216 lines
8.5 KiB
Text
216 lines
8.5 KiB
Text
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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
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= Introduction
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Pascal's Wager says that you should believe in God out of a utilitarian
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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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$ "Prob"(o_k) $
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be the probability of the outcome $o_k$ occurring.
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We calculate the *Cartesian product* $A times O$ which contains ordered pairs
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of the form $(a_k, o_k)$ representing every possible combination of action and
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outcome.
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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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$ "Util"((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) "Prob"(o_i) dot "Util"((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 react 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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= 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]) but concludes 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 _evil_ or _weird_ gods. These gods
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may grant eternal afterlifes for other reasons, or perhaps even punish people
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with eternal suffering for belief in the wrong god. This introduces _negative
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utilities_, since being punished for all of eternity in hell is much worse than
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simply dying and not receiving any afterlife at all.
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Suppose that there is an Weird God who punishes anyone who even believes in a
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deity at all, and does nothing to those who don't.
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#figure(
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caption: [Weird God],
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align(
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center,
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table(
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columns: (auto, auto, auto, auto, auto),
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table.header(
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[],
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[Christian God exists ($50%$)],
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[No god exists ($25%$)],
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[Weird God exists ($25%$)],
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[Expected utility],
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),
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[ Believe in God ], [$infinity$], [2], [$-infinity$], [$?$],
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[Believe in Weird God], [1], [2], [$-infinity$], [$-infinity$],
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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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[2],
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[4],
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[2],
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),
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),
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)
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We've added the Weird God to the decision matrix. Believing in a Weird God and
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the Christian God existing means you missed out on an eternal afterlife, so
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we've assigned it the same utility as not believing and the Christian God
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existing. The same reasoning applies for believing and no gods existing. And of
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course, believing in the Weird God and them actually existing gives you an
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eternal afterlife in hell, so it has $-infinity$ utility, which means the
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expected utility of believing in Weird God is $-infinity$.
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But what if you believe in the Christian God, and the Weird God actually
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exists? Clearly you get sent to hell for eternity, resulting in a utility of
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$-infinity$. But how do we calculate the expected utility? We can't just do
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$infinity + (-infinity)$, as that's an indeterminate value. There are an
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infinite amount of real numbers, and an infinite amount of integers. Subtracing
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these infinities, however, is entirely meaningless. Indeed, it makes no sense
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to add or subtract our infinite expected utilities. The entire calculation of
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our expected utilities is meaningless, and so BG2 cannot be true, since we've
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shown that the decision matrix approach used to justify it becomes unworkable
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with the introduction of negative utility and $-infinity$.
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= Paper Logistics
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There are #total-words words in this paper.
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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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