This is perhaps the most formidable of all the logical fallacies, or at least on of the most annoying. This happens whenever anyone says that you can’t prove that their claim is false, therefore the claim is true. This tactic can be employed accidentally, but any honest person will relinquish their claim when it is pointed out to them that they are making a conjecture based on a lack of knowledge. A “believer” (in woo, religion, etc…) will hold to and defend their claim anyway even when you point out that logic doesn’t allow you to draw conclusions from facts you don’t know. This reasoning, at it’s worst, can look like this:
Person A: We have no idea who committed the murder, but someone must have done it. Let’s arrest Fred.
Person B: Really? Do you know it was him?
Person A: No. But you can’t prove that it’s not him!
This fallacy isn’t always so in-your-face. Often it hides behind special terms. For example, the phrase God of the Gaps is used to describe this fallacy whenever someone uses God to fill a gap in current scientific knowledge. “Oh, you don’t know how life got started on Earth? Well, no need to look for evidence to come to a conclusion. Just assume God did it!” Richard Dawkins famously said that he is against religion because it teaches people to be satisfied with not knowing answers to questions. This is the type of reasoning he’s talking about.
A related fallacy (which I bring up here because I see it as a kind of sub-type of the Appeal to Ignorance) is the Special Pleading Fallacy. This happens whenever someone’s reasoning goes along the lines of, “There’s a reason that I’m right, you’re just too stupid to see it.” This happens a lot in arguments with religious people too (see the trend?) whenever they can’t explain why God did something.
Person A: So why did God kill so many babies with that hurricane?
Person B: Look, just because I can’t come up with an explanation doesn’t mean that you’ve disproved my hypothesis (that a god exists). Maybe God just works in mysterious ways that you can’t understand.
The Special Pleading is the most common reply I hear to the problem of evil and is one of the twelve types of possible responses. Special pleading can be employed a lot by other types of believers as well. It’s the most common response from proponents of homeopathy, for example. “So how come homeopathy has to ignore several laws of physics in order to work?” “It must work by some mysterious way that we don’t know about/that is outside of science.”