MichaelStJules comments on MichaelStJules’s Quick takes

[MichaelStJules]

If we think

1. it’s always better to improve the welfare of an existing person (or someone who would exist anyway) than to bring others into existence, all else equal, and

2. two outcomes are (comparable and) equivalent if they have the same distribution of welfare levels (but possibly different identities; this is often called Anonymity),

then not only would we reject Mere Addition (the claim that adding good lives, even those which are barely worth living but still worth living, is never bad), but the following would be true:

Given any two nonempty populations $A$ and $B$, if any individual in $B$ is worse off than any individual in $A$, then $A\cup B$ is worse than $A$. In other words, we shouldn’t add to a population any individual who isn’t at least as well off as the best off in the population, all else equal.

Intuitively, adding someone with worse welfare than someone who would exist anyway is equivalent to reducing the existing individual’s welfare and adding someone with better welfare than them; you just swap their welfares.

More formally, suppose $a$, a member of the original population $A$ with welfare $u$, is better off than $b$, a member of the added population $B$ with welfare $v$, so $u>v$. Then consider

$A^{\prime }$ which is $A$, but has $b$ instead of $a$, with welfare $u$.

$B^{\prime }$ which is $B$, but has $a$ instead of $b$, with welfare $v$.

Then, $A$ is better than $A^{\prime }\cup B^{\prime }$ , by the first hypothesis, because the latter has all the same individuals from $A$ (and extras from $B$) with exactly the same welfare levels, except for $a$ (from $A$ and $B^{\prime }$) who is worse off with welfare $v$ (from $B^{\prime }$) instead of $u$ (from $A$). So $A\succ A^{\prime }\cup B^{\prime }$.

And $A^{\prime }\cup B^{\prime }$ is equivalent to $A\cup B$ , by the second hypothesis, because the only difference is that we’ve swapped the welfare levels of $a$ and $b$. So $A^{\prime }\cup B^{\prime }\simeq A\cup B$.

So, by transitivity (and the independence of irrelevant alternatives),

$$A\succ A\cup B.$$

[MichaelStJules]

If welfare is real-valued (specifically from an interval $I\subseteq R$), then Maximin (maximize the welfare of the worst off individual) and theories which assign negative value to the addition of individuals with non-maximal welfare satisfy the properties above.

Furthermore, if along with welfare from a real interval and property 1 in the previous comment (2. Anonymity is not necessary), the following two properties also hold:

3. Extended Continuity, a modest definition of continuity for a theory comparing populations with real-valued welfares which must be satisfied by any order representable by a real-valued function that is continuous with respect to the welfares of the individuals in each population, and

4. Strong Pareto (according to one equivalent definition, under transitivity and the independence of irrelevant alternatives): if two outcomes with the same individuals in their populations differ only by the welfare of one individual, then the outcome in which that individual is better off is strictly better than the other,

then the theory must assign negative value to the addition of individuals with non-maximal welfare (and no positive value to the addition of individuals with maximal welfare) as long as any individual in the initial population has non-maximal welfare. In other words, the theory must be antinatalist in principle, although not necessarily in practice, since all else is rarely equal.

Proof : Suppose $A$ is any population with an individual $a$ with some non-maximal welfare $u$ and consider adding an individual $b$ who would also have some non-maximal welfare $v$. Denote, for all $ϵ>0$ small enough ($0<ϵ<{ϵ}_0$),

$A+ϵ{\chi }_a$: the population $A$, but where individual $a$ has welfare $u+ϵ$ (which exists for all sufficiently small $ϵ>0$, since $u$ is non-maximal, and welfare comes from an interval).

Also denote

$B$: the population containing only $b$, with non-maximal welfare $v$, and

$C$: the population containing only $b$, but with some welfare $w>v$ ($v$ is non-maximal, so there must be some greater welfare level).

Then

$$\begin{array}{cc}1.& A+ϵ{\chi }_a\succ A\cup C,\text{for all}ϵ,0<ϵ<{ϵ}_0,\text{and}\\ 2.& A\cup C\succ A\cup B,\end{array}$$

where the first inequality follows from the hypothesis that it’s better to improve the welfare of an existing individual than to add any others, and the second inequality follows from Strong Pareto, because the only difference is $b$’s welfare.

Then, by Extended Continuity and the first inequality for all (sufficiently small) $ϵ>0$, we can take the limit (infimum) of $A+ϵ{\chi }_a$ as $ϵ\to 0$ to get

$$A⪰A\cup C,$$

so, it’s no better to add $b$ even if they would have maximal welfare, and by transitivity (and the independence of irrelevant alternatives) with $2.A\cup C\succ A\cup B$,

$$A\succ A\cup B,$$

so it’s strictly worse to add $b$ with non-maximal welfare. This completes the proof.