Computational and mathematical modeling has revealed that cultural evolution may have played a key role in the evolution of language. This paper explores the hypothesis that processes of cultural transmission have to a large extent shaped language to fit domain-general constraints deriving from the human brain. An implication of this view is that much of the neural hardware involved in language is not specific to it. But how could language have evolved to be as complex as it is without language-specific constraints? Based on computational modeling of the cultural evolution of language, this paper proposes that language has evolved to rely on a multitude of probabilistic information sources for its acquisition, allowing it to be as expressive as possible while still being learnable by domain-general learning mechanisms. Empirical predictions are derived from this perspective regarding the role of phonological cues in the learning of basic aspects of syntax. These predictions are corroborated by results from corpus analyses, computational modeling, and human experimentation, suggesting that the integration of phonological cues with other types of information is integral to the computational architecture of our language capacity. It concludes by considering how computational modeling of cultural evolution can help us understand the evolution of language.