<p>Biodiversity is commonly described as having multiple different facets, or aspects. However, there is no clear account of the sense in which these aspects fit together and contribute to an overarching concept. Neither the proponents nor the critics of the multi-faceted view of biodiversity have provided a general analysis of what it would mean for the various facets to be combinable. This article attempts to provide such an analysis. It argues that the combination of aspects of biodiversity is in many ways analogous to the combination of preference-rankings in social choice theory, and faces obstacles that are similar to Arrow’s famous impossibility theorem. We derive a similar impossibility result, and discuss its implications for the study of biodiversity.</p>