Overview: At least since Derek Parfit’s Reasons and Persons, philosophers have been searching for a satisfactory population axiology: a theory of the value of populations. Unfortunately, the project has proved difficult. Some claim that it’s impossible. Several philosophers offer impossibility theorems which seem to prove that no population axiology can satisfy each of a small number of adequacy conditions. Of these impossibility theorems, Gustaf Arrhenius’s six theorems are perhaps the most compelling.
However, it’s recently been pointed out that each of Arrhenius’s theorems depends on a dubious assumption: Finite Fine-Grainedness. This assumption states, roughly, that you can get from a very positive welfare level to a very negative welfare level via a finite number of slight decreases in welfare. Lexical population axiologies deny Finite Fine-Grainedness, and so can satisfy all of Arrhenius’s plausible adequacy conditions. These lexical views have other advantages as well. They cohere nicely with most people’s intuitions in cases like Haydn and the Oyster, and they offer a neat way of avoiding the Repugnant Conclusion.
In this paper, I rework Arrhenius’s impossibility theorems so that lexical views do not escape them. I point out that, since all of our population-affecting actions have a non-zero probability of bringing about more than one distinct population, it is population prospect axiologies that are of practical relevance. I then prove impossibility theorems which state that no population prospect axiology can satisfy each of a small number of adequacy conditions. These theorems do not depend on Finite Fine-Grainedness, so even lexical views violate at least one of their conditions.
How we should respond to these theorems is another question. Though I don't say it in the paper, I believe that the Total View is as satisfactory as population prospect axiologies get. We should accept the Repugnant Conclusion (and even the Very Repugnant Conclusion) because each of the alternatives is even worse.