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#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 argument makes heavy use of the concepts of utility and expected utility.
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Utility is essentially the usefulness of an action, or to what degree it helps
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increase "good," like happiness, pleasure, benefit, and decrease "bad," like
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suffering or harm. Given a set of possible actions and distinct possible
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outcomes, each action may be assigned an "expected utility" by pairing the
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action with each possible outcome and assigning every action-outcome pair some
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amount of utility. Using the probabilities of each outcome occurring, we can
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compute a weighted average that gives the expected utility of the action.
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More precisely, let us define a set of $n$ actions
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$ {a_1, a_2, ..., a_n} in A $
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where $a_k$ represents the $k^"th"$ action, and a set of $m$ outcomes
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$ {o_1, o_2, ..., o_m} in O $
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where $o_k$ represents the $k^"th"$ outcome. Additionally, let
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$ rho (o_k) $
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be the probability of the outcome $o_k$ occurring.
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We compute the *Cartesian product* $A times O$ which contains ordered pairs of
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the form $(a_k, o_k)$ representing every possible combination of action and
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outcome. Formally,
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$ A times O = {(a_j, o_i) | j in {1,2,...,n}, i in {1,2,...,m}} $
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We assign each action-outcome pair its utility as we deem fit. The function
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$ U ((a_k, o_k)) $
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gives the utility of an ordered action-outcome pair $(a_k, o_k)$.
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Then, to determine the expected utility for an action $a_k$, we select all of
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the ordered pairs with $a_k$ in the first position, multiply their utility by
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the probability of their corresponding outcome occurring, and sum of all of
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these products. In precise terms, given $m$ possible outcomes, then:
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$ "Expected utility of" a_k = sum_(i=1)^(m) rho (o_i) dot U ((a_k, o_i)) $
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In order to make this more clear, we construct a so-called "decision matrix"
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where we can easily assign a utility value for each action-outcome pair and
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then calculate the expected utility.
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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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== The argument for betting on God
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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. If you believe in God, but God
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doesn't exist, you've led a pious life without gaining much in return. If you
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don't believe in God, and God doesn't exist, then you have it slightly better
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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.
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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. So, the expected utility for not
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believing is $0.5 times 1 + 0.5 times 3 = 2$, and the expected utility is $0.5
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times infinity + 0.5 times 2 = infinity$. If, according to BG1, you should pick
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the option with greatest expected utility, you should clearly choose to believe
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in God, because the expected utility is $infinity$.
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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 it
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makes no sense to even calculate expected utilities of believing in God. 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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#pagebreak()
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= Many Gods
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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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#[
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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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Our new matrix has a problem: how do we calculate the expected utility?
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$infinity + (-infinity)$, is an indeterminate value. We cannot really perform
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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.
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#linebreak()
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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, $U_p$, such that
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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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Binary file not shown.
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@ -1,14 +1,13 @@
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#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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#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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#align(
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@ -19,7 +18,11 @@
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Perm: A2V4847
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],
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[
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Word Count: #total-words #footnote[Figure computed programmatically during document compilation. Discounts content in tables and the AI contribution statement.]<wordcount-exclude>
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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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]<wordcount-exclude>
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],
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),
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)
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@ -35,31 +38,16 @@ guarantees the greatest expected utility.
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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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#pad(
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x: 16pt,
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[
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#set par(first-line-indent: 0pt)
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#smallcaps[The Argument for Betting on God]:
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#pad(
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y: -2pt,
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[(BG1) One should always choose the option with the greatest expected utility
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],
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)
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#pad(
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y: -2pt,
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[(BG2) Believing in God has a greater expected utility than not believing in God
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],
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)
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#pad(
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y: -2pt,
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[(BG3) So, you should believe in God
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],
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)
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],
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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 an action to bring you the most
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utility (i.e. be the most useful), why wouldn't you choose to do it?
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utility (i.e. be the most useful), it's rational 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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@ -149,7 +137,9 @@ reason. For instance, suppose there exists an _Evil God_ who sends anyone who
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believes in any god to hell for eternity, and does nothing to atheists.
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Let us modify our decision matrix to model an outcome where the Evil God
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exists. #pagebreak()
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exists.
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#pagebreak()
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#[
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#set figure()
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@ -201,44 +191,24 @@ makes no mathematical or physical sense.]. Clearly, this notion is meaningless
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and we 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 argument:
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Consider the following Indeterminate Utilities argument:
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#pad(
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x: 16pt,
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[
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#set par(first-line-indent: 0pt)
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#smallcaps[The Indeterminate Utilities argument]:
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#pad(
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y: -1pt,
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[(IU1) If the expected utility of believing in god is undefined, then we
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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 or not believing
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in god.],
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)
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#pad(
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y: -1pt,
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[(IU2) The expected utility of believing in god is undefined.],
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)
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#pad(
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y: -1pt,
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[(IU3) So, we cannot compare the expected utilities of believing in god or
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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 or
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not believing in god.
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],
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)
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#pad(
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y: -1pt,
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[(IU4) If we cannot compare the expected utilities of believing in god or
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[If we cannot compare the expected utilities of believing in god or
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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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)
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#pad(
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y: -1pt,
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[(IU5) So, we cannot determine if believing in god has a higher expected
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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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)],
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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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@ -306,12 +276,12 @@ make a similar argument for believing in god.
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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 argument
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relies on. Namely, any number multiplied by $infinity$ is still $infinity$, so
|
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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
|
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#cite(<Korman2022-KORLFA>, supplement: [p. 40]), that the probabilities are
|
||||
possibly incorrect, since the numbers don't matter anyways.
|
||||
The problem with this argument is that infinity has a special property the
|
||||
argument relies on. Namely, any number multiplied by $infinity$ is still
|
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$infinity$, so the exact probabilities we set for the existence of God don't
|
||||
matter. This is important for defending against the objection the author
|
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mentions on #cite(<Korman2022-KORLFA>, supplement: [p. 40]), that the
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probabilities are possibly incorrect, since the numbers don't matter anyways.
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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
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|
|
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21
work/2024/phil-1/paper-1/prelude.typ
Normal file
21
work/2024/phil-1/paper-1/prelude.typ
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|
|||
#set cite(style: "institute-of-electrical-and-electronics-engineers")
|
||||
#set text(fractions: true)
|
||||
|
||||
#let indented-argument(title: "", abbreviation: "", ..args) = [
|
||||
#set par(first-line-indent: 0pt)
|
||||
|
||||
#pad(left: 12pt, smallcaps(title))
|
||||
|
||||
#let arg-numbering = (..nums) => nums.pos().map(n => (
|
||||
"(" + abbreviation + str(n) + ")"
|
||||
)).join()
|
||||
|
||||
#enum(
|
||||
numbering: arg-numbering,
|
||||
indent: 16pt,
|
||||
tight: false,
|
||||
..args.pos(),
|
||||
)
|
||||
]
|
||||
|
||||
|
||||
Loading…
Reference in a new issue