Incompatible properties arguments attempt to show a logical contradiction in the concept in question. It shows that an entity (or something) cannot exist because of a certain contradiction. A good example of this is the properties of the Christian God/Theism. One can prove that this god does not exist by showing contradictions in some of its properties. For example, Omniscience and Free Will. A being/entity cannot be omniscient and at the same time free (acquires free will). It shows that that entity can't simply exist because of the two contradictory properties. Another example is the Transcendence and Omnipresence properties of the theist god. The incompatible properties argument can also show that two objects cannot logically exist simultaneously. For example, the god of Islam cannot co-exist with the God of Christians.
Another way to prove a negative is by carefully looking and seeing. The idea is that if a thing exists, one must be able to detect its existence directly or indirectly (not directly observable but the object causes effects which are directly observable). For example, suppose a friend of mine claims that there is a dog in my room but when I went to my room I found no dog. Since I found no dog, then I can conclude that there was no dog in my room. This same method allows us to know that things such as unicorns, Loch Ness Monster, Bigfoot do not exist. Of course one could argue in my example above that that dog may have left when I arrived in my room. But based on the lack of proof/evidence that there was a really dog and that based on the limits of my understanding of anything (I could be mistaken on what I saw) I still proved a negative in such a case. Again, proof here is not the same as mathematical proof.
Lastly, another good way (found also here) to know that a person is justified in believing that a thing, say X, does not exist if all of the conditions below are met:
- the area where evidence would appear, if there were any, has been comprehensively examined, and
- all of the available evidence that X exists is inadequate, and
- X is the sort of entity that, if X exists, then it would show.