255 lines
11 KiB
Text
255 lines
11 KiB
Text
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
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#import "@preview/wordometer:0.1.3": word-count, total-words
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#show: ams-article.with(
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title: [On Pascal's Wager],
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bibliography: bibliography("refs.bib"),
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)
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#set cite(style: "institute-of-electrical-and-electronics-engineers")
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#set text(fractions: true)
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#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))
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= Introduction
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The argument for Betting on God, or better known as Pascal's Wager, says that
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you should believe in God, regardless of other evidence, purely out of
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self-interest. In this paper, I will challenge this argument by assessing the
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premise that believing in a particular God always guarantees the greatest
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expected utility.
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The author uses a so-called "decision matrix" to compute the expected utility
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of each combination of action and possible outcome. The possible actions are
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placed on the rows, and the possible outcomes are placed on the columns, except
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for the last column, which is the calculated expected utility. At each
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intersection of a row and column, we put the utility we gain from that
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combination of action and outcome.
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Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
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<Korman2022-KORLFA>) which gives the expected utility for believing or not
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believing in God.
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#show table.cell.where(x: 0): strong
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#show table.cell.where(y: 0): strong
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#figure(
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caption: [Pascal's Wager],
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align(
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center,
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table(
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columns: (auto, auto, auto, auto),
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table.header(
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[],
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[God exists ($50%$)],
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[God doesn't exist ($50%$)],
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[Expected utility],
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),
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[ Believe in God ], [$infinity$], [2], [$infinity$],
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[
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Don't believe in God
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],
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[1],
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[3],
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[2],
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),
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),
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)
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Note that the numerical utility values themselves have no meaning, and they are
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meant to be viewed relative to each other. Utility doesn't literally provide an
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empirical measure of "usefulness" or "happiness."
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The author's argument for belief in God #cite(supplement: [p. 38],
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<Korman2022-KORLFA>) goes as follows:
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$
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&"(BG1) One should always choose the option with the greatest expected utility" \
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&"(BG2) Believing in God has a greater expected utility than not believing in God" \
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&"(BG3) So, you should believe in God"
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$
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BG1 should be generally uncontroversial. If you expect an action to bring you
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the most utility (i.e. be the most useful), why wouldn't you do it?
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BG2 is also substantiated by the decision matrix. All 4 action-outcome pairs
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are assigned a utility with the following logic.
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If you believe in God, but God doesn't exist, you've led a pious life without
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gaining much in return, so we say that has a utility of 2.
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If you don't believe in God, and God doesn't exist, then you have it slightly
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better than the previous scenario. You haven't wasted your time on religious
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activities (like going to church) and end up with the same fate as the
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believers, so let's give it a utility of 3.
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If God does in fact exist, however, then believing in God gives you an
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_unlimited_ amount of utility. You end up in an afterlife of eternal bliss and
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pleasure, more valuable than anything you could gain on earth. That means that
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the worst scenario is not believing in God and God existing, because you've
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just missed out on the eternal afterlife. Let's assign the first scenario a
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utility of $infinity$ and the second a utility of 1.
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So, the expected utility for not believing is $0.5 times 1 + 0.5 times 3 = 2$,
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and the expected utility is $0.5 times infinity + 0.5 times 2 = infinity$. If,
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according to BG1, you should pick the option with greatest expected utility,
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you should clearly choose to believe in God, because the expected utility is
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$infinity$.
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Additionally, notice that the actual probability of God existing doesn't
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matter, because any non-zero value multiplied by $infinity$ is still
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$infinity$, and so as long as you believe there is a _non-zero chance_ that God
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exists, the infinite expected utility of believing remains. Adjusting the
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probabilities may increase or decrease the expected utility of not believing in
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God, but not believing in God will never give you the opportunity of attaining
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the afterlife of infinite utility, so it can never reach the infinite expected
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utility of believing in God.
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I will show that Pascal's Wager fails because BG2 fails. Namely, we cannot know
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whether or not believing in God has the greatest expected utility because the
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decision matrix approach fails when more possible outcomes are introduced. In
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section 2, I present my objection to BG2, and in section 3, I will address a
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few possible responses to my objection.
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= Unlimited Suffering
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Maybe there are more gods than just the one that sends you to an eternal
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afterlife for believing. The author addresses this in
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#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), concluding that even if
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other gods exist, it is still preferable to choose any specific god who may
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grant you an eternal afterlife of pleasure than to not believe, since the
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expected utility of belief is still $infinity$. Essentially, the argument makes
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no claims as to _which_ god you choose, but says that you should believe in
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_some_ god.
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However, this leaves out the possibility of gods who punish you for believing
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in the wrong god. These gods may grant eternal afterlifes for other reasons, or
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perhaps even punish people with eternal suffering for belief in the wrong god.
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This introduces _negative utilities_, since being punished for all of eternity
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in hell is much worse than simply dying and not receiving any afterlife at all.
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Let us modify our decision matrix to accommodate an outcome where we believed
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in the wrong god. There are two scenarios: either we believe in the wrong god,
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but the true god is _forgiving_, so we are not punished, or we believe in the
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wrong god, and the true god happens to be _spiteful_ and punishes us with
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eternal damnation.
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#pagebreak()
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#[
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#set figure()
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#figure(
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caption: [Other gods existing],
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table(
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columns: (auto, auto, auto, auto, auto, auto),
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align: center,
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table.header(
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[],
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[Correct god exists ($25%$)],
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[No god exists ($25%$)],
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[Wrong god, spiteful ($25%$)],
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[Wrong god, forgiving ($25%$)],
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[E.U.],
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),
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[ Believe in God ], [$infinity$], [3], [$-infinity$], [1], [$?$],
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[
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Don't believe in God
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],
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[2],
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[4],
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[2],
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[2],
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[2.5],
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),
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)
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]<other-gods-table>
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We've added the new options to our matrix. #smallcaps[Wrong god, spiteful]
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represents the outcome where we are punished for believing in the wrong god,
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and #smallcaps[Wrong god, forgiving] represents the outcome where we are not
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punished, but we still missed out on the afterlife. This is slightly worse than
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being an atheist and missing out. If you are an atheist, then the outcome is
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the same no matter which god exists: you miss out on heaven. Again, the exact
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numbers don't matter too much when working with the infinities. However, we now
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have the possibility of the worst case of all: eternal punishment for believing
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in the wrong god. If eternal bliss in heaven has a utility of $infinity$, then
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it follows that we should represent eternal punishment in hell with a utility
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of $-infinity$.
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There is a problem: how do we calculate the expected utility of believing in
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god? $infinity + (-infinity)$, is an indeterminate value. We cannot really
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perform algebraic operations on $infinity$. Indeed, it makes no sense to add or
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subtract our infinite expected utilities.
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Since the author uses this decision matrix approach to justify BG2, it now
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fails. Once negative infinities are introduced, calculating expected utilities
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in the usual method becomes meaningless. It is not that BG2 is necessarily
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_wrong_, it just cannot be decided either way with the decision matrix. If BG2
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cannot be determined, then BG3 is also indeterminate.
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= Addressing Objections
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== Believing in a god is still preferable to atheism
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One might argue that believing in a god that rewards believers is always
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preferable to atheism since you at least have the _opportunity_ to receive
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eternity in heaven. Perhaps there exists a god who punishes non-believers with
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eternal damnation. Then, even without the exact expected utility calculation,
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it's clear that the expected utility of believing in some god must be higher
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than believing in none as you stand to gain more. Either as a theist or
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atheist, you run the risk of eternal punishment, but you only have the
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opportunity to go to heaven by believing in some god rather than none.
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Fair, the possibility that you are punished for believing in the wrong god
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doesn't imply that you should be an atheist either. Indeed, there may be a god
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that punishes atheists. However, there could also exist a god who sends
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everyone to heaven regardless. Or perhaps they only send atheists to heaven.
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Either way, there is also the possibility of attaining the infinite afterlife
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in heaven by being an atheist, so it's still impossible to say that the
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expected utility of believing in god is must be higher.
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== Finite utilities
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We can avoid the issues with $infinity$ in utility calculations by simply not
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using it. Instead, simply say that the utility of going to heaven is an
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immensely large finite number. The utility of going to hell is likewise a very
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negative number. Now, we no longer run into the issue of being unable to
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compare utilities. All of our expected utility calculations will succeed, and
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given sufficiently large utilities, we should be able to make similar arguments
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for believing in god.
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The problem with this argument is that we now open our expected utilities up to
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individual subjective determination. A core feature of the previous argument
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involving infinite utilities is that they can effectively bypass numerical
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comparison. If, instead, finite utilities were used, then each person may
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assign different utilities to each possible outcome based on their own beliefs.
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Also, the probabilities are no longer irrelevant, so they must be analyzed as
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well. This greatly complicates the decision matrix.
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An implied feature of BG2 is that believing in god has a greater expected
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utility for _everyone_. Suppose there is someone who believes that the
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suffering of being condemned to hell for eternity is worse (in absolute terms)
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than the joy of being rewarded with heaven for eternity is good. In precise
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terms, given the utility of being rewarded with an eternity in heaven $U_r$,
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and the utility of being punished with an eternity in hell, $U_p$, we have
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$ abs(U_p) > U_r $
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Then, substituting these values for $infinity$ and $-infinity$ in
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#link(<other-gods-table>)[Table 2], it's actually possible to obtain an
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expected utility of believing in god that is less than the expected utility of
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not believing. We can no longer say that BG2 is universally true for
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_everyone_, so it no longer holds.
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#[
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= Paper Logistics
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There are #total-words words in this paper, discounting this section as well
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as any content in tables.
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== AI Contribution Statement
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#quote[I did not use AI in the writing of this paper.]
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]<wordcount-exclude>
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