Most people who argue about population ethics think the hard part is figuring out how much a life is worth. Turns out, that might be the easy part.
Here's the thing — there's a result in moral philosophy that quietly wrecks a whole family of theories we use to think about whether having more people is good or bad. That's why it's called the Arrhenius sixth impossibility theorem, and if you've never heard of it, you're not alone. But if you care about climate policy, procreation, or long-termism, it's already shaping arguments you've read.
The short version is this: critical-level views — the theories that say a life is worth adding only if it's above a certain quality threshold — run into a mathematical wall. And that wall isn't a matter of opinion. It's a theorem Still holds up..
What Is the Arrhenius Sixth Impossibility Theorem
So what are we even talking about? Let's skip the textbook voice and just walk through it Small thing, real impact..
Critical-level views are a type of totalist population ethics. In real terms, if their life is below that line, adding them makes the world worse. They say: adding a person to the world is good if their life is better than some fixed "critical level" of well-being. If it's exactly at the line, it's neutral Which is the point..
Gustaf Arrhenius, a philosopher at the Institute for Futures Studies, spent years mapping out what population axioms everyone agrees on — and then showing you can't have them all. He produced a series of "impossibility theorems." The sixth one is the brutal one for critical-level views.
Not obvious, but once you see it — you'll see it everywhere Worth keeping that in mind..
Critical-Level Views in Plain Language
Imagine a dial. In practice, every life has a happiness score. The critical level is the mark on the dial where a life stops being a net negative to add. Above it? Go ahead, have the kid. Below it? You've made things worse by existing.
That sounds reasonable. It avoids the "more people at any cost" problem of pure total utilitarianism. And it avoids the "only prevent suffering" quietism of some anti-natalists That alone is useful..
The Sixth Theorem Specifically
The sixth impossibility theorem shows that no critical-level view can satisfy a small set of apparently innocent conditions about how we compare populations. Conditions like: if everyone is better off in B than in A, B is better. Or: adding a tiny benefit to a person shouldn't flip the whole moral ranking of two worlds Worth keeping that in mind..
Arrhenius proved that if you accept a few of these sane-sounding rules, critical-level views collapse into contradiction. Not "they're unpopular." Contradiction. As in, the math says they can't be consistent That's the part that actually makes a difference. Which is the point..
Why It Matters
Why does this matter? Because most people skip it.
Every time someone says "we should only have kids if they'll have a decent life," they're leaning on a critical-level intuition. Every time a policy paper weights future generations by quality of life, same thing. The theorem says those intuitions, taken seriously as a full theory, don't hold together.
And look — this isn't just philosophy-class navel-gazing. This leads to the UK's climate advice, fertility incentives in wealthy nations, and effective-altruism debates about the far future all touch population ethics. If the cleanest version of "only add good lives" is mathematically impossible, we're building on sand.
What Goes Wrong Without Noticing
In practice, people patch the theory. They say "well, critical level is zero" — but that's just average utilitarianism in disguise, and it has its own problems (like thinking a massive population of barely-worth-living lives is great). So naturally, or they make the critical level variable. Arrhenius's later work shows that moves you to other impossibility results, not out of the woods Worth keeping that in mind..
Real talk: most applied ethicists don't engage the sixth theorem. They cite Parfit's Repugnant Conclusion and stop. Day to day, it doesn't just say "your view has a weird outcome. But the sixth theorem is sharper. " It says "your view can't exist as a coherent ranking.
How It Works
Let's get into the mechanics. I'll keep it grounded.
The Axioms Everyone Likes
Arrhenius uses axioms like:
- Strong Pareto: If every person in population B is at least as well off as in A, and someone is better off, B is better.
- Interpersonal Addition: Adding a person with positive well-being to a population makes it better, all else equal.
- Non-Extreme Priority: You don't infinitely weight the worst off in a way that breaks comparisons.
- Averaging Independence (loosely): How good a population is shouldn't flip based on tiny tweaks to one life.
These aren't radical. A normal person reads them and says "yeah, obviously."
The Critical-Level Structure
A critical-level view assigns value: V(P) = sum over persons (well-being_i − c), where c is the critical level. In practice, if the sum is positive, the population is good to create. If negative, bad Simple, but easy to overlook..
Now combine that formula with the axioms. So arrhenius shows you get a contradiction. The simplest version: you can construct two populations where axiom A says B is better, but the critical-level sum says A is better. And another pair where the reverse happens. The view can't rank every population consistently.
Why the Math Bites
Here's what most people miss. Now, the problem isn't the number you pick for c. Day to day, it's that any fixed c creates rankings that violate the mild axioms when you scale populations up or down. Big populations with mid lives vs small populations with great lives — the subtraction of c per head distorts things in ways the axioms forbid.
I know it sounds simple — but it's easy to miss that "subtract a constant per person" is exactly what breaks consistency. You'd think a constant is safe. It isn't.
A Sketch of the Proof Idea
Take a world A with a few very happy people. Consider this: world B adds millions at just-above-critical level. Critical-level math says B is hugely better (millions of small positives). But an axiom about not letting tiny per-person gains dominate can say A is better if those millions are only marginally above c. Think about it: flip the numbers, get the opposite verdict. No critical level escapes both Nothing fancy..
Common Mistakes
Honestly, this is the part most guides get wrong.
Mistake 1: Thinking It Only Refutes Extremes
People hear "impossibility theorem" and assume it's about some crazy view nobody holds. Now, no. The sixth theorem hits the moderate, common-sense critical-level view. The one your smart friend defends at dinner.
Mistake 2: Believing a Lower Critical Level Fixes It
Drop c to zero? Plus, that's average-utilitarian-ish and fails other axioms (Arrhenius's earlier theorems). Drop it negative? You're back to "more lives always better" repugnance. Because of that, the theorem isn't about the level. It's about the structure.
Mistake 3: Confusing It With the Repugnant Conclusion
The Repugnant Conclusion is a weird outcome. One says "your view implies something ugly.The sixth theorem is a consistency proof. Different beasts. " The other says "your view can't be a view." Worth knowing the difference.
Mistake 4: Thinking Personal Intuition Is Safe
You might say "I don't need a formal theory, I just judge each case.Worth adding: " Fine for you. But institutions need consistent rules. And the theorem shows even your scattered intuitions, if they include critical-level thinking plus the mild axioms, are inconsistent somewhere And it works..
Practical Tips
What actually works when you're stuck with this?
Tip 1: Use Critical-Level Views Locally, Not Globally
For small-scale decisions — should this couple have a kid given their situation? Don't pretend it scales to civilization-level population policy. On the flip side, — a critical-level heuristic is fine. The theorem bites at scale It's one of those things that adds up..
Tip 2: Read the Source Before Debating
Arrhenius's book Population Ethics lays it out. If you're writing about fertility or futures, cite the sixth theorem correctly. Misusing it is worse than not knowing it Simple, but easy to overlook..
Tip 3: Pair With Sufficientarianism Carefully
Some think sufficientarian views (enough is enough, no credit for extra) dodge it. They have their own issues. But mixing a soft critical level with a cap can approximate sane policy without claiming theorem-proof purity.
Tip 4: Be Honest About Inconsistency
If your org's values imply a critical level
, acknowledge that there will be edge cases where your stated principles collide. Write down where the friction shows up—say, in funding decisions between preventing severe deprivation and adding mildly positive lives—and let stakeholders see the trade-off rather than hiding behind a falsely unified framework Which is the point..
Tip 5: Separate Descriptive From Normative Claims
It helps to say plainly when you are describing what people tend to value (more lives, fewer tragedies) versus prescribing what they ought to value. The sixth theorem is a normative constraint on theory-building, not a prediction about behavior. Keeping that line clear stops pointless arguments about "but everyone thinks having kids is good.
Conclusion
The sixth theorem of population ethics is not a curiosity for philosophers alone. Worth adding: it shows that any critical-level view, combined with a few weak and reasonable axioms, cannot deliver consistent verdicts across all population scenarios. Still, practical coping means using critical-level thinking where it is harmless, citing the result accurately, and admitting the limits of any rule we adopt. On top of that, the mistakes around it—dismissing it as extremism, lowering the threshold, confusing it with the Repugnant Conclusion, or trusting private intuition—obscure how close the problem sits to ordinary moral reasoning. We may not escape the impossibility, but we can stop pretending it is someone else's problem.
