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@ -2,7 +2,7 @@
#import "@preview/wordometer:0.1.3": word-count, total-words
#show: ams-article.with(
title: [On Pascal's Wager],
title: [The Argument for Betting on God and the Possibility of Infinite Suffering],
bibliography: bibliography("refs.bib"),
)
@ -19,7 +19,7 @@
Perm: A2V4847
],
[
Word Count: #total-words #footnote[Discounting content in tables and the AI contribution statement]
Word Count: #total-words #footnote[Figure computed programmatically during document compilation. Discounts content in tables and the AI contribution statement.]<wordcount-exclude>
],
),
)
@ -27,18 +27,49 @@
= 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 for Betting on God says that you should believe in God, regardless
of other evidence, purely out of self-interest. In this paper, I challenge this
argument by assessing the premise that believing in a particular God always
guarantees the greatest 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.
The author's argument for belief in God #cite(supplement: [p. 38],
<Korman2022-KORLFA>) goes as follows:
#pad(
x: 16pt,
[
#set par(first-line-indent: 0pt)
#smallcaps[The Argument for Betting on God]:
#pad(
y: -2pt,
[(BG1) One should always choose the option with the greatest expected utility
],
)
#pad(
y: -2pt,
[(BG2) Believing in God has a greater expected utility than not believing in God
],
)
#pad(
y: -2pt,
[(BG3) So, you should believe in God
],
)
],
)
BG1 should be uncontroversial. If you expect an action to bring you the most
utility (i.e. be the most useful), why wouldn't you choose to do it?
To justify BG2, 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 place the utility we gain
from that combination of action and outcome. The expected utility for a given
action is computed by multiplying the utility of each action-outcome pair in
that row by the probability of the corresponding outcome occurring, and summing
up all of those values.
Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
<Korman2022-KORLFA>) which gives the expected utility for believing or not
@ -48,7 +79,7 @@ believing in God.
#show table.cell.where(y: 0): strong
#figure(
caption: [Pascal's Wager],
caption: [Author's decision matrix],
align(
center,
table(
@ -75,190 +106,226 @@ 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:
$
&"(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, 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, 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. Let's assign the first scenario a
utility of $infinity$ and the second a utility of 1.
We assign the various finite utilities as we see fit, based on how much each
scenario benefits us. In the case where God does exist, and you believed in
God, then you are rewarded with an eternal afterlife of bliss and pleasure in
heaven. This reward is infinitely greater than any possible reward on earth, so
it has a utility of $infinity$.
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
clearly you should 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.
The exact utilities don't matter much, since any finite utility you could gain
for atheism cannot possibly be greater than the infinite expected utility of
believing in God. Also, as the author points out on #cite(<Korman2022-KORLFA>,
supplement: [p. 40]), the exact probabilities don't matter either since
multiplying them by $infinity$ still results in the expected utility of
$infinity$.
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 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.
I will show that the Argument for Betting on God fails because BG2 fails. In
section 2, I argue you cannot determine whether or not believing in God has the
greatest expected utility because the decision matrix approach fails when
possible outcomes involving infinitely negative utilities are introduced. In
section 3, I address a possible response to this objection.
= Unlimited Suffering
= Possibility of Infinite 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]), 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.
It is possible there are more gods than just the one that sends you to an
eternal afterlife for believing? The author partially addresses this in
#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), using the example of Zeus.
Zeus will only reward those who believe in him with an eternal afterlife of
pleasure. So, if you believe in the wrong god, you don't go to the afterlife.
The author concludes either believing in Zeus or the Christian God still has
expected utilities of $infinity$, while being an atheist does has a finite
expected utility. Therefore, it is still preferable to believe in _some_ god
that may grant you an eternal afterlife, although no argument is made for
_which_ 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.
However, this leaves out the possibility of gods who punish you for some
reason. For instance, suppose there exists an _Evil God_ who sends anyone who
believes in any god to hell for eternity, and does nothing to atheists.
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()
Let us modify our decision matrix to model an outcome where the Evil God
exists. #pagebreak()
#[
#set figure()
#figure(
caption: [Other gods existing],
caption: [Possibility of an Evil God],
table(
columns: (auto, auto, auto, auto, auto, auto),
columns: (auto, auto, auto, auto, auto),
align: center,
table.header(
[],
[Correct god exists ($25%$)],
[No god exists ($25%$)],
[Wrong god, spiteful ($25%$)],
[Wrong god, forgiving ($25%$)],
[Correct god exists ($33.3%$)],
[No god exists ($33.3%$)],
[Evil God exists ($33.3%$)],
[E.U.],
),
[ Believe in God ], [$infinity$], [3], [$-infinity$], [1], [$?$],
[ Believe in some God ], [$infinity$], [1], [$-infinity$], [$?$],
[
Don't believe in God
Don't believe in any God
],
[2],
[3],
[4],
[2],
[2],
[2.5],
[4.5],
),
)
]<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 $-infinity$.
We've added the new option to our matrix. For the sake of argument, let's say
each option has an equally likely outcome. Again, the exact probabilities don't
really matter when we're multiplying them by infinity.
The utilities are mostly the same as before. Not believing in any god and the
Evil God existing is now the best case for the atheist since they avoided
infinite suffering. However, the theist now faces 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 $-infinity$.
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.
god? We have $0.333 times infinity + 0.333 times 1 + 0.333 times -infinity$.
What is $infinity - infinity$? A naive answer might be 0, but infinity is not a
number in the traditional sense. It makes no sense to add or subtract infinite
values. For instance, try and subtract the total amount of integers
($infinity$) from the total amount of real numbers (also $infinity$)
#footnote[Famously, this infinity is "larger" than the infinite number of
integers in the sense of cardinality (G. Cantor). But subtracting them still
makes no mathematical or physical sense.]. Clearly, this notion is meaningless
and we cannot obtain a solution. So, we consider $infinity - infinity$ an
_indeterminate form_. So, the expected utility is now _undefined_.
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 we cannot claim that BG3 is true.
Consider the following argument:
#pad(
x: 16pt,
[
#set par(first-line-indent: 0pt)
#smallcaps[The Indeterminate Utilities argument]:
#pad(
y: -1pt,
[(IU1) If the expected utility of believing in god is undefined, then we
cannot compare the expected utilities of believing in god or not believing
in god.],
)
#pad(
y: -1pt,
[(IU2) The expected utility of believing in god is undefined.],
)
#pad(
y: -1pt,
[(IU3) So, we cannot compare the expected utilities of believing in god or
not believing in god.
],
)
#pad(
y: -1pt,
[(IU4) If we cannot compare the expected utilities of believing in god or
not believing in god, then we cannot determine if believing in god has a
higher expected utility than not believing in god.
],
)
#pad(
y: -1pt,
[(IU5) So, we cannot determine if believing in god has a higher expected
utility than not believing in god. ],
)],
)<wordcount-exclude>
We just showed why the premise IU2 is true, and the conclusion IU5 is in direct
contradiction with BG2. So, if IU5 holds, then BG2 must fail.
It's important to note that the Indeterminate Utilities argument doesn't say
that the _opposite_ of BG2 is true. It doesn't argue that the expected utility
of being an atheist is greater. In fact, it doesn't say anything about the
expected utilities, except that they cannot all be compared. If they cannot all
be calculated, then they cannot be compared. So, we can't say whether believing
in god has a greater expected utility or not.
= Addressing Objections
== Believing in a god is still preferable to atheism
// == 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.
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.
== The Evil God is not plausible
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.
One might argue that it is not plausible there is an Evil God who punishes all
theists, including their own believers. Many religions present a god that
rewards believers and at most punishes disbelievers. None of the major world
religions propose an Evil God who punishes all believers. It's much more likely
that a benevolent god exists than an evil one.
I contend that it doesn't matter whether or not the Evil God is less plausible
than a benevolent god. Surely, if a rational atheist who is unconvinced by all
the world's scriptures can still concede that there is at least a non-zero
chance that some god exists, the rational theist should also concede that there
is a non-zero chance that the Evil God exists. All it takes is that non-zero
chance, no matter how small, because multiplying it by $-infinity$ still
results in the undefined expected utility.
== 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.
One might argue that we can avoid using $infinity$ to ensure that all expected
utility calculations are defined. Instead, suppose that the utility of going to
heaven is just an immensely large finite number. The utility of going to hell
is likewise a very negative number. All of our expected utility calculations
will be defined, and given sufficiently large utilities, we should be able to
make a similar argument 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.
// 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
The problem with this argument is that infinity has a special property argument
relies on. Namely, any number multiplied by $infinity$ is still $infinity$, so
the exact probabilities we set for the existence of God don't matter. This is
important for defending against the objection the author mentions on
#cite(<Korman2022-KORLFA>, supplement: [p. 40]), that the probabilities are
possibly incorrect, since the numbers don't matter anyways.
$ abs(U_p) > U_r $
If, instead, only finite utilities were used, then the theist must contend with
the concern that the probabilities in the matrix are wrong. There could
conceivably exist a matrix with probabilities for a benevolent god and an Evil
God such that the expected utility of atheism is actually higher. The issue is
we cannot say for sure what the probabilities of the benevolent god and the
Evil God existing are. If we cannot know what the actual probabilities are,
then we cannot know the final outcome of our matrix. So, without knowing the
final outcome of the matrix, we still cannot determine whether or not believing
in god has greater expected utility, and BG2 still fails.
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.
#pagebreak()
#[
= AI Contribution Statement
#quote[I did not use AI in the writing of this paper.]
#quote[I did not use AI whatsoever in the writing of this paper.]
]<wordcount-exclude>