Absence of evidence is not evidence of absence

"Absence of evidence is not evidence of absence" is a rebuttal to an argument of the form "there is no evidence for X, therefore X does not exist." Logically, the latter argument is invalid; however, a similar, more careful line of inductive reasoning can nevertheless lead to a reliable conclusion. Thus the objection may be warranted or unwarranted, depending on the circumstances.

A simple example

 * An explorer lands on a remote island and sees no buildings or other signs of human habitation.
 * The explorer therefore concludes that the island is uninhabited.

The conclusion here is unwarranted. There may be inhabitants on the other side of the island or in dwellings that are otherwise hidden from view. In this case, absence of evidence of habitation is not evidence that there are no inhabitants.

However, consider the following scenario:
 * An explorer lands on a remote island and spends weeks carefully searching the entire island for signs of habitation.
 * Finding no evidence, the explorer concludes that the island is uninhabited.

In this case, the explorer's conclusion is justified. If there were inhabitants on the island, there should be some evidence of their existence somewhere on the island. If there are no signs where there should have been some, then it is likely that there are no inhabitants.

In summary: Note that the "diligent search" must be of a kind that could realistically be expected to produce positive results if P were indeed true.
 * When presented with some proposition P, we should initially keep an open mind. But if a diligent search fails to produce evidence that P is true, then we can rationally conclude that P is false.

Scientific investigation
Science often works in a similar manner, using inference based not only on observed but also unobserved phenomena.

An example:
 * A scientist develops hypothesis H explaining a certain observed phenomenon P1.
 * The scientist reasons that if H is true, then a different phenomenon P2 should also be observed under certain conditions.
 * When this prediction is tested, phenomenon P2 is not observed.
 * Repeated tests still fail to produce phenomenon P2.
 * The scientist concludes that H was not the proper explanation of phenomenon P1.

Note that even here, by strict deductive logic the conclusion still does not follow. However, when dealing with what is likely to be true, this kind of inductive logic can still be relied upon — provided the conclusions are recognized as being subject to revision upon further investigation.

Absence of evidence for God
The question of whether God exists is considered very important to many people, and has been for millennia. Thousands of people have assiduously searched for evidence that a god exists. Nonetheless, there is still no good evidence for a god. Many atheists thus feel justified in not believing that any gods exist, and in fact some feel justified in concluding that no gods exist.