## 1. Background



It is important to contrast Pascal's argument with various
putative 'proofs' of the existence of God that had come
before it. Anselm's ontological argument, Aquinas'
'five ways', Descartes' ontological and cosmological
arguments, and so on, purport to prove that God exists. Pascal is
apparently unimpressed by such attempted justifications of theism:
"Endeavour ... to convince yourself, not by increase of
proofs of God..." Indeed, he insists that "we do not
know if He is ...". Pascal's project, then, is
radically different. He aims to show that *we ought to
believe* in God, rather than that *God
exists*. And he seeks to provide *prudential*
reasons rather than *evidential* reasons for believing in
God. To put it simply, we should wager that God exists because it is
the *best bet*. Ryan 1994 finds precursors to this line of
reasoning in the writings of Plato, Arnobius, Lactantius, and others;
we might add Ghazali to his list--see Palacios 1920. Franklin
2018 presents striking parallels to Pascal's Wager by Sirmond
and Chillingworth from 1637 and 1638 respectively, thus predating
Pascal by a few years. But what is distinctive is Pascal's
explicitly decision-theoretic formulation of the reasoning. In fact,
Hacking 1975 describes the Wager as "the first well-understood
contribution to decision theory" (viii). Thus, we should pause
briefly 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 assign
*utilities* to such outcomes, numbers that represent the degree
to which the agent values them. It is typical to present these numbers
in a decision table, with the columns corresponding to the various
relevant states of the world, and the rows corresponding to the
various possible actions that the agent can perform.



In *decisions under uncertainty*, nothing more is
given--in particular, the agent does not assign subjective
probabilities to the states of the world. Still, sometimes rationality
dictates a unique decision nonetheless. Consider, for example, a case
that will be particularly relevant here. Suppose that you have two
possible actions, \(A\_1\) and \(A\_2\), and the worst outcome
associated with \(A\_1\) is at least as good as the best outcome
associated with \(A\_2\); suppose also that in at least one state of
the 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 you
to perform
 \(A\_1\).[1]



**Example**. A magician will toss a coin. You know
nothing about the coin--it might be a normal coin, it might be
two-headed, it might be two-tailed, and it might be biased to any
degree. Suppose that in this state of complete ignorance about the
coin, you do not assign any probability whatsoever to its landing
heads. Suppose that you can either bet on heads or on tails; it costs
nothing to bet, and you will win $1 if you bet correctly. But I will
pay you an extra $1 if you bet on heads (I especially like bets on
heads). Your possible total pay-offs are given by this decision
table:





|  |  |  |
| --- | --- | --- |
|  | ***Coin lands heads*** | ***Coin lands tails*** |
| ***Bet on heads*** | 2 | 1 |
| ***Bet on tails*** | 0 | 1 |



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



In *decisions under risk*, the agent assigns subjective
probabilities to the various states of the world. Assume that the
states of the world are independent of what the agent does. A figure
of merit called the *expected utility*, or the
*expectation* of a given action can be calculated by a simple
formula: for each state, multiply the utility that the action produces
in that state by the state's probability; then, add these
numbers. According to decision theory, rationality requires you to
perform the action of maximum expected utility (if there is one).



**Example**. Suppose that the utility of money is linear
in number of dollars: you value money at exactly its face value.
Suppose now that you know that a *fair* coin will be
tossed, 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 the
decision table:




|  |  |  |
| --- | --- | --- |
|  | ***Coin lands heads*** | ***Coin lands tails*** |
| ***Play*** | 2 | -1 |
| ***Do not play*** | 0 | 0 |



The expectation of playing is (2 x 1/2)  + (-1 x 1/2) =
1/2. This exceeds the expectation of not playing--namely
0--so you should play.



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




|  |  |  |
| --- | --- | --- |
|  | ***Coin lands heads*** | ***Coin lands tails*** |
| ***Play*** | 1 | -1 |
| ***Do not play*** | 0 | 0 |



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 in
Pascal's arguments. It should be admitted that there are certain
exegetical problems in presenting these arguments. Pascal never
finished the *Pensees*, but rather left them in the form
of notes of various sizes pinned together. Hacking 1972 describes the
"Infinite--nothing" as consisting of "two
pieces of paper covered on both sides by handwriting going in all
directions, full of erasures, corrections, insertions, and
afterthoughts"
 (24).[2]
 This may explain why certain passages are notoriously difficult to
interpret, as we will see. Furthermore, our formulation of the
arguments in the parlance of modern Bayesian decision theory might
appear somewhat anachronistic. For example, Pascal did not distinguish
between what we would now call *objective* and
*subjective* probability, although it is clear that it is the
latter that is relevant to his arguments. To some extent,
"Pascal's Wager" now has a life of its own, and our
presentation of it here is perfectly standard. Still, we will closely
follow Pascal's text, supporting our reading of his arguments as
much as possible. (See also Golding 1994 for another detailed analysis
of Pascal's reasoning, broken down into more steps than the
presentation here.)



There is the further problem of dividing the *Infinite-nothing*
into separate arguments. We will locate three arguments that each
conclude that rationality requires you to wager for God, although they
interleave in the
 text.[3]
 Finally, there is some disagreement over just what "wagering
for God" involves--is it *believing* in God, or
merely *engendering* belief? We will conclude with a discussion
of what Pascal meant by this.


## 2. The Argument from Superdominance



Pascal maintains that we are incapable of knowing whether God exists
or not, yet we must "wager" one way or the other. Reason
cannot settle which way we should incline, but a consideration of the
relevant outcomes supposedly can. Here is the first key passage:



> 
> "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... 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.
> 



There are exegetical problems already here, partly because Pascal
appears to contradict himself. He speaks of "the true" as
something that you can "lose", and "error" as
something "to shun". Yet he goes on to claim that if you
lose the wager that God is, then "you lose nothing".
Surely in that case you "lose the true", which is just to
say that you have made an error, and since this is something "to
shun", it is presumably a cost. Pascal believes, of course,
that the existence of God is "the true"--but
*that* 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 than
the other". Rather, by Pascal's own account, it is because
"[r]eason can decide nothing here". (If it could, then it
might well be shocked--namely, if you chose in a way contrary to
it.)



Following McClennen 1994, Pascal's argument seems to be best
captured as presenting the following decision table:




|  |  |  |
| --- | --- | --- |
|  | ***God exists*** | ***God does not exist*** |
| ***Wager for God*** | Gain all | Status quo |
| ***Wager against God*** | Misery | Status quo |



Wagering for God superdominates wagering against God: the worst
outcome associated with wagering for God (status quo) is at least as
good as the best outcome associated with wagering against God (status
quo); and if God exists, the result of wagering for God is strictly
better than the result of wagering against God. (The fact that the
result is *much* better does not matter yet.) Pascal draws the
conclusion at this point that you should wager for God.



Without any assumption about your probability assignment to
God's existence, the argument is invalid. Rationality does
*not* require you to wager for God if you assign probability 0
to God existing, as a strict atheist might. And Pascal does not
explicitly rule this possibility out until a later passage, when he
assumes that you assign positive probability to God's existence;
yet this argument is presented as if it is self-contained. His claim
that "[r]eason can decide nothing here" may suggest that
Pascal regards this as a decision under uncertainty, which is to
assume that you do *not* assign probability at all to
God's existence. If that is a further premise, then the argument
is apparently valid; but that premise contradicts his subsequent
assumption that you assign positive probability. See McClennen for a
reading 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 wager
> too much."
> 



The thought seems to be that if I wager for God, and God does not
exist, then I really do lose something. In fact, Pascal himself speaks
of *staking* *something* when one wagers for God, which
presumably one loses if God does not exist. (We have already mentioned
'the true' as one such thing; Pascal also seems to regard
one's worldly life as another.) In that case, the table is
mistaken in presenting the two outcomes under 'God does not
exist' as if they were the same, and we do not have a case of
superdominance after all.



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


## 3. The Argument From Expectation



He continues:



> 
> 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.
> 



Understanding "equal risk" here as "equal
probability", the probability of gain (winning the wager) and of
loss (losing the wager) must each be 1/2. His hypothetically
speaking of "two lives" and "three lives" may
strike one as odd. It is helpful to bear in mind Pascal's
interest in gambling (which after all provided the initial motivation
for his study of probability) and to take the gambling model quite
seriously here. Indeed, the Wager is permeated with gambling
metaphors: "game", "stake", "heads or
tails", "cards" and, of course, "wager".
Now, recall our calculation of the expectations of the two dollar and
three dollar gambles. Pascal apparently assumes now that utility is
linear in number of *lives*, that wagering for God costs
"one life", and then reasons analogously to the way we did
in our expectation calculations above! This is, as it were, a warm-up.
Since wagering for God is rationally required even in the hypothetical
case in which one of the prizes is three lives, then all the more it
is rationally required in the actual case, in which one of the prizes
is an *eternity* of life (salvation).



So Pascal has now made two striking assumptions:



     (1). The probability of God's existence is
1/2.



     (2). Wagering for God brings *infinite*
reward if God exists.



Morris 1994 is sympathetic to (1), while Hacking 1972 finds it
"a monstrous premiss" (189). One way to defend it is via
the classical interpretation of probability, according to which all
possibilities are given equal weight. The interpretation seems
attractive for various gambling games, which by design involve an
evidential symmetry with respect to their outcomes; and Pascal even
likens 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 of
winning the lottery; but either you win the lottery or you
don't, so each of these possibilities has probability 1/2?!)
Pascal's argument for (1) is presumably that "[r]eason can
decide nothing here". (In the lottery ticket case, reason can
decide *something*.) But it is not clear that complete
ignorance should be modeled as equiprobability. Morris imagines,
rather, an agent who does have evidence for and against the existence
of God, but it is equally balanced. In any case, it \(is\) clear that
there are people in Pascal's audience who do not assign
probability 1/2 to God's existence. This argument, then, does
not speak to them.



However, Pascal realizes that the value of 1/2 actually plays no real
role in the argument, thanks to (2). This brings us to the third, and
by 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, 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.
Pascal's talk of winning two, or three, lives is a little
misleading. By his own decision theoretic lights, you would
*not* act stupidly "by refusing to stake one life against
three at a game in which out of an infinity of chances there is one
for you"--in fact, you should not stake more than an
infinitesimal amount in that case (an amount that is bigger than 0,
but smaller than every positive real number). The point, rather, is
that the prospective prize is "an infinity of an infinitely
happy life". In short, if God exists, then wagering for God
results in infinite utility.



What about the utilities for the other possible outcomes? There is
some dispute over the utility of "misery". Hacking
interprets this as "damnation" (188), and Pascal does
later speak of "hell" as the outcome in this case. Martin
1983 among others assigns this a value of *negative infinity*.
Sobel 1996, on the other hand, is one author who takes this value to
be finite. There is some textual support for this reading: "The
justice of God must be vast like His compassion. Now justice to the
outcast is less vast ... than mercy towards the elect". As
for the utilities of the outcomes associated with God's
non-existence, Pascal tells us that "what you stake is
finite". This suggests that whatever these values are, they are
finite.



Pascal's guiding insight is that the argument from expectation
goes through equally well *whatever* your probability for
God's existence is, provided that it is non-zero and finite
(non-infinitesimal)--"a chance of gain against a finite
number of chances of
 loss".[4]



Pascal's assumptions about utilities and probabilities are now
in place. In another landmark moment in this passage, he next presents
a formulation of expected utility theory. When gambling, "every
player stakes a certainty to gain an uncertainty, and yet he stakes a
finite certainty to gain a finite uncertainty, without transgressing
against reason". How much, then, should a player be prepared to
stake without transgressing against reason? Here is Pascal's
answer: "... the uncertainty of the gain is proportioned to
the certainty of the stake according to the proportion of the chances
of gain and loss ..." It takes some work to show that this
yields expected utility theory's answer exactly, but it is work
well 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: the
first concerns the decision table of rewards, the second concerns
the probability that you should give to God's existence, and the
third is a maxim about rational decision-making. Specifically:


1. Either God exists or God does not exist, and you can either wager for
God or wager against God. The utilities of the relevant possible
outcomes are as follows, where \(f\_1, f\_2\), and \(f\_3\) are numbers
whose values are not specified beyond the requirement that they be
finite:



|  |  |  |
| --- | --- | --- |
|  | ***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 God
existing to be positive (and finite).
3. Rationality requires you to perform the act of maximum expected
utility (when there is one).
4. Conclusion 1. Rationality requires you to wager for
God.
5. Conclusion 2. You should wager for God.



We have a decision under risk, with probabilities assigned to the ways
the world could be, and utilities assigned to the outcomes. In
particular, we represent the infinite utility associated with
salvation as '\(\infty\)'. We assume that the real line is
extended to include the element '\(\infty\)', and that the
basic arithmetical operations are extended as follows:



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

For all real numbers \(r\): \(\infty \times r = \infty\) if \(r \gt
0\).



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

 
\[ \mathrm{E}(\text{wager for God}) = \infty \times p + f\_1 \times(1 - p) = \infty \]


That is, your expected utility of belief in God is infinite--as
Pascal puts it, "our proposition is of infinite force". On
the other hand, your expected utility of wagering against God is



\[ \mathrm{E}(\text{wager against God}) = f\_2 \times p + f\_3 \times(1 - p) \]


This is
 finite.[6]
 By premise 3, rationality requires you to perform the act of maximum
expected utility. Therefore, rationality requires you to wager for
God.



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 regarded
as the most important of them, 'the many Gods objection'
(see also the link to footnote 7).



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



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



2. The utility of salvation could not be infinite. One might
argue that the very notion of infinite utility is suspect--see
for example Jeffrey 1983 and McClennen
 1994.[7]
 Hence, the objection continues, whatever the utility of salvation
might be, it must be finite. Strict finitists, who are suspicious of
the notion of infinity in general, will agree--see Dummett 1978
and Wright 1987. Or perhaps the notion of infinite utility makes
sense, but an infinite reward could only be finitely appreciated by a
human being.



3. There should be more than one infinity in the table. There
are also critics of the Wager who, far from objecting to infinite
utilities, want to see *more* of them in the table. For
example, it might be thought that a forgiving God would bestow
infinite utility upon wagerers-for and wagerers-against
alike--Rescher 1985 is one author who entertains this
possibility. Or it might be thought that, on the contrary, wagering
against an existent God results in *negative* infinite utility.
(As we have noted, some authors read Pascal himself as saying as
much.) 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 God
does not exist, but that we are reincarnated *ad infinitum*,
and that the total utility we receive is an infinite sum that diverges
to infinity or to negative infinity.



4. The table should have more rows. Perhaps there is more
than one way to wager for God, and the rewards that God bestows vary
accordingly. For instance, God might not reward infinitely those who
strive to believe in Him only for the very mercenary reasons that
Pascal gives, as James 1956 has observed. One could also imagine
distinguishing belief based on faith from belief based on evidential
reasons, and posit different rewards in each case.



5. The table should have more columns: the many Gods
objection. If Pascal is really right that reason can decide
nothing here, then it would seem that various other theistic
hypotheses are also live options. Pascal presumably had in mind the
Catholic conception of God--let us suppose that this is the God
who either 'exists' or 'does not exist'. By
excluded middle, this is a partition. The objection, then, is that the
partition is not sufficiently fine-grained, and the '(Catholic)
God does not exist' column really subdivides into various
*other* theistic hypotheses. The objection could equally run
that Pascal's argument 'proves too much': by
parallel reasoning we can 'show' that rationality requires
believing 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 various
ways. Mackie 1982 writes, "the church within which alone
salvation is to be found is not necessarily the Church of Rome, but
perhaps that of the Anabaptists or the Mormons or the Muslim Sunnis or
the worshippers of Kali or of Odin" (203). Cargile 1966 shows
just how easy it is to multiply theistic hypotheses: for each real
number \(x\), consider the God who prefers contemplating \(x\) more
than any other activity. It seems, then, that such 'alternative
gods' are a dime a dozen--or \(\aleph\_1\), for that
matter.



In response, some authors argue that in such a competition among
various possible deities for one's belief, some are more
probable than others. Although there may be ties among the expected
utilities--all infinite--for believing in various ones among
them, their respective probabilities can be used as tie-breakers.
Schlesinger (1994, 90) offers this principle: "In cases where
the mathematical expectations are infinite, the criterion for choosing
the outcome to bet on is its probability". (Note that this
principle is not found in the Wager itself, although it might be
regarded as a friendly addition. Askell 2018 proposes a similar
principle.) Are there reasons, then, for assigning higher probability
to some Gods than others? Jordan (1994a, 107) suggests that some
outlandish theistic hypotheses may be dismissed for having "no
backing of tradition". Similarly, Schlesinger maintains that
Pascal is addressing readers who "have a notion of what genuine
religion is about" (88), and we might take that to suggest that
Cargile's imagined Gods, for example, may be correspondingly
assigned lower probability than Pascal's God. Franklin (2018,
41) writes that "[Pascal's] rhetoric is addressed to real
agents, namely "men of the world" in the Paris of 1660" for
whom "the spectrum of religious theories to which they attached
grounded subjective non-zero probability consisted of just Catholicism
and atheism". Saka (2018, 190-191) replies to various authors
who 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 of
alleged Satanism ... and they knew, from their acquaintance with the
foregoing, that still other religions could readily be
hypothesized." But still there is the issue of what
probabilities should be assigned to alternative deities. Lycan
and Schlesinger 1989 give more theoretical reasons for favoring
Pascal's God over others in one's probability assignments.
They begin by noting the familiar problem in science of
underdetermination of theory by evidence. Faced with a multiplicity of
theories that all fit the observed data equally well, we favor the
simplest such theory. They go on to argue that simplicity
considerations similarly favor a conception of God as
"absolutely perfect", "which is theologically unique
in that it implies all the other predicates traditionally ascribed to
God" (104), and we may add that this conception is
Pascal's. Conceptions of rival Gods, by contrast, leave open
various questions about their nature, the answering of which would
detract from their simplicity, and thus their probability.



Finally, Bartha 2012 models one's probability assignments to
various theistic hypotheses as evolving over time according to a
'deliberational dynamics' somewhat analogous to the
dynamics of evolution by natural selection. So understood,
Pascal's Wager is not a single decision, but rather a sequence
of decisions in which one's probabilities update sequentially in
proportion to how choiceworthy each God appeared to be in the previous
round. (This relies on a sophisticated handling of infinite utilities
in terms of utility ratios given in his 2007; see below.) He argues
that a given probability assignment is choiceworthy only if it is an
equilibrium of this deliberational dynamics. He shows that certain
assignments are choiceworthy by this criterion, thus providing a kind
of 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 are
straightforward; the second two are more technical, and can be found
by following the link to footnote 9.



1. Undefined probability for God's existence. Premise 1
presupposes that you should *have* a probability for
God's existence in the first place. However, perhaps you could
rationally *fail* to assign it a probability--your
probability that God exists could remain *undefined*. We cannot
enter here into the thorny issues concerning the attribution of
probabilities to agents. But there is some support for this response
even in Pascal's own text, again at the pivotal claim that
"[r]eason 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..." The
thought could be that any probability assignment is inconsistent with
a state of "epistemic nullity" (in Morris' 1986
phrase): to assign a probability at all--even 1/2--to
God's existence is to feign having evidence that one in fact
totally lacks. For unlike a coin that we know to be fair, this
metaphorical 'coin' is 'infinitely far' from
us, hence apparently completely unknown to us. Perhaps, then,
rationality actually requires us to *refrain* from assigning a
probability to God's existence (in which case at least the
Argument from Superdominance would apparently be valid). Or perhaps
rationality does not require it, but at least *permits* it.
Either way, the Wager would not even get off the ground.



2. Zero probability for God's existence. Strict atheists
may insist on the rationality of a probability assignment of 0, as
Oppy 1990 among others points out. For example, they may contend that
reason alone *can* settle that God does not exist, perhaps by
arguing that the very notion of an omniscient, omnipotent,
omnibenevolent being is contradictory. Or a Bayesian might hold that
rationality places no constraint on probabilistic judgments beyond
coherence (or conformity to the probability calculus). Then as long as
the strict atheist assigns probability 1 to God's non-existence
alongside his or her assignment of 0 to God's existence, no norm
of rationality has been violated.



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



\[ \infty \times 0 = 0 \]


For then the expectation calculations become:

 
\[\begin{align\*}
\mathrm{E}(\text{wager for God}) &= \infty \times 0 + f\_1 \times(1 - 0) \\
 &= f\_1 \\
 &\\
 \mathrm{E}(\text{wager against God}) &= f\_2 \times 0 + f\_3 \times(1 - 0) \\
 &= f\_3 
\end{align\*}\]


And nothing in the argument implies that \(f\_1 \gt f\_3\). (Indeed,
this inequality is questionable, as even Pascal seems to allow.) In
short, 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 theoretic
assumption that rationality requires one to perform the act of maximum
expected utility (when there is one). Now *perhaps* this is an
analytic truth, in which case we could grant it to Pascal without
further discussion--perhaps it is *constitutive* of
rationality to maximize expectation, as some might say. But this
premise has met serious objections. The Allais 1953 and Ellsberg 1961
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.)[10]



Various refinements of expected utility theory have been suggested as
a result of such problems. For example, we might consider expected
*differences* between the pay-offs of options, and prefer one
option to another if and only if the expected difference of the former
relative to the latter is positive--see Hajek and Nover
2006, Hajek 2006, Colyvan 2008, and Colyvan & Hajek
2016. 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
2007. 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. Nonetheless, the door is opened to some
suitable reformulation of it that might serve Pascal's purposes.
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. This objection is especially relevant, since Pascal admits
that perhaps you "must renounce reason" in order to follow
his advice. 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
1994.)


### 5.4 Is the Argument Valid?



A number of authors who have been otherwise critical of the Wager have
explicitly conceded that the Wager is valid--e.g. Mackie 1982,
Rescher 1985, Mougin and Sober 1994, and most emphatically, Hacking
1972. That is, these authors agree with Pascal that wagering for God
really is rationally mandated by Pascal's decision table in
tandem with positive probability for God's existence, and the
decision theoretic account of rational action.



However, Duff 1986 and Hajek 2003 argue that the argument is in
fact invalid. 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. Consider the mixed
strategy: "Toss a fair coin: heads, you wager for God; tails,
you wager against God". By Pascal's lights, with
probability 1/2 your expectation will be infinite, and with
probability 1/2 it will be finite. The expectation of the entire
strategy is:

 
\[ \frac{1}{2} \times \infty + \frac{1}{2} \times [f\_2 \times p + f\_3 \times(1 - p)] = \infty \]


That is, the 'coin toss' strategy has the same expectation
as outright wagering for God. But the probability 1/2 was incidental
to the result. Any mixed strategy that gives positive and finite
probability to wagering for God will likewise have infinite
expectation: "wager for God iff a fair die lands 6",
"wager for God iff your lottery ticket wins", "wager
for God iff a meteor quantum tunnels its way through the side of your
house", and so on.



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. So even if rationality requires you to perform the act
of maximum expected utility when there is one, here there isn't
one. 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![11]



Monton 2011 defends Pascal's Wager against this line of
objection. 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. Monton's thought
is that your expected utility now changes; it is no longer infinite,
but rather that of an atheist or agnostic who has no prospect of the
infinite reward for wagering for God. 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 2012 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. Think, for example, of rolling a 4-sided die, then
a 9-sided die, and in general an \((n+1)^2\)-sided die on the
\(n\)th trial ..., a strategy for which the
probability that you will eventually wager for God is only 1/2, as
Robertson shows. However, Easwaran and Monton 2012 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? By Monton's argument, it seems you should roll a die
again. 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. To repeat,
the objection's upshot is that even granting Pascal all his
premises, still wagering for God is not rationally required. But we
have seen numerous reasons *not* to grant all his premises.
Very well then; let's not. Indeed, let's suppose that you
give *tiny probability p* to them all being true, where \(p\)
is positive and finite. So you assign probability \(p\) to your
decision problem being exactly as Pascal claims it to be. But if it
is, according to the mixed strategies objection, all hell breaks
loose. Yet again, \(p\) multiplied by infinity gives infinity. 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. There are some subtleties that we have elided over; for example,
if you also assign positive and finite probability to a source of
*negative* infinite utility, then the expected utilities
instead become \(\infty\) - \(\infty\), which is undefined. 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. We
might call this *Pascal's Revenge*. See Hajek
(2015) for more discussion.



Jackson and Rogers (2019), developing points in Jackson (2016), argue
that the mixed-strategies objection is a "structural, but not
substantive" (61) objection to Pascal's Wager. They provide
cases in which it is clearly rational to prefer one infinite good to
another. They suggest a reformulation of how prospects of infinite
rewards should be compared. (This also provides a response to the many
Gods objection.) Hajek (2003 and especially 2018) offers many
valid reformulations of the Wager with more nuanced representations of
the utility of salvation, such that the lower the probability of
wagering for God, the lower the expected utility. See 
 footnote 11 for
further discussion.


### 5.5 Moral Objections to Wagering for God



Let us grant Pascal's first conclusion for the sake of the
argument: rationality requires you to wager for God. The second
conclusion, that you *should* wager for God, does not obviously
follow. 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. He admits that if you do not believe in
God, his recommended course of action "will deaden your
acuteness" (This is Trotter's translation. Pascal's
original French wording is "vous abetira", whose
literal translation is even more startling: "will make you a
beast".) One way of putting the argument is that wagering for
God may require you to corrupt yourself, thus violating a Kantian duty
to yourself. Clifford 1877 argues that an individual's believing
something on insufficient evidence harms society by promoting
credulity. Penelhum 1971 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 1994 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. (Note, however, that Pascal
leaves open in the Wager whether the payoff for non-believers \(is\)
just; indeed, as far as his argument goes, it may be extremely
unjust.)



Finally, Voltaire protests that there is something unseemly about the
whole Wager: "That article seems a bit indecent and childish;
that notion of gambling, of losses and winnings, does not suit the
gravity of the subject" (Voltaire 1778 [1961, 123]). This does
not so much support wagering against God, as dismissing all talk of
'wagerings' altogether. Schlesinger (1994, 84) canvasses a
sharpened formulation of this objection: an appeal to greedy,
self-interested motivations is incompatible with "the quest for
piety" that is essential to religion. He replies that the
pleasure of salvation that Pascal's Wager countenances is
"of the most exalted kind", and that if seeking it counts
as greed at all, then it is "the manifestation of a noble greed
that is to be acclaimed" (85). Franklin (1998, 2018) regards
Voltaire as caricaturing Pascal's Wager and missing his key point that
"you must wager". Franklin argues that given that the
choice Pascal presents is unavoidable, it should be made on the basis
of a rational calculation, and that his conclusion concerns
*action*, 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 in
store for us. In his "End of this address" he writes
regarding wagering for God:



> 
> Now, what harm will befall you in taking this side? You will be
> faithful, humble, grateful, generous, a sincere friend, truthful.
> Certainly you will not have those poisonous pleasures, glory and
> luxury; but will you not have others? I will tell you that you will
> thereby gain in this life, and that, at each step you take on this
> road, you will see so great certainty of gain, so much nothingness in
> what you risk, that you will at last recognise that you have wagered
> for something certain and infinite, for which you have given
> nothing.



This passage makes two further striking claims regarding wagering for
God: you will "gain in this life", and "you have
wagered for something certain". The decision table can be
presented as follows (with the outcomes ranked):




|  |  |  |
| --- | --- | --- |
|  | ***God exists*** | ***God does not exist*** |
| ***Wager for God*** | 
Gain all

(Best) | 
Gain in earthly life

(Second best) |
| ***Wager against God*** | 
Misery

(Worst) | 
Earthly life

(Third best) |



This is again a decision under uncertainty (in our technical
sense)--it involves no considerations of probability. Indeed,
"the wager" dissolves, twice over: utilities alone
definitively settle that you should wager for God, and in any case it
is not really a gamble at all, since your gain is certain! (That is my
understanding of "you have wagered for something certain";
it is not God's existence itself that
is "certain".) The worst outcome associated with
wagering for God (gain in earthly life) is *strictly
better* than the best outcome associated with wagering
against God (earthly life). It follows immediately that you should
wager for God. Hajek (2018) calls this an argument
from *superduperdominance*. This is a valid argument for
wagering for God, even if we allow that God's existence might be
impossible.



Pascal has come full circle back to the first wager, and he now goes
even beyond it. The solution to *this* decision
problem is trivial. The status of this "wager"--the
soundness or otherwise of this argument--turns on whether one's
prospects 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 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.g. Quinn 1994, and Jordan 1994a.
But perhaps one cannot simply believe in God at will; and rationality
cannot require the impossible. Pascal is well aware of this objection:
"[I] am so made that I cannot believe. What, then, would you
have me do?", says his imaginary interlocutor. However, he
contends that one can take steps to cultivate such belief:



> 
> 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. ...
> 
> 
> 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. Critics may question the psychology of belief
formation that Pascal presupposes, pointing out that one could strive
to believe (perhaps by following exactly Pascal's prescription),
yet fail. 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.



According to Pascal, 'wagering for God' and
'wagering against God' are contradictories, as there is no
avoiding wagering one way or another: "you must wager. It is not
optional." The decision to wager for or against God is one that
you make at a time--at \(t\), say. 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. What Pascal intends by
'wagering for God' is an ongoing action--indeed, one
that continues until your death--that involves your adopting a
certain set of practices and living the kind of life that fosters
belief in God. The decision problem for you at \(t\), then, is whether
you should embark on this course of action; to fail to do so is to
wager against God at \(t\).


## 8. The Continuing Influence of Pascal's Wager



Pascal's Wager vies with Anselm's Ontological Argument for
being the most famous argument in the philosophy of religion. 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 1980 and Skyrms 1980 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. It raises subtle issues about the extent to
which one's beliefs can be a matter of the will, and the ethics
of belief.



Reasoning reminiscent of Pascal's Wager, often with an explicit
acknowledgment of it, also informs a number of debates in moral
philosophy, both theoretical and applied. Kenny 1985 suggests that
nuclear Armageddon has negative infinite utility, and some might say
the same for the loss of even a single human life. Stich 1978
criticizes an argument that he attributes to Mazzocchi, that there
should be a total ban on recombinant DNA research, since such research
could lead to the "Andromeda scenario" of creating a
killer strain of bacterial culture against which humans are helpless;
the ban, moreover, should be enforced if the "Andromeda scenario
has even the smallest possibility of occurring" (191), in
Mazzocchi's words. This is plausibly read, then, as an
assignment of negative infinite utility to the Andromeda scenario.
More recently, Colyvan, Cox, and Steele 2010 discuss Pascal's
Wager-like problems for certain deontological moral theories, in which
violations of duties are assigned negative infinite utility. Colyvan,
Justus and Regan 2011 canvas difficulties associated with assigning
infinite value to the natural environment. Bartha and DesRoches 2017
respond, with an appeal to relative utility theory. Stone 2007 argues
that a version of Pascal's Wager applies to sustaining patients
who are in a persistent vegetative state; see Varelius 2013 for a
dissenting view. Pascal's Wager has even been appealed to in the
medical debate over whether antibiotics should be used to prevent a
certain kind of inflammation in the heart (Shaw and Conway 2010).



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