Theories have language acquisition have often had difficulties with understanding how children can learn language from only positive input. They observe what sentences can occur; but seem fairly insensitive to information about what sentences cannot occur (if, indeed, much such information is available). In particular, it has often been viewed as paradoxical that language learners can avoid overgeneral grammars, which allow ungrammatical sentences---because these grammars 'fit' all the observed input to the child. This has been one motivation for nativist views of language acquisition. I discuss some recent results, from joint work with Paul Vitanyi, that show that, in principle, language learning from positive evidence is possible, in a statistical sense. This suggests that empiricist views of language acquisition, and perhaps learning in other domains, may be more feasible than is frequently assumed.
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