The lexical knowledge of NLP systems shouldbe tested (i) for their internal consistency(avoiding groundedness issues) and (ii) bothfor content words and logical words. In thispaper we propose a new method to test the understandingof the hypernymy relationship bymeasuring its antisymmetry according to themodels. Previous studies often rely only on thedirect question (e.g., A robin is a ...), where weargue a correct answer could only rely on collocationalcues, rather than hierarchical cues. We show how to control for this, and how it isimportant. We develop a method to ask similarquestions about logical words that encode anentailment-like relation (e.g., because or therefore).Our results show important weaknessesof BERT-like models on these semantic tasks.