286 lines
12 KiB
Text
286 lines
12 KiB
Text
|
#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>
|