In this paper we present new results on applying topological data analysis to discourse structures. We show that topological information, extracted from the relationships between sentences can be used in inference, namely it can be applied to the very difficult legal entailment given in the COLIEE 2018 data set. Previous results of Doshi and Zadrozny (2018) and Gholizadeh et al. (2018) show that topological features are useful for classification. The applications of computational topology to entailment are novel in our view provide a new set of tools for discourse semantics: computational topology can perhaps provide a bridge between the brittleness of logic and the regression of neural networks. We discuss the advantages and disadvantages of using topological information, and some open problems such as explainability of the classifier decisions.