It is widely assumed that human learning and the structure of human languages are intimately related. This relationship is frequently suggested to derive from a language-specific biological endowment, which encodes universal, but communicatively arbitrary, principles of language structure (a Universal Grammar or UG). How might such a UG have evolved? We argue that UG could not have arisen either by biological adaptation or non-adaptationist genetic processes, resulting in a logical problem of language evolution. Specifically, as the processes of language change are much more rapid than processes of genetic change, language constitutes a “moving target” both over time and across different human populations, and, hence, cannot provide a stable environment to which language genes could have adapted. We conclude that a biologically determined UG is not evolutionarily viable. Instead, the original motivation for UG – the mesh between learners and languages – arises because language has been shaped to fit the human brain, rather than vice versa. Following Darwin, we view language itself as a complex and interdependent “organism,” which evolves under selectional pressures from human learning and processing mechanisms. That is, languages themselves are shaped by severe selectional pressure from each generation of language users and learners. This suggests that apparently arbitrary aspects of linguistic structure may result from general learning and processing biases deriving from the structure of thought processes, perceptuo-motor factors, cognitive limitations, and pragmatics.