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@ -8,10 +8,17 @@
bibliography: bibliography("refs.bib"),
)
#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))
#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(
@ -28,7 +35,7 @@
#footnote[
Figure computed programmatically during document compilation. Discounts
content in tables and the AI contribution statement.
]<wordcount-exclude>
]
],
),
),
@ -38,11 +45,11 @@
= 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.
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 #cite(supplement: [p. 38],
The author's argument for belief in God on #cite(supplement: [p. 38],
<Korman2022-KORLFA>) goes as follows:
#indented-argument(
@ -53,21 +60,29 @@ The author's argument for belief in God #cite(supplement: [p. 38],
[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.
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 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.
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 utility for believing or not
<Korman2022-KORLFA>) which gives the expected utilities for believing or not
believing in God.
#show table.cell.where(x: 0): strong
@ -97,51 +112,59 @@ believing in 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."
Note that utility doesn't provide an empirical measure of "usefulness" or
"happiness," and should be viewed as a relative measurement.
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 $infinity$.
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.
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
$infinity$.
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 $infinity$.
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
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 possible response to this objection.
section 3, I address a few possible responses 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.
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 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.
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.
@ -158,7 +181,7 @@ exists.
table.header(
[],
[Correct god exists ($33.3%$)],
[No god exists ($33.3%$)],
[No god or wrong god ($33.3%$)],
[Evil God exists ($33.3%$)],
[E.U.],
),
@ -179,19 +202,18 @@ 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 $-infinity$.
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
$-infinity$.
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 $infinity$)
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
@ -204,13 +226,13 @@ Consider the following Indeterminate Utilities 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
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 or
[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 or
[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.
],
@ -254,26 +276,26 @@ utility, it is a false premise.
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.
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.
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.
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 that the utility of going to
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, and given sufficiently large utilities, we should be able to
make a similar argument for believing in god.
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
@ -284,21 +306,24 @@ make a similar argument for believing in god.
// 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.
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, 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.
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()

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..args.pos(),
)
]