We introduce two reference-free metrics for quality evaluation of taxonomies in the absence of labels. The first metric evaluates robustness by calculating the correlation between semantic and taxonomic similarity, addressing error types not considered by existing metrics. The second uses Natural Language Inference to assess logical adequacy. Both metrics are tested on five taxonomies and are shown to correlate well with F1 against ground truth taxonomies. We further demonstrate that our metrics can predict downstream performance in hierarchical classification when used with label hierarchies.