This paper proposes a critical revision of the proposal of van Heijenoort to confront a systematic logic with the natural language and questions such as the existence of a universal domain, the subject-predicative structure of sentences, and semantical problems. This paper analyses the historical context of the systematic logic.

Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey van Heijenoort's contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort's is not widely known, and much of it remains unpublished.