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+#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],
+<Korman2022-KORLFA>) 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],
+<Korman2022-KORLFA>) 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(<Korman2022-KORLFA>, 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
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+++ b/work/2024/phil-1/paper-1/refs.bib
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+@book{Korman2022-KORLFA,
+	author = {Daniel Z. Korman},
+	editor = {},
+	publisher = {The PhilPapers Foundation},
+	title = {Learning From Arguments: An Introduction to Philosophy},
+	year = {2022}
+}