Deciding whether a semantically ambiguous word is homonymous or polysemous is equivalent to establishing whether it has any pair of senses that are semantically unrelated. We present novel methods for this task that leverage information from multilingual lexical resources. We formally prove the theoretical properties that provide the foundation for our methods. In particular, we show how the One Homonym Per Translation hypothesis of Hauer and Kondrak (2020a) follows from the synset properties formulated by Hauer and Kondrak (2020b). Experimental evaluation shows that our approach sets a new state of the art for homonymy detection.