alexandria/work/2024/phil-1/paper-1/main.typ

#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
#import "@preview/wordometer:0.1.3": word-count, total-words

#import "prelude.typ": indented-argument

#show: ams-article.with(
  title: [The Argument for Betting on God and the Possibility of Infinite Suffering],
  bibliography: bibliography("refs.bib"),
)

#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))

#align(
  center,
  pad(
    x: 20%,
    table(
      columns: (1fr, 1fr),
      [
        Perm: A2V4847
      ],
      [
        Word Count: #total-words
        #footnote[
          Figure computed programmatically during document compilation. Discounts
          content in tables and the AI contribution statement.
        ]<wordcount-exclude>
      ],
    ),
  ),
)


= Introduction

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's argument for belief in God #cite(supplement: [p. 38],
<Korman2022-KORLFA>) goes as follows:

#indented-argument(
  title: "The Argument for Betting on God",
  abbreviation: "BG",
  [One should always choose the option with the greatest expected utility.],
  [Believing in God has a greater expected utility than not believing in God.],
  [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), it's rational 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
believing in God.

#show table.cell.where(x: 0): strong
#show table.cell.where(y: 0): strong

#figure(
  caption: [Author's decision matrix],
  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],
    ),
  ),
)

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."

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,
clearly you should choose to believe in God, because the expected utility is
$infinity$.

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 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.

= Possibility of Infinite Suffering

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 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 model an outcome where the Evil God
exists.

#pagebreak()

#[
  #set figure()
  #figure(
    caption: [Possibility of an Evil God],
    table(
      columns: (auto, auto, auto, auto, auto),
      align: center,
      table.header(
        [],
        [Correct god exists ($33.3%$)],
        [No god exists ($33.3%$)],
        [Evil God exists ($33.3%$)],
        [E.U.],
      ),

      [ Believe in some God ], [$infinity$], [1], [$-infinity$], [$?$],
      [
        Don't believe in any God
      ],
      [2],
      [3],
      [4],
      [4.5],
    ),
  )
]<other-gods-table>

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? 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_.

Consider the following Indeterminate Utilities argument:

#indented-argument(
  title: "The Indeterminate Utilities argument",
  abbreviation: "IU",
  [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.],
  [The expected utility of believing in god is undefined.],
  [So, we cannot compare the expected utilities of believing in god or
    not believing in god.
  ],
  [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.
  ],
  [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 be compared. If they can't be
compared, then we can't say for certain which option has the higher expected
utility. Since BG2 claims that believing in god must have the higher expected
utility, it is a false premise.

= 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.

== The Evil God is not plausible

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

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 infinity has a special property the
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.

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.

#pagebreak()

#[
  = AI Contribution Statement

  #quote[I did not use AI whatsoever in the writing of this paper.]
]<wordcount-exclude>