Proving a Negative

I always thought that proving a negative is impossible. I just learned and read that it is not. It would be difficult though to prove a negative because one must have omniscience to do so. But "proving" in proving a negative does not necessarily mean proving beyond reasonable doubt or a mathematical proof. One must only need reasons or evidence (or maybe the burden of proof) to prove a negative and not necessarily a proof beyond reasonable doubt. There are two (plus one below) ways in which one can prove the nonexistence of something. One is to prove that it cannot exist because it leads to a contradiction (incompatible properties argument) and the other is by carefully looking and seeing (empirical evidence).

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:

1. the area where evidence would appear, if there were any, has been comprehensively examined, and
2. all of the available evidence that X exists is inadequate, and
3. X is the sort of entity that, if X exists, then it would show.
   

So all in all, based on the three ways above one can really prove a negative or that prove that there is no god or Santa Claus or Loch Ness Monster or a planet between Earth and Venus.