From 3b8a213ab271dad33567da0b7dceac47b6c13d44 Mon Sep 17 00:00:00 2001 From: Youwen Wu Date: Tue, 15 Oct 2024 20:21:05 -0700 Subject: [PATCH] add paper 1 --- work/2024/phil-1/paper-1/main.typ | 215 ++++++++++++++++++++++++++++++ work/2024/phil-1/paper-1/refs.bib | 7 + 2 files changed, 222 insertions(+) create mode 100644 work/2024/phil-1/paper-1/main.typ create mode 100644 work/2024/phil-1/paper-1/refs.bib diff --git a/work/2024/phil-1/paper-1/main.typ b/work/2024/phil-1/paper-1/main.typ new file mode 100644 index 0000000..266bf0e --- /dev/null +++ b/work/2024/phil-1/paper-1/main.typ @@ -0,0 +1,215 @@ +#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof +#import "@preview/wordometer:0.1.3": word-count, total-words + +#show: ams-article.with( + title: [On Pascal's Wager], + bibliography: bibliography("refs.bib"), +) + +#set cite(style: "institute-of-electrical-and-electronics-engineers") +#set text(fractions: true) + +#show: word-count + += Introduction + +Pascal's Wager says that you should believe in God out of a utilitarian +self-interest. In this paper, I will challenge this argument by assessing the +premise that believing in a particular God always guarantees the greatest +expected utility. + +The argument makes heavy use of the concepts of utility and expected utility. +Utility is essentially the usefulness of an action, or to what degree it helps +increase "good," like happiness, pleasure, benefit, and decrease "bad," like +suffering or harm. Given a set of possible actions and distinct possible +outcomes, each action may be assigned an "expected utility" by pairing the +action with each possible outcome and assigning every action-outcome pair some +amount of utility. Using the probabilities of each outcome occurring, we can +compute a weighted average that gives the expected utility of the action. + +More precisely, let us define a set of $n$ actions + +$ a_1, a_2, ..., a_n in A $ + +where $a_k$ represents the $k^"th"$ action, and a set of $m$ outcomes + +$ o_1, o_2, ..., o_m in O $ + +where $o_k$ represents the $k^"th"$ outcome. Additionally, let + +$ "Prob"(o_k) $ +be the probability of the outcome $o_k$ occurring. + +We calculate the *Cartesian product* $A times O$ which contains ordered pairs +of the form $(a_k, o_k)$ representing every possible combination of action and +outcome. + +$ A times O = {(a_j, o_i) | j in {1,2,...,n}, i in {1,2,...,m}} $ + +We assign each action-outcome pair its utility as we deem fit. The function + +$ "Util"((a_k, o_k)) $ + +gives the utility of an ordered action-outcome pair $(a_k, o_k)$. + +Then, to determine the expected utility for an action $a_k$, we select all of +the ordered pairs with $a_k$ in the first position, multiply their utility by +the probability of their corresponding outcome occurring, and sum of all of +these products. In precise terms, given $m$ possible outcomes, then: + +$ "Expected utility of" a_k = sum_(i=1)^(m) "Prob"(o_i) dot "Util"((a_k, o_i)) $ + +In order to make this more clear, we construct a so-called "decision matrix" +where we can easily assign a utility value for each action-outcome pair and +then calculate the expected utility. + +Here is the decision matrix the author proposes on #cite(supplement: [p. 38], +) which gives the expected utility for believing or not +believing in God. + +#show table.cell.where(x: 0): strong +#show table.cell.where(y: 0): strong +#figure( + caption: [Pascal's Wager], + align( + center, + table( + columns: (auto, auto, auto, auto), + table.header( + [], + [God exists ($50%$)], + [God doesn't exist ($50%$)], + [Expected utility], + ), + + [ Believe in God ], [$infinity$], [2], [$infinity$], + [ + Don't believe in God + ], + [1], + [3], + [$2$], + ), + ), +) + +== The argument for betting on God + +The author's argument for belief in God #cite(supplement: [p. 38], +) goes as follows: + +$ + &"(BG1) One should always choose the option with the greatest expected utility" \ + &"(BG2) Believing in God has a greater expected utility than not believing in God" \ + &"(BG3) So, you should believe in God" +$ + +BG1 should be generally uncontroversial. If you expect an action to bring you +the most utility (i.e. be the most useful), why wouldn't you do it? + +BG2 is also substantiated by the decision matrix. All 4 action-outcome pairs +are assigned a utility with the following logic. If you believe in God, but God +doesn't exist, you've led a pious life without gaining much in return. If you +don't believe in God, and God doesn't exist, then you have it slightly better +than the previous scenario. You haven't wasted your time on religious +activities (like going to church) and end up with the same fate as the +believers. + +If God does in fact exist, however, then believing in God gives you an +_unlimited_ amount of utility. You end up in an afterlife of eternal bliss and +pleasure, more valuable than anything you could gain on earth. That means that +the worst scenario is not believing in God and God existing, because you've +just missed out on the eternal afterlife. So, the expected utility for not +believing is $0.5 times 1 + 0.5 times 3 = 2$, and the expected utility is $0.5 +times infinity + 0.5 times 2 = infinity$. If, according to BG1, you should pick +the option with greatest expected utility, you should clearly choose to believe +in God, because the expected utility is $infinity$. + +Additionally, notice that the actual probability of God existing doesn't +matter, because any non-zero value multiplied by $infinity$ is still +$infinity$, and so as long as you believe there is a _non-zero chance_ that God +exists, the infinite expected utility of believing remains. Adjusting the +probabilities may increase or decrease the expected utility of not believing in +God, but not believing in God will never give you the opportunity of attaining +the afterlife of infinite utility, so it can never react the infinite expected +utility of believing in God. + +I will show that Pascal's Wager fails because BG2 fails. Namely, we cannot know +whether or not believing in God has the greatest expected utility because it +makes no sense to even calculate expected utilities of believing in God. In +section 2, I present my objection to BG2, and in section 3, I will address a +few possible responses to my objection. + += Many Gods + +Maybe there are more gods than just the one that sends you to an eternal +afterlife for believing. The author addresses this in +#cite(, supplement: [pp. 43-44]) but concludes that even if +other gods exist, it is still preferable to choose any specific god who may +grant you an eternal afterlife of pleasure than to not believe, since the +expected utility of belief is still $infinity$. Essentially, the argument makes +no claims as to _which_ god you choose, but says that you should believe in +_some_ god. + +However, this leaves out the possibility of _evil_ or _weird_ gods. These gods +may grant eternal afterlifes for other reasons, or perhaps even punish people +with eternal suffering for belief in the wrong god. This introduces _negative +utilities_, since being punished for all of eternity in hell is much worse than +simply dying and not receiving any afterlife at all. + +Suppose that there is an Weird God who punishes anyone who even believes in a +deity at all, and does nothing to those who don't. + +#figure( + caption: [Weird God], + align( + center, + table( + columns: (auto, auto, auto, auto, auto), + table.header( + [], + [Christian God exists ($50%$)], + [No god exists ($25%$)], + [Weird God exists ($25%$)], + [Expected utility], + ), + + [ Believe in God ], [$infinity$], [2], [$-infinity$], [$?$], + [Believe in Weird God], [1], [2], [$-infinity$], [$-infinity$], + [ + Don't believe in God + ], + [1], + [2], + [4], + [2], + ), + ), +) + +We've added the Weird God to the decision matrix. Believing in a Weird God and +the Christian God existing means you missed out on an eternal afterlife, so +we've assigned it the same utility as not believing and the Christian God +existing. The same reasoning applies for believing and no gods existing. And of +course, believing in the Weird God and them actually existing gives you an +eternal afterlife in hell, so it has $-infinity$ utility, which means the +expected utility of believing in Weird God is $-infinity$. + +But what if you believe in the Christian God, and the Weird God actually +exists? Clearly you get sent to hell for eternity, resulting in a utility of +$-infinity$. But how do we calculate the expected utility? We can't just do +$infinity + (-infinity)$, as that's an indeterminate value. There are an +infinite amount of real numbers, and an infinite amount of integers. Subtracing +these infinities, however, is entirely meaningless. Indeed, it makes no sense +to add or subtract our infinite expected utilities. The entire calculation of +our expected utilities is meaningless, and so BG2 cannot be true, since we've +shown that the decision matrix approach used to justify it becomes unworkable +with the introduction of negative utility and $-infinity$. + += Paper Logistics + +There are #total-words words in this paper. + +== AI Contribution Statement + +#quote[I did not use AI in the writing of this paper.] diff --git a/work/2024/phil-1/paper-1/refs.bib b/work/2024/phil-1/paper-1/refs.bib new file mode 100644 index 0000000..25f6815 --- /dev/null +++ b/work/2024/phil-1/paper-1/refs.bib @@ -0,0 +1,7 @@ +@book{Korman2022-KORLFA, + author = {Daniel Z. Korman}, + editor = {}, + publisher = {The PhilPapers Foundation}, + title = {Learning From Arguments: An Introduction to Philosophy}, + year = {2022} +}