An appeal to ignorance takes the fact that something has not been proven as a sufficient reason for believing that it is false.

The logical form of an appeal to ignorance is as follows:

(1) There is no evidence that p.
Therefore:
(2) It is not the case that p.

Arguments of this form are fallacious except in those cases where if p were true then we would have evidence for it. Consider the following argument, a simplified version of the atheist philosopher Ted Drange’s “argument from nonbelief”:

(1) There is no evidence that a God whose primary concern is to reveal himself to us exists.
Therefore:
(2) It is not the case that such a God exists.

This argument is not a fallacious appeal to ignorance. If there were a God whose primary concern is to reveal himself to us, then we would expect to have evidence of his existence. The absence of such evidence therefore would support the claim that there is no such God.

The following argument, however, does commit the appeal to ignorance fallacy:

(1) No one has been able to prove that God exists.
Therefore:
(2) God does not exist.

This argument takes an absence of a certain level of evidence for God’s existence to be sufficient reason to believe in God’s non-existence. The existence of God, however, is perfectly consistently with our being unable to prove the existence of God, and so our inability to prove the existence of God would not entail his non-existence.