A Dilemma for Lexical and Archimedean Views in Population Axiology

According to lexical views in population axiology, there are good lives x and y such that some number of lives equally good as x is not worse than any number of lives equally good as y. Such views can avoid the Repugnant Conclusion without violating Transitivity or Separability, but they imply a dilemma: either some good life is better than any number of slightly worse lives, or else the ‘at least as good as’ relation on populations is radically incomplete, in a sense to be explained. One might judge that the Repugnant Conclusion is preferable to each of these horns and hence embrace an Archimedean view. This is, roughly, the claim that quantity can always substitute for quality: each population is worse than a population of enough good lives. However, Archimedean views face an analogous dilemma: either some good life is better than any number of slightly worse lives, or else the ‘at least as good as’ relation on populations is radically and symmetrically incomplete, in a sense to be explained. Therefore, the lexical dilemma gives us little reason to prefer Archimedean views. Even if we give up on lexicality, problems of the same kind remain.