337 lines
14 KiB
Text
337 lines
14 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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#import "prelude.typ": indented-argument
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#show: ams-article.with(
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title: [The Argument for Betting on God and the Possibility of Infinite Suffering],
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bibliography: bibliography("refs.bib"),
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)
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#show: word-count.with(exclude: (
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heading,
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<wordcount-exclude>,
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table,
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figure,
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footnote,
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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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#set table(inset: 8pt, align: center)
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#align(
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center,
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pad(
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x: 20%,
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table(
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columns: (1fr, 1fr),
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[
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Perm: A2V4847
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],
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[
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Word Count: #total-words
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#footnote[
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Figure computed programmatically during document compilation. Discounts
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content in tables and the AI contribution statement.
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]
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],
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),
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),
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)
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= Introduction
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The argument for Betting on God says that you should believe in God, regardless
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of other evidence, purely out of rational self-interest. In this paper, I
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challenge this argument by assessing the premise that believing in a particular
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God always guarantees the greatest expected utility.
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The author's argument for belief in God on #cite(supplement: [p. 38],
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<Korman2022-KORLFA>) goes as follows:
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#indented-argument(
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title: "The Argument for Betting on God",
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abbreviation: "BG",
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[One should always choose the option with the greatest expected utility.],
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[Believing in God has a greater expected utility than not believing in God.],
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[So, you should believe in God.],
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)
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BG1 should be uncontroversial. If you expect that an action will bring you the
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most utility (i.e. be the most useful), it's rational to choose to do it.
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// To justify BG2, the author uses a so-called "decision matrix" to compute the
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// expected utility of each combination of action and possible outcome. The
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// possible actions are placed on the rows, and the possible outcomes are placed
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// on the columns, except for the last column, which is the calculated expected
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// utility. At each intersection of a row and column, we place the utility gained
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// from that combination of action and outcome. The expected utility for a given
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// action is computed by multiplying the utility of each action-outcome pair in
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// that action's row by the probability of the corresponding outcome occurring,
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// and summing up all of those values.
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To justify BG2, the author uses a so-called "decision matrix" to compute the
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expected utility of either belief or disbelief in God. Both possible actions
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are placed on the first column, and the possible outcomes (God existing or God
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not existing) are placed on the first row. The last column of the matrix
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represents the expected utility of the action in its corresponding row. At each
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intersection of action and outcome, we write the utility gained from that
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action-outcome combination.
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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 utilities 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: [Author's decision matrix],
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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 utility doesn't provide an empirical measure of "usefulness" or
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"happiness," and should be viewed as a relative measurement.
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We assign each action-outcome combination utilities as we see fit, based on how
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much each scenario benefits us. You'll see shortly that the exact values we set
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for the finite utilities don't matter when infinite utility is introduced.
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In the specific case where God does exist, and you believed in God, you
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are rewarded with an eternal afterlife of bliss and pleasure in heaven. This
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reward is infinitely greater than any possible reward on earth, so it has a
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utility of $infinity$.
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To calculate the expected utility of a given action, we first multiply the
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utility gained from each action-outcome combination in the action's row by the
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probability of the corresponding outcome occurring. We then sum up all of these
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values to obtain the final expected utility.
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So, the expected utility for disbelief is $0.5 times 1 + 0.5 times 3 = 2$, and
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the expected utility for belief is $0.5 times infinity + 0.5 times 2 =
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infinity$. If, according to BG1, you should pick the option with greatest
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expected utility, you should clearly choose to believe in God, because the
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expected utility is $infinity$.
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Also, as the author points out on #cite(<Korman2022-KORLFA>, supplement: [p.
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40]), the exact probabilities don't matter either since multiplying even the
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smallest percentage by $infinity$ still results in the expected utility of
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$infinity$.
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I will show that the Argument for Betting on God fails because BG2 fails. In
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section 2, I argue you cannot determine whether or not believing in God has the
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greatest expected utility because the decision matrix approach fails when
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possible outcomes involving infinitely negative utilities are introduced. In
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section 3, I address a few possible responses to this objection.
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= Possibility of Infinite Suffering
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I propose that there is the possibility of more gods than just the Christian one that
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sends you to an eternal afterlife for believing. The author partially addresses
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this concern on #cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), using the
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example of Zeus. Zeus will only reward those who believe in him specifically
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with an eternal afterlife. So, if you believe in the wrong god, you don't go to
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the afterlife. The author concludes believing in either Zeus or the Christian
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God still result in expected utilities of $infinity$, while being an atheist
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always has a finite expected utility. Therefore, you should always believe in
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_some_ god that could grant you an eternal afterlife, although no argument is
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made for _which_ god.
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However, this leaves out the possibility of a god who instead punishes you for
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eternity. For instance, suppose there exists an _Evil God_ who sends any theist
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to hell for eternity, and does nothing to atheists. That is, the Evil God will
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punish anyone who believes in _any_ god, including those who believe in the
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Evil God themselves.
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Let us modify our decision matrix to model an outcome where the Evil God
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exists.
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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: [Possibility of an Evil God],
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table(
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columns: (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 ($33.3%$)],
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[No god or wrong god ($33.3%$)],
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[Evil God exists ($33.3%$)],
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[E.U.],
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),
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[ Believe in some God ], [$infinity$], [1], [$-infinity$], [$?$],
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[
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Don't believe in any God
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],
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[2],
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[3],
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[4],
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[4.5],
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),
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)
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]<other-gods-table>
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We've added the new option to our matrix. For the sake of argument, let's say
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each option has an equally likely outcome. Again, the exact probabilities don't
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really matter when we're multiplying them by infinity.
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The utilities are mostly the same as before. However, the theist now faces the
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possibility of the worst case of all: eternal punishment if the Evil God
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exists. If eternal bliss in heaven has a utility of $infinity$, then it follows
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that we should represent eternal punishment in hell with a utility of
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$-infinity$.
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Let us attempt to calculate the expected utility of believing in god using our
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usual method. We have $0.333 times infinity + 0.333 times 1 + 0.333 times
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-infinity$. What is $infinity - infinity$? A naive answer might be 0, but
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infinity is not a number in the traditional sense. It makes no sense to add or
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subtract infinite values. For instance, try and subtract the total amount of
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integers ($infinity$) from the total amount of real numbers (also $infinity$)
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#footnote[Minor digression: famously, the infinity of $RR$ is "larger" than
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the infinity of $ZZ$ in the sense of cardinality, where $frak(c) > aleph_0$ (G.
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Cantor). However, our familiar algebraic operations of $+$ and $-$ are still
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not defined on them. Perhaps we could pursue a line of reasoning to rigorously
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define algebra with infinity using the hyperreals $attach(RR, tl: *)$, but that
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is out of the scope of this paper.]. Clearly, this notion is meaningless and we
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cannot obtain a solution. So, we consider $infinity - infinity$ an
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_indeterminate form_. So, the expected utility is now _undefined_.
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Consider the following Indeterminate Utilities argument:
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#indented-argument(
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title: "The Indeterminate Utilities argument",
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abbreviation: "IU",
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[If the expected utility of believing in god is undefined, then we
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cannot compare the expected utilities of believing in god and not believing
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in god.],
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[The expected utility of believing in god is undefined.],
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[So, we cannot compare the expected utilities of believing in god and
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not believing in god.
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],
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[If we cannot compare the expected utilities of believing in god and
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not believing in god, then we cannot determine if believing in god has a
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higher expected utility than not believing in god.
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],
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[So, we cannot determine if believing in god has a higher expected
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utility than not believing in god. ],
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)<wordcount-exclude>
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We just showed why the premise IU2 is true, and the conclusion IU5 is in direct
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contradiction with BG2. So, if IU5 holds, then BG2 must fail.
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It's important to note that the Indeterminate Utilities argument doesn't say
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that the _opposite_ of BG2 is true. It doesn't argue that the expected utility
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of being an atheist is greater. In fact, it doesn't say anything about the
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expected utilities, except that they cannot be compared. If they can't be
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compared, then we can't say for certain which option has the higher expected
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utility. Since BG2 claims that believing in god must have the higher expected
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utility, it is a false premise.
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= Addressing Objections
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// == Believing in a god is still preferable to atheism
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//
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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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//
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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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== The Evil God is not plausible
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One might argue that it is not plausible there is an Evil God who punishes all
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theists, including their own believers. Many religions present a god that
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rewards believers and at most punishes disbelievers, yet none of the major
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world religions propose an Evil God who punishes all believers
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indiscriminately. It's much more likely that a benevolent god exists than an
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evil one.
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Notice that it doesn't actually matter how plausible the Evil God is. If a
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rational atheist should concede there is at least a non-zero chance some god
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exists, then there must also be a non-zero chance the Evil God exists. After
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all, can you say for sure that the Evil God doesn't exist? All it takes is that
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non-zero chance, no matter how small, because multiplying it by $-infinity$
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still results in the undefined expected utility.
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== Finite utilities
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One might argue that we can avoid using $infinity$ to ensure that all expected
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utility calculations are defined. Instead, suppose the utility of going to
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heaven is just an immensely large finite number. The utility of going to hell
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is likewise a very negative number. All of our expected utility calculations
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will be defined, since infinity is not used. Given sufficiently large
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utilities, we should be able to make a similar argument 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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The problem with this argument is that infinity has a special property the
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argument relies on that no finite numbers have. Namely, any number multiplied
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by $infinity$ is still $infinity$, so the exact probabilities we set for the
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existence of God don't matter. This is important for defending against the
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objection that the probabilities are possibly incorrect which the author
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mentions on #cite(<Korman2022-KORLFA>, supplement: [p. 40]). If the exact
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numbers don't matter due to $infinity$, it doesn't matter if they might be
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wrong (as long as they are non-zero).
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If, instead, only finite utilities were used, the concern that the
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probabilities in the matrix are wrong cannot be resolved with the same argument
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as before. There could conceivably exist a matrix with probabilities for a
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benevolent god and an Evil God such that the expected utility of atheism is
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actually higher. The issue is we cannot say for sure what the probabilities of
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the benevolent god and the Evil God existing are. If we cannot know what the
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actual probabilities are, then we cannot know the final outcome of our matrix.
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So, without knowing the final outcome of the matrix, we still cannot determine
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whether or not believing in god has greater expected utility, and BG2 still
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fails.
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#pagebreak()
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#[
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= AI Contribution Statement
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#quote[I did not use AI whatsoever in the writing of this paper.]
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]<wordcount-exclude>
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