An axiology is a theory of betterness — a ranking of possible worlds by how good they are. When worlds can differ in who and how many people exist, every candidate ranking collides with an intuition we are loath to give up. This page lets you build populations, watch six axiologies disagree about them, and walk through the classic results — the Repugnant Conclusion, the Mere Addition Paradox, the Sadistic Conclusion — that culminate in Arrhenius's impossibility theorems.
Throughout, each person has a welfare level on a cardinal scale with a natural zero. Welfare 0 is the neutral level: a life neither good nor bad for the one living it. Positive welfare = a life worth living; negative welfare = a life worth not living (net suffering). A life "barely worth living" sits just above zero. In the diagrams below a world is drawn as blocks: width = number of people, height = welfare per person, so a block's signed area = that group's total welfare, and the dashed line is the neutral level.
Drag a block's right edge to change how many people are in that group, or its top edge to change their welfare. Add or remove groups with the buttons. Both worlds share the same axes, so equal areas mean equal total welfare. The table recomputes live: watch a single pair of worlds get ranked in opposite directions by different theories of value.
| Axiology | Value · A | Value · B | Ranks better | Reads the numbers as… |
|---|
What to look for. The default worlds — a small élite at high welfare (A) versus a large crowd at low but positive welfare (B) — are ranked A-better by average, maximin, critical-level and (for enough dampening) variable-value, yet B-better by the total sum — and, at the default priority strength, by prioritarianism too (a concave transform still rewards B's far larger population). That disagreement is the seed of every paradox below. Push the critical level or γ up and watch views cross over.
Parfit (1984). “For any possible population of at least ten billion people, all with a very high quality of life, there must be some much larger imaginable population whose existence, if other things are equal, would be better, even though its members have lives that are barely worth living.” Slide from world A (few people, high welfare) toward world Z (astronomically many, each life barely worth living), holding the interpolation so that total welfare rises while average welfare falls. Total utilitarianism must call every step an improvement on A; average utilitarianism must call every step a loss.
Height = welfare per person (honest linear scale). Width uses a log scale because populations span 100 → 100,000; areas here are not to scale — read the totals from the readout and the charts.
Two panels, two honest y-axes — never a shared dual axis. The dashed line marks world A's level. Total only ever climbs above it; average only ever sinks below.
Parfit (1984). The Repugnant Conclusion can be argued for without assuming total utilitarianism, using three worlds and a chain of hard-to-deny comparisons. Step through them.
Arrhenius (2000). The theories that dodge the Repugnant Conclusion tend to buy their escape with something worse: they can rank adding a few miserable lives as better than adding many good ones. Start from a fixed background population C, then choose an addition.
Parfit (1984) (anticipated by Schwartz, Adams and Kavka). Many choices — social policies, when to conceive — change which people come to exist, not just how well off a fixed group is. A "depletion" policy might leave future people worse off than an alternative would have — yet those very people owe their existence to it; had we chosen otherwise, different people would exist. So the policy is worse for no one: no one who exists under it would have been better off had we chosen differently (they simply would not have been). This guts person-affecting intuitions — “an outcome can be worse only if it is worse for someone” — which is why person-affecting views struggle to say depletion is wrong, and why they fail completeness/transitivity in the matrix below.
Each view, its value function, the intuition it honours, and the bullet it cannot dodge. n = population wᵢ = welfare of person i
This is the punchline. List a handful of individually compelling adequacy conditions along the top and the axiologies down the side. Hover a cell for the reason. Every row has at least one ✗ — no axiology satisfies them all. That is the shape (though not the letter) of Arrhenius's impossibility theorems: the intuitions we bring to population ethics are jointly unsatisfiable.