add paper 1
This commit is contained in:
parent
6f693b9da5
commit
3b8a213ab2
GPG key ID:
865658ED1FE61EC3
2 changed files with 222 additions and 0 deletions
215
work/2024/phil-1/paper-1/main.typ
Normal file
215
work/2024/phil-1/paper-1/main.typ
Normal file
|
|
@ -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],
|
||||||
|
<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.]
|
||||||
7
work/2024/phil-1/paper-1/refs.bib
Normal file
7
work/2024/phil-1/paper-1/refs.bib
Normal file
|
|
@ -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}
|
||||||
|
}
|
||||||
Loading…
Reference in a new issue