State-of-the-art deep-learning-based approaches to Natural Language Processing (NLP) are credited with various capabilities that involve reasoning with natural language texts. %However, reasoning in this setting is often ill-defined and shallow. In this paper we carry out a large-scale empirical study investigating the detection of formally valid inferences in controlled fragments of natural language for which the satisfiability problem becomes increasingly complex. We find that, while transformer-based language models perform surprisingly well in these scenarios, a deeper analysis reveals that they appear to overfit to superficial patterns in the data rather than acquiring the logical principles governing the reasoning in these fragments.