.The Emergence ofProbability, Cambridge: Cambridge University Press.
• Hájek, Alan, 1997. “The Illogic of Pascal’sWager”,Proceedings of the 10th Logica InternationalSymposium, T. Childers et al. (eds.), Filosophia, The Instituteof Philosophy of the Academy of Sciences of the Czech Republic,239–249.
• –––, 2000. “Objecting Vaguely toPascal’s Wager”,Philosophical Studies, 98(1):1–16.
• –––, 2003. “Waging War on Pascal’sWager”,Philosophical Review, 112(1): 27–56.
• –––, 2006. “Some Reminiscences on RichardJeffrey, and Some Reflections onThe Logic ofDecision”,Philosophy of Science, 73(5):947–958.
• –––, 2012. “Blaise and Bayes”, inJake Chandler and Victoria S. Harrison (eds.),Probability in thePhilosophy of Religion, Oxford: Oxford University Press,167–186.
• –––, 2015. “Pascal’s UltimateGamble”, in Alex Byrne, Joshua Cohen, Gideon Rosen, and SeanaShiffrin (eds.),The Norton Introduction to Philosophy, NewYork: Norton.
• –––, 2018. “The (In)validity ofPascal’s Wager”, in Bartha and Pasternack (eds.) 2018,123–147.
• Hájek, Alan and Harris Nover, 2006. “PerplexingExpectations”,Mind, 115 (July): 703–720.
• Herzberg, Frederik, 2011. “Hyperreal Expected Utilities andPascal’s Wager”,Logique et Analyse, 213:69–108.
• Jackson, Liz, 2016. “Wagering Against DivineHiddenness”,The European Journal for Philosophy ofReligion, 8(4): 85–109.
• Jackson, Liz, and Andrew Rogers 2019. “SalvagingPascal’s Wager”,Philosophia Christi, 21(1):59–84.
• James, William, 1956. “The Will to Believe”, inThe Will to Believe and Other Essays in Popular Philosophy,New York: Dover Publications.
• Jeffrey, Richard C., 1983.The Logic of Decision, 2ndedition, Chicago: University of Chicago Press.
• Jordan, Jeff, 1994a. “The Many Gods Objection”, inJordan (ed.) 1994b, 101–113.
• –––, 2006.Pascal’s Wager:Pragmatic Arguments and Belief in God, Oxford: Oxford UniversityPress.
• –––, 2018. “The Wager and WilliamJames”, in Bartha and Pasternack (eds.) 2018, 101–119.
• Jordan, Jeff (ed.), 1994b.Gambling on God: Essays onPascal’s Wager, Lanham, MD: Rowman & Littlefield.
• Joyce, James M., 2005. “How Probabilities ReflectEvidence”,Philosophical Perspectives, 19:153–179.
• Lewis, David, 1980. “A Subjectivist’s Guide toObjective Chance”, in Richard C. Jeffrey (ed.),Studies inInductive Logic and Probability (Volume II), Berkeley and LosAngeles: University of California Press; reprinted in Lewis 1986.
• –––, 1986,Philosophical Papers: VolumeII, Oxford: Oxford University Press.
• Lindstrom, Tom, 1988. “Invitation to Non-StandardAnalysis”, in Cutland (ed.) 1988, 1–139.
• Lycan, William, and George Schlesinger. 1989. “You Bet YourLife”, in Joel Feinberg (ed.),Reason andResponsibility, 7th edition, Belmont CA: Wadsworth; also in the8th, 9th, 10th editions; reprinted in Tom Beauchamp, Joel Feinberg,and James M. Smith (eds.),Philosophy and the HumanCondition, 2nd edition, Englewood Cliffs, NJ: Prentice Hall,1989.
• Mackie, J. L., 1982.The Miracle of Theism, Oxford:Oxford University Press.
• Martin, Michael 1983. “Pascal’s Wager as an Argumentfor Not Believing in God”,Religious Studies, 19:57–64.
• –––, 1990.Atheism: a PhilosophicalJustification, Philadelphia: Temple University Press.
• McClennen, Edward, 1994. “Pascal’s Wager and FiniteDecision Theory”, in Jordan (ed.) 1994b, 115–137.
• Monton, Bradley, 2011. “Mixed Strategies Can’t EvadePascal’s Wager”,Analysis, 71:642–645.
• Morris, Thomas V., 1986. “Pascalian Wagering”,Canadian Journal of Philosophy, 16: 437–54.
• –––, 1994. “Wagering and theEvidence”, in Jordan (ed.) 1994b, 47–60.
• Mougin, Gregory, and Elliott Sober, 1994. “Betting AgainstPascal’s Wager”,Noûs, XXVIII:382–395.
• Nelson, Edward, 1987.Radically Elementary ProbabilityTheory (Annals of Mathematics Studies, 117), Princeton: PrincetonUniversity Press.
• Nelson, Mark T., 1991. “Utilitarian Eschatology”,American Philosophical Quarterly, 28: 339–347.
• Ng, Yew-Kwang, 1995. “Infinite Utility and VanLiedekerke’s Impossibility: A Solution”,AustralasianJournal of Philosophy, 73: 408–411.
• Oppy, Graham, 1990. “On Rescher on Pascal’sWager”,International Journal for Philosophy ofReligion, 30: 159–68.
• –––, 2018. “Infinity in Pascal’sWager”, in Bartha and Pasternack 2018, 260–277.
• Palacios, M. Asin, 1920.Los Precedentes Musulmanes del‘Pari’ de Pascal, Santander: Boletín de laBiblioteca Menéndez Pelayo.
• Pascal, Blaise, 1670,Pensées, translated by W. F.Trotter, London: Dent, 1910.
• Penelhum, Terence, 1971.Religion and Rationality, NewYork: Random House.
• Pivato, Michael, 2014. “Additive Representation of SeparablePreferences over Infinite Products”,Theory andDecision, 77: 31–83.
• Quinn, Philip L., 1994. “Moral Objections to PascalianWagering”, in Jordan (ed.) 1994b, 61–81.
• Rescher, Nicholas, 1985.Pascal’s Wager, NotreDame: South Bend, IN: Notre Dame University Press.
• Rinard, Susanna, 2018. “Pascal’s Wager and ImpreciseProbability”, in Bartha and Pasternack (eds.) 2018, 278–292.
• Robinson, Abraham, 1966.Non-Standard Analysis,Amsterdam: North Holland.
• Rota, Michael, 2016. “A Better Version of Pascal’sWager”,American Catholic Philosophical Quarterly,90(3): 415–439.
• Rota, Michael, 2017. “Pascal’s Wager”,Philosophy Compass, 12:e12404.
• Ryan, John, 1945. “The Wager in Pascal and Others”,New Scholasticism, 19(3): 233–50; reprinted in Jordan(ed.) 1994b, 11–19.
• Saka, Paul, 2018. “Rationality and the Wager”, inBartha and Pasternack (eds.) 2018, 187–208.
• Schlesinger, George, 1994. “A Central TheisticArgument”, in Jordan (ed.) 1994b, 83–99.
• Shaw, David and David Conway, 2010. “Pascal’s Wager,Infective Endocarditis and the ‘No-Lose’ Philosophy inMedicine”,Heart, 96(1): 15–18.
• Skalia, H. J., 1975.Non-Archimedean Utility Theory,Dordrecht: D. Reidel.
• Skyrms, Brian, 1980,Causal Necessity, New Haven: YaleUniversity Press.
• Sobel, Howard, 1996. “Pascalian Wagers”,Synthese, 108: 11–61.
• Sorensen, Roy, 1994. “Infinite Decision Theory”, inJordan (ed.) 1994b, 139–159.
• Stone, Jim, 2007. “Pascal’s Wager and the PersistentVegetative State”,Bioethics, 21(2): 84–92.
• Swinburne, R. G., 1969. “The Christian Wager”,Religious Studies, 4: 217–28.
• Vallentyne, Peter, 1993. “Utilitarianism and InfiniteUtility”,Australasian Journal of Philosophy, 71:212–217.
• –––, 1995. “Infinite Utility: Anonymityand Person-Centredness”,Australasian Journal ofPhilosophy, 73: 413–420.
• Vallentyne, Peter and Shelly Kagan, 1997. “Infinite Valueand Finitely Additive Value Theory”,The Journal ofPhilosophy, XCIV(1): 5–27
• Van Liedekerke, Luc, 1995. “Should Utilitarians Be CautiousAbout an Infinite Future?”,Australasian Journal ofPhilosophy, 73(3): 405–407.
• Varelius, Jukka, 2013. “Pascal’s Wager and DecidingAbout the Life-Sustaining Treatment of Patients in PersistentVegetative State”,Neuroethics, 6: 277–285.
• Voltaire, F. M. A., 1778 [1961].Philosophical Letters(Letters Concerning the English Nation). Trans. E. Dilworth.Bobbs-Merrill, 17th letter, remark V.
• Weirich, Paul, 1984. “The St. Petersburg Gamble andRisk”,Theory and Decision, 17: 193–202.
• Wenmackers, Sylvia, 2018. “Infinitesimal Probabilities andPascal’s Wager”, in Bartha and Pasternack (eds.) 2018,293–314.
• Wright, Crispin, 1987. “Strict Finitism”, inRealism, Meaning and Truth, Oxford: Blackwell.

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Other Internet Resources

• Pascal's Wager: A Pragmatic Argument for Belief in God, by Liz Jackson, at1000 Word Philosophy.
• Theistic Belief and Religious Uncertainty by Jeff Jordan.
• Pascal’s Wager, by Paul Saka, in theInternet Encyclopedia ofPhilosophy.
• Pascal’s Wager, maintained by Stephen R. Welch, at infidels.org.

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Acknowledgments

I thank Bronwyn Finnigan and Liz Jackson for helpful comments.