The recent success of deep neural network techniques in natural language processing rely heavily on the so-called distributional hypothesis. We suggest that the latter can be understood as a simplified version of the classic structuralist hypothesis, at the core of a program aiming at reconstructing grammatical structures from first principles and analysis of corpora. Then, we propose to reinterpret the structuralist program with insights from proof theory, especially associating paradigmatic relations and units with formal types defined through an appropriate notion of interaction. In this way, we intend to build original conceptual bridges between computational logic and classic structuralism, which can contribute to understanding the recent advances in NLP.