Distributional semantics creates vectorspace representations that capture many forms of semantic similarity, but their relation to semantic entailment has been less clear. We propose a vector-space model which provides a formal foundation for a distributional semantics of entailment. Using a mean-field approximation, we develop approximate inference procedures and entailment operators over vectors of probabilities of features being known (versus unknown). We use this framework to reinterpret an existing distributionalsemantic model (Word2Vec) as approximating an entailment-based model of the distributions of words in contexts, thereby predicting lexical entailment relations. In both unsupervised and semi-supervised experiments on hyponymy detection, we get substantial improvements over previous results.