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#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
#import "@preview/wordometer:0.1.3": word-count, total-words
#show: ams-article.with(
title: [On Pascal's Wager],
bibliography: bibliography("refs.bib"),
)
#set cite(style: "institute-of-electrical-and-electronics-engineers")
#set text(fractions: true)
#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))
= Introduction
The argument for Betting on God, or better known as Pascal's Wager, says that
you should believe in God, regardless of other evidence, purely out of
self-interest. In this paper, I will challenge this argument by assessing the
premise that believing in a particular God always guarantees the greatest
expected utility.
The argument makes heavy use of the concepts of utility and expected utility.
Utility is essentially the usefulness of an action, or to what degree it helps
increase "good," like happiness, pleasure, benefit, and decrease "bad," like
suffering or harm. Given a set of possible actions and distinct possible
outcomes, each action may be assigned an "expected utility" by pairing the
action with each possible outcome and assigning every action-outcome pair some
amount of utility. Using the probabilities of each outcome occurring, we can
compute a weighted average that gives the expected utility of the action.
More precisely, let us define a set of $n$ actions
$ {a_1, a_2, ..., a_n} in A $
where $a_k$ represents the $k^"th"$ action, and a set of $m$ outcomes
$ {o_1, o_2, ..., o_m} in O $
where $o_k$ represents the $k^"th"$ outcome. Additionally, let
$ rho (o_k) $
be the probability of the outcome $o_k$ occurring.
We compute the *Cartesian product* $A times O$ which contains ordered pairs of
the form $(a_k, o_k)$ representing every possible combination of action and
outcome. Formally,
$ A times O = {(a_j, o_i) | j in {1,2,...,n}, i in {1,2,...,m}} $
We assign each action-outcome pair its utility as we deem fit. The function
$ U ((a_k, o_k)) $
gives the utility of an ordered action-outcome pair $(a_k, o_k)$.
Then, to determine the expected utility for an action $a_k$, we select all of
the ordered pairs with $a_k$ in the first position, multiply their utility by
the probability of their corresponding outcome occurring, and sum of all of
these products. In precise terms, given $m$ possible outcomes, then:
$ "Expected utility of" a_k = sum_(i=1)^(m) rho (o_i) dot U ((a_k, o_i)) $
In order to make this more clear, we construct a so-called "decision matrix"
where we can easily assign a utility value for each action-outcome pair and
then calculate the expected utility.
Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
<Korman2022-KORLFA>) which gives the expected utility for believing or not
believing in God.
#show table.cell.where(x: 0): strong
#show table.cell.where(y: 0): strong
#figure(
caption: [Pascal's Wager],
align(
center,
table(
columns: (auto, auto, auto, auto),
table.header(
[],
[God exists ($50%$)],
[God doesn't exist ($50%$)],
[Expected utility],
),
[ Believe in God ], [$infinity$], [2], [$infinity$],
[
Don't believe in God
],
[1],
[3],
[2],
),
),
)
== The argument for betting on God
The author's argument for belief in God #cite(supplement: [p. 38],
<Korman2022-KORLFA>) goes as follows:
$
&"(BG1) One should always choose the option with the greatest expected utility" \
&"(BG2) Believing in God has a greater expected utility than not believing in God" \
&"(BG3) So, you should believe in God"
$
BG1 should be generally uncontroversial. If you expect an action to bring you
the most utility (i.e. be the most useful), why wouldn't you do it?
BG2 is also substantiated by the decision matrix. All 4 action-outcome pairs
are assigned a utility with the following logic. If you believe in God, but God
doesn't exist, you've led a pious life without gaining much in return. If you
don't believe in God, and God doesn't exist, then you have it slightly better
than the previous scenario. You haven't wasted your time on religious
activities (like going to church) and end up with the same fate as the
believers.
If God does in fact exist, however, then believing in God gives you an
_unlimited_ amount of utility. You end up in an afterlife of eternal bliss and
pleasure, more valuable than anything you could gain on earth. That means that
the worst scenario is not believing in God and God existing, because you've
just missed out on the eternal afterlife. So, the expected utility for not
believing is $0.5 times 1 + 0.5 times 3 = 2$, and the expected utility is $0.5
times infinity + 0.5 times 2 = infinity$. If, according to BG1, you should pick
the option with greatest expected utility, you should clearly choose to believe
in God, because the expected utility is $infinity$.
Additionally, notice that the actual probability of God existing doesn't
matter, because any non-zero value multiplied by $infinity$ is still
$infinity$, and so as long as you believe there is a _non-zero chance_ that God
exists, the infinite expected utility of believing remains. Adjusting the
probabilities may increase or decrease the expected utility of not believing in
God, but not believing in God will never give you the opportunity of attaining
the afterlife of infinite utility, so it can never reach the infinite expected
utility of believing in God.
I will show that Pascal's Wager fails because BG2 fails. Namely, we cannot know
whether or not believing in God has the greatest expected utility because it
makes no sense to even calculate expected utilities of believing in God. In
section 2, I present my objection to BG2, and in section 3, I will address a
few possible responses to my objection.
#pagebreak()
= Many Gods
Maybe there are more gods than just the one that sends you to an eternal
afterlife for believing. The author addresses this in
#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]) concluding that even if
other gods exist, it is still preferable to choose any specific god who may
grant you an eternal afterlife of pleasure than to not believe, since the
expected utility of belief is still $infinity$. Essentially, the argument makes
no claims as to _which_ god you choose, but says that you should believe in
_some_ god.
However, this leaves out the possibility of gods who punish you for believing
in the wrong god. These gods may grant eternal afterlifes for other reasons, or
perhaps even punish people with eternal suffering for belief in the wrong god.
This introduces _negative utilities_, since being punished for all of eternity
in hell is much worse than simply dying and not receiving any afterlife at all.
Let us modify our decision matrix to accommodate an outcome where we believed
in the wrong god. There are two scenarios: either we believe in the wrong god,
but the true god is _forgiving_, so we are not punished, or we believe in the
wrong god, and the true god happens to be _spiteful_ and punishes us with
eternal damnation.
#[
#set figure()
#figure(
caption: [Other gods existing],
table(
columns: (auto, auto, auto, auto, auto, auto),
align: center,
table.header(
[],
[Correct god exists ($25%$)],
[No god exists ($25%$)],
[Wrong god, spiteful ($25%$)],
[Wrong god, forgiving ($25%$)],
[E.U.],
),
[ Believe in God ], [$infinity$], [3], [$-infinity$], [1], [$?$],
[
Don't believe in God
],
[2],
[4],
[2],
[2],
[2.5],
),
)
]<other-gods-table>
We've added the new options to our matrix. #smallcaps[Wrong god, spiteful]
represents the outcome where we are punished for believing in the wrong god,
and #smallcaps[Wrong god, forgiving] represents the outcome where we are not
punished, but we still missed out on the afterlife. This is slightly worse than
being an atheist and missing out. If you are an atheist, then the outcome is
the same no matter which god exists: you miss out on heaven. Again, the exact
numbers don't matter too much when working with the infinities. However, we now
have the possibility of the worst case of all: eternal punishment for believing
in the wrong god. If eternal bliss in heaven has a utility of $infinity$, then
it follows that we should represent eternal punishment in hell with a utility
of $-infinity$.
Our new matrix has a problem: how do we calculate the expected utility?
$infinity + (-infinity)$, is an indeterminate value. We cannot really perform
algebraic operations on $infinity$. Indeed, it makes no sense to add or
subtract our infinite expected utilities.
Since the author uses this decision matrix approach to justify BG2, it now
fails. Once negative infinities are introduced, calculating expected utilities
in the usual method becomes meaningless.
#linebreak()
= Addressing Objections
== Believing in a god is still preferable to atheism
One might argue that believing in a god that rewards believers is always
preferable to atheism since you at least have the _opportunity_ to receive
eternity in heaven. Perhaps there exists a god who punishes non-believers with
eternal damnation. Then, even without the exact expected utility calculation,
it's clear that the expected utility of believing in some god must be higher
than believing in none as you stand to gain more. Either as a theist or
atheist, you run the risk of eternal punishment, but you only have the
opportunity to go to heaven by believing in some god rather than none.
Fair, the possibility that you are punished for believing in the wrong god
doesn't imply that you should be an atheist either. Indeed, there may be a god
that punishes atheists. However, there could also exist a god who sends
everyone to heaven regardless. Or perhaps they only send atheists to heaven.
Either way, there is also the possibility of attaining the infinite afterlife
in heaven by being an atheist, so it's still impossible to say that the
expected utility of believing in god is must be higher.
== Finite utilities
We can avoid the issues with $infinity$ in utility calculations by simply not
using it. Instead, simply say that the utility of going to heaven is an
immensely large finite number. The utility of going to hell is likewise a very
negative number. Now, we no longer run into the issue of being unable to
compare utilities. All of our expected utility calculations will succeed, and
given sufficiently large utilities, we should be able to make similar arguments
for believing in god.
The problem with this argument is that we now open our expected utilities up to
individual subjective determination. A core feature of the previous argument
involving infinite utilities is that they can effectively bypass numerical
comparison. If, instead, finite utilities were used, then each person may
assign different utilities to each possible outcome based on their own beliefs.
Also, the probabilities are no longer irrelevant, so they must be analyzed as
well. This greatly complicates the decision matrix.
An implied feature of BG2 is that believing in god has a greater expected
utility for _everyone_. Suppose there is someone who believes that the
suffering of being condemned to hell for eternity is worse (in absolute terms)
than the joy of being rewarded with heaven for eternity is good. In precise
terms, given the utility of being rewarded with an eternity in heaven $U_r$,
and the utility of being punished with an eternity in hell, $U_p$, such that
$ abs(U_p) > U_r $
Then, substituting these values for $infinity$ and $-infinity$ in
#link(<other-gods-table>)[Table 2], it's actually possible to obtain an
expected utility of believing in god that is less than the expected utility of
not believing. We can no longer say that BG2 is universally true for
_everyone_, so it no longer holds.
#[
= Paper Logistics
There are #total-words words in this paper, discounting this section as well
as any content in tables.
== AI Contribution Statement
#quote[I did not use AI in the writing of this paper.]
]<wordcount-exclude>

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@ -9,59 +9,22 @@
#set cite(style: "institute-of-electrical-and-electronics-engineers")
#set text(fractions: true)
#show: word-count
#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))
= Introduction
Pascal's Wager says that you should believe in God out of a utilitarian
The argument for Betting on God, or better known as Pascal's Wager, says that
you should believe in God, regardless of other evidence, purely out of
self-interest. In this paper, I will challenge this argument by assessing the
premise that believing in a particular God always guarantees the greatest
expected utility.
The argument makes heavy use of the concepts of utility and expected utility.
Utility is essentially the usefulness of an action, or to what degree it helps
increase "good," like happiness, pleasure, benefit, and decrease "bad," like
suffering or harm. Given a set of possible actions and distinct possible
outcomes, each action may be assigned an "expected utility" by pairing the
action with each possible outcome and assigning every action-outcome pair some
amount of utility. Using the probabilities of each outcome occurring, we can
compute a weighted average that gives the expected utility of the action.
More precisely, let us define a set of $n$ actions
$ a_1, a_2, ..., a_n in A $
where $a_k$ represents the $k^"th"$ action, and a set of $m$ outcomes
$ o_1, o_2, ..., o_m in O $
where $o_k$ represents the $k^"th"$ outcome. Additionally, let
$ "Prob"(o_k) $
be the probability of the outcome $o_k$ occurring.
We calculate the *Cartesian product* $A times O$ which contains ordered pairs
of the form $(a_k, o_k)$ representing every possible combination of action and
outcome.
$ A times O = {(a_j, o_i) | j in {1,2,...,n}, i in {1,2,...,m}} $
We assign each action-outcome pair its utility as we deem fit. The function
$ "Util"((a_k, o_k)) $
gives the utility of an ordered action-outcome pair $(a_k, o_k)$.
Then, to determine the expected utility for an action $a_k$, we select all of
the ordered pairs with $a_k$ in the first position, multiply their utility by
the probability of their corresponding outcome occurring, and sum of all of
these products. In precise terms, given $m$ possible outcomes, then:
$ "Expected utility of" a_k = sum_(i=1)^(m) "Prob"(o_i) dot "Util"((a_k, o_i)) $
In order to make this more clear, we construct a so-called "decision matrix"
where we can easily assign a utility value for each action-outcome pair and
then calculate the expected utility.
The author uses a so-called "decision matrix" to compute the expected utility
of each combination of action and possible outcome. The possible actions are
placed on the rows, and the possible outcomes are placed on the columns, except
for the last column, which is the calculated expected utility. At each
intersection of a row and column, we put the utility we gain from that
combination of action and outcome.
Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
<Korman2022-KORLFA>) which gives the expected utility for believing or not
@ -69,6 +32,7 @@ believing in God.
#show table.cell.where(x: 0): strong
#show table.cell.where(y: 0): strong
#figure(
caption: [Pascal's Wager],
align(
@ -88,12 +52,14 @@ believing in God.
],
[1],
[3],
[$2$],
[2],
),
),
)
== The argument for betting on God
Note that the numerical utility values themselves have no meaning, and they are
meant to be viewed relative to each other. Utility doesn't literally provide an
empirical measure of "usefulness" or "happiness."
The author's argument for belief in God #cite(supplement: [p. 38],
<Korman2022-KORLFA>) goes as follows:
@ -108,22 +74,28 @@ BG1 should be generally uncontroversial. If you expect an action to bring you
the most utility (i.e. be the most useful), why wouldn't you do it?
BG2 is also substantiated by the decision matrix. All 4 action-outcome pairs
are assigned a utility with the following logic. If you believe in God, but God
doesn't exist, you've led a pious life without gaining much in return. If you
don't believe in God, and God doesn't exist, then you have it slightly better
than the previous scenario. You haven't wasted your time on religious
are assigned a utility with the following logic.
If you believe in God, but God doesn't exist, you've led a pious life without
gaining much in return, so we say that has a utility of 2.
If you don't believe in God, and God doesn't exist, then you have it slightly
better than the previous scenario. You haven't wasted your time on religious
activities (like going to church) and end up with the same fate as the
believers.
believers, so let's give it a utility of 3.
If God does in fact exist, however, then believing in God gives you an
_unlimited_ amount of utility. You end up in an afterlife of eternal bliss and
pleasure, more valuable than anything you could gain on earth. That means that
the worst scenario is not believing in God and God existing, because you've
just missed out on the eternal afterlife. So, the expected utility for not
believing is $0.5 times 1 + 0.5 times 3 = 2$, and the expected utility is $0.5
times infinity + 0.5 times 2 = infinity$. If, according to BG1, you should pick
the option with greatest expected utility, you should clearly choose to believe
in God, because the expected utility is $infinity$.
just missed out on the eternal afterlife. Let's assign the first scenario a
utility of $infinity$ and the second a utility of 1.
So, the expected utility for not believing is $0.5 times 1 + 0.5 times 3 = 2$,
and the expected utility is $0.5 times infinity + 0.5 times 2 = infinity$. If,
according to BG1, you should pick the option with greatest expected utility,
you should clearly choose to believe in God, because the expected utility is
$infinity$.
Additionally, notice that the actual probability of God existing doesn't
matter, because any non-zero value multiplied by $infinity$ is still
@ -131,85 +103,153 @@ $infinity$, and so as long as you believe there is a _non-zero chance_ that God
exists, the infinite expected utility of believing remains. Adjusting the
probabilities may increase or decrease the expected utility of not believing in
God, but not believing in God will never give you the opportunity of attaining
the afterlife of infinite utility, so it can never react the infinite expected
the afterlife of infinite utility, so it can never reach the infinite expected
utility of believing in God.
I will show that Pascal's Wager fails because BG2 fails. Namely, we cannot know
whether or not believing in God has the greatest expected utility because it
makes no sense to even calculate expected utilities of believing in God. In
whether or not believing in God has the greatest expected utility because the
decision matrix approach fails when more possible outcomes are introduced. In
section 2, I present my objection to BG2, and in section 3, I will address a
few possible responses to my objection.
= Many Gods
= Unlimited Suffering
Maybe there are more gods than just the one that sends you to an eternal
afterlife for believing. The author addresses this in
#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]) but concludes that even if
#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), concluding that even if
other gods exist, it is still preferable to choose any specific god who may
grant you an eternal afterlife of pleasure than to not believe, since the
expected utility of belief is still $infinity$. Essentially, the argument makes
no claims as to _which_ god you choose, but says that you should believe in
_some_ god.
However, this leaves out the possibility of _evil_ or _weird_ gods. These gods
may grant eternal afterlifes for other reasons, or perhaps even punish people
with eternal suffering for belief in the wrong god. This introduces _negative
utilities_, since being punished for all of eternity in hell is much worse than
simply dying and not receiving any afterlife at all.
However, this leaves out the possibility of gods who punish you for believing
in the wrong god. These gods may grant eternal afterlifes for other reasons, or
perhaps even punish people with eternal suffering for belief in the wrong god.
This introduces _negative utilities_, since being punished for all of eternity
in hell is much worse than simply dying and not receiving any afterlife at all.
Suppose that there is an Weird God who punishes anyone who even believes in a
deity at all, and does nothing to those who don't.
Let us modify our decision matrix to accommodate an outcome where we believed
in the wrong god. There are two scenarios: either we believe in the wrong god,
but the true god is _forgiving_, so we are not punished, or we believe in the
wrong god, and the true god happens to be _spiteful_ and punishes us with
eternal damnation.
#pagebreak()
#[
#set figure()
#figure(
caption: [Weird God],
align(
center,
caption: [Other gods existing],
table(
columns: (auto, auto, auto, auto, auto),
columns: (auto, auto, auto, auto, auto, auto),
align: center,
table.header(
[],
[Christian God exists ($50%$)],
[Correct god exists ($25%$)],
[No god exists ($25%$)],
[Weird God exists ($25%$)],
[Expected utility],
[Wrong god, spiteful ($25%$)],
[Wrong god, forgiving ($25%$)],
[E.U.],
),
[ Believe in God ], [$infinity$], [2], [$-infinity$], [$?$],
[Believe in Weird God], [1], [2], [$-infinity$], [$-infinity$],
[ Believe in God ], [$infinity$], [3], [$-infinity$], [1], [$?$],
[
Don't believe in God
],
[1],
[2],
[4],
[2],
),
[2],
[2.5],
),
)
]<other-gods-table>
We've added the Weird God to the decision matrix. Believing in a Weird God and
the Christian God existing means you missed out on an eternal afterlife, so
we've assigned it the same utility as not believing and the Christian God
existing. The same reasoning applies for believing and no gods existing. And of
course, believing in the Weird God and them actually existing gives you an
eternal afterlife in hell, so it has $-infinity$ utility, which means the
expected utility of believing in Weird God is $-infinity$.
We've added the new options to our matrix. #smallcaps[Wrong god, spiteful]
represents the outcome where we are punished for believing in the wrong god,
and #smallcaps[Wrong god, forgiving] represents the outcome where we are not
punished, but we still missed out on the afterlife. This is slightly worse than
being an atheist and missing out. If you are an atheist, then the outcome is
the same no matter which god exists: you miss out on heaven. Again, the exact
numbers don't matter too much when working with the infinities. However, we now
have the possibility of the worst case of all: eternal punishment for believing
in the wrong god. If eternal bliss in heaven has a utility of $infinity$, then
it follows that we should represent eternal punishment in hell with a utility
of $-infinity$.
But what if you believe in the Christian God, and the Weird God actually
exists? Clearly you get sent to hell for eternity, resulting in a utility of
$-infinity$. But how do we calculate the expected utility? We can't just do
$infinity + (-infinity)$, as that's an indeterminate value. There are an
infinite amount of real numbers, and an infinite amount of integers. Subtracing
these infinities, however, is entirely meaningless. Indeed, it makes no sense
to add or subtract our infinite expected utilities. The entire calculation of
our expected utilities is meaningless, and so BG2 cannot be true, since we've
shown that the decision matrix approach used to justify it becomes unworkable
with the introduction of negative utility and $-infinity$.
There is a problem: how do we calculate the expected utility of believing in
god? $infinity + (-infinity)$, is an indeterminate value. We cannot really
perform algebraic operations on $infinity$. Indeed, it makes no sense to add or
subtract our infinite expected utilities.
Since the author uses this decision matrix approach to justify BG2, it now
fails. Once negative infinities are introduced, calculating expected utilities
in the usual method becomes meaningless. It is not that BG2 is necessarily
_wrong_, it just cannot be decided either way with the decision matrix. If BG2
cannot be determined, then BG3 is also indeterminate.
= Addressing Objections
== Believing in a god is still preferable to atheism
One might argue that believing in a god that rewards believers is always
preferable to atheism since you at least have the _opportunity_ to receive
eternity in heaven. Perhaps there exists a god who punishes non-believers with
eternal damnation. Then, even without the exact expected utility calculation,
it's clear that the expected utility of believing in some god must be higher
than believing in none as you stand to gain more. Either as a theist or
atheist, you run the risk of eternal punishment, but you only have the
opportunity to go to heaven by believing in some god rather than none.
Fair, the possibility that you are punished for believing in the wrong god
doesn't imply that you should be an atheist either. Indeed, there may be a god
that punishes atheists. However, there could also exist a god who sends
everyone to heaven regardless. Or perhaps they only send atheists to heaven.
Either way, there is also the possibility of attaining the infinite afterlife
in heaven by being an atheist, so it's still impossible to say that the
expected utility of believing in god is must be higher.
== Finite utilities
We can avoid the issues with $infinity$ in utility calculations by simply not
using it. Instead, simply say that the utility of going to heaven is an
immensely large finite number. The utility of going to hell is likewise a very
negative number. Now, we no longer run into the issue of being unable to
compare utilities. All of our expected utility calculations will succeed, and
given sufficiently large utilities, we should be able to make similar arguments
for believing in god.
The problem with this argument is that we now open our expected utilities up to
individual subjective determination. A core feature of the previous argument
involving infinite utilities is that they can effectively bypass numerical
comparison. If, instead, finite utilities were used, then each person may
assign different utilities to each possible outcome based on their own beliefs.
Also, the probabilities are no longer irrelevant, so they must be analyzed as
well. This greatly complicates the decision matrix.
An implied feature of BG2 is that believing in god has a greater expected
utility for _everyone_. Suppose there is someone who believes that the
suffering of being condemned to hell for eternity is worse (in absolute terms)
than the joy of being rewarded with heaven for eternity is good. In precise
terms, given the utility of being rewarded with an eternity in heaven $U_r$,
and the utility of being punished with an eternity in hell, $U_p$, we have
$ abs(U_p) > U_r $
Then, substituting these values for $infinity$ and $-infinity$ in
#link(<other-gods-table>)[Table 2], it's actually possible to obtain an
expected utility of believing in god that is less than the expected utility of
not believing. We can no longer say that BG2 is universally true for
_everyone_, so it no longer holds.
#[
= Paper Logistics
There are #total-words words in this paper.
There are #total-words words in this paper, discounting this section as well
as any content in tables.
== AI Contribution Statement
#quote[I did not use AI in the writing of this paper.]
]<wordcount-exclude>