"Absence of evidence isn't evidence of absence."
You've probably heard this line many times before, and you've probably heard it abused almost all of those times. It's a particular favorite of people who want to leave open the possibility of their pet supernatural (or just unconventional) belief which has absolutely no evidence supporting it. However, there are a few problems with this line of reasoning.
This statement contains within it a common linguistic assumption which has indirectly led to many logical errors and misconceptions. The statement can more clearly and accurately be stated as "Absence of evidence isn't necessarily evidence of absence." This should be contrasted with the meaning of "Absence of evidence is never evidence of absence." The dropping of "necessarily" from the initial (true) statement changes both its denotational and connotational meaning, but it's something that happens in casual speach, especially when dropping it leaves a line that's much catchier.
The reason it's crucial to leave in "necessarily" is that the statement has a big exception to it. This exception is for when you've actually looked for evidence - something that's actually happened in most cases where this mantra is being used to defend someone's belief, thus making their use of it fallacious. When you've appropriately looked for evidence for a claim and didn't find any, you can put this into the Modus Tollens argument form (slightly modified) to use this as evidence that the claim is false.
The basic form of Modus Tollens is:
If A, then C.
Therefore, not A.
In this case, we modify it to turn A into a union of A and B, getting the form:
If A and B, then C.
B and not C.
Therefore, not both A and B.
Therefore, not A.
In this case, we're using A as some claim, B as a means of investigating that claim, and C as possible evidence that could be found to support that claim.
Let's go through an example to illustrate how this works, such as the claim that there's a full-size rhinoceros in the room. Now, we'll normally have no evidence that there is a rhinoceros in the room, and it's actually quite simple to extend this into evidence that there is no rhinoceros in the room:
If there is a full-size rhinoceros in the room (A) and I look around the room in every area large enough to hold a rhinoceros (B), then I will see a rhinoceros in one of these areas (C).
I looked around the room in every area large enough to hold a rhinoceros (B), and I did not see a rhinoceros in any of them (not C).
Therefore, either I did not look thoroughly enough, or there is no rhinoceros in the room (not both A and B).
However, since I did look thoroughly enough (B), there is no rhinoceros in the room (not A).
This principle is quite powerful, and it in fact lies beneath one of the fundamental properties of science: Falsification. Putting arguments into this form is exactly what allows us to test and possibly falsify them. If we assume that absense of evidence isn't evidence of absense, then we've thrown the possibility of falsifying almost anything right out the window.