There is much debate over the degree to which language learning is governed by innate language-specific biases, or acquired through cognition-general principles. Here we examine the probabilistic language acquisition hypothesis on three levels: We outline a theoretical result showing that probabilistic learning in the limit is possible for a very general class of languages. We then describe a practical computational framework, which can be used to quantify natural language learnability of a wide variety of linguistic constructions. Finally, we present an experiment which tests the learnability predictions for a variety of linguistic constructions, for which learnability has been much debated. We find that our results support the possibility that these linguistic constructions are acquired probabilistically from cognition-general principles.