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,
  figure,
  footnote,
))

#set cite(style: "institute-of-electrical-and-electronics-engineers")
#set text(fractions: true)
#set table(inset: 8pt, align: center)


#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.
        ]
      ],
    ),
  ),
)


= Introduction

The argument for Betting on God says that you should believe in God, regardless
of other evidence, purely out of rational 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 on #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 that an action will bring you the
most utility (i.e. be the most useful), it's rational to 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 gained
// 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 action's row by the probability of the corresponding outcome occurring,
// and summing up all of those values.

To justify BG2, the author uses a so-called "decision matrix" to compute the
expected utility of either belief or disbelief in God. Both possible actions
are placed on the first column, and the possible outcomes (God existing or God
not existing) are placed on the first row. The last column of the matrix
represents the expected utility of the action in its corresponding row. At each
intersection of action and outcome, we write the utility gained from that
action-outcome combination.

Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
<Korman2022-KORLFA>) which gives the expected utilities 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 utility doesn't provide an empirical measure of "usefulness" or
"happiness," and should be viewed as a relative measurement.

We assign each action-outcome combination utilities as we see fit, based on how
much each scenario benefits us. You'll see shortly that the exact values we set
for the finite utilities don't matter when infinite utility is introduced.

In the specific case where God does exist, and you believed in God, 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 $infinity$.


To calculate the expected utility of a given action, we first multiply the
utility gained from each action-outcome combination in the action's row by the
probability of the corresponding outcome occurring. We then sum up all of these
values to obtain the final expected utility.

So, the expected utility for disbelief is $0.5 times 1 + 0.5 times 3 = 2$, and
the expected utility for belief 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 $infinity$.

Also, as the author points out on #cite(<Korman2022-KORLFA>, supplement: [p.
40]), the exact probabilities don't matter either since multiplying even the
smallest percentage 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 few possible responses to this objection.

= Possibility of Infinite Suffering

I propose that there is the possibility of more gods than just the Christian one that
sends you to an eternal afterlife for believing. The author partially addresses
this concern on #cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), using the
example of Zeus. Zeus will only reward those who believe in him specifically
with an eternal afterlife. So, if you believe in the wrong god, you don't go to
the afterlife. The author concludes believing in either Zeus or the Christian
God still result in expected utilities of $infinity$, while being an atheist
always has a finite expected utility. Therefore, you should still believe in
_some_ god that could grant you an eternal afterlife, although no argument is
made for _which_ god.

However, this leaves out the possibility of gods who instead punish you for
eternity. For instance, suppose there exists an _Evil God_ who sends any theist
to hell for eternity, and does nothing to atheists. That is, the Evil God will
punish anyone who believes in _any_ god, including those who believe in the
Evil God themselves.

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 or wrong god ($33.3%$)],
        [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. However, the theist now faces the
possibility of the worst case of all: eternal punishment if the Evil God
exists. 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
$-infinity$.

Let us attempt to calculate the expected utility of believing in god using our
usual method. 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 and not believing
    in god.],
  [The expected utility of believing in god is undefined.],
  [So, we cannot compare the expected utilities of believing in god and
    not believing in god.
  ],
  [If we cannot compare the expected utilities of believing in god and
    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, yet none of the major
world religions propose an Evil God who punishes all believers
indiscriminately. It's much more likely that a benevolent god exists than an
evil one.

Notice that it doesn't actually matter how plausible the Evil God is. If a
rational atheist should concede there is at least a non-zero chance some god
exists, then there must also be a non-zero chance the Evil God exists. After
all, can you say for sure that the Evil God doesn't exist? 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 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, since infinity is not used. 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 that no finite numbers have. 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 that the probabilities are possibly incorrect which the author
mentions on #cite(<Korman2022-KORLFA>, supplement: [p. 40]). If the exact
numbers don't matter due to $infinity$, it doesn't matter if they might be
wrong (as long as they are non-zero).

If, instead, only finite utilities were used, the concern that the
probabilities in the matrix are wrong cannot be resolved with the same argument
as before. 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>