This article provides a preliminary semantic framework for Dependency Grammar in which lexical words are semantically defined as contextual distributions (sets of contexts) while syntactic dependencies are compositional operations on word distributions. More precisely, any syntactic dependency uses the contextual distribution of the dependent word to restrict the distribution of the head, and makes use of the contextual distribution of the head to restrict that of the dependent word. The interpretation of composite expressions and sentences, which are analyzed as a tree of binary dependencies, is performed by restricting the contexts of words dependency by dependency in a left-to-right incremental way. Consequently, the meaning of the whole composite expression or sentence is not a single representation, but a list of contextualized senses, namely the restricted distributions of its constituent (lexical) words. We report the results of two large-scale corpus-based experiments on two different natural language processing applications: paraphrasing and compositional translation.