Absence of proof is not proof of absence is the rebuttal to a logical fallacy called lack of imagination. It is offered in response to a claim that a given thing or idea cannot exist because it has never been detected, and it usually takes the form that human beings cannot perceive all of the universe of things or ideas all of the time.
Ayn Rand has argued for the existence of copyrights, or government-granted monopolies on the expression in an original work of authorship, using a "lack of imagination" argument:
Claim: Copyright must exist, or authors will not create works.
Evidence: No system other than copyright can reward authors enough.
Reasonable doubt: Human thought does not cover all of the business models all of the time. In fact, I know of two business models other than copyright for publication of a work: patronage by advertisers and trade secret licensing.
The Poincaré Conjecture: First try
The Poincaré conjecture states that any compact manifold that is simply connected (that is, doesn't have any holes in it) is homeomorphic to a sphere of the same dimension.
This has been proven true for manifolds of all dimensions except for 3-dimensional manifolds (3-spheres in 4-space).
Claim: The Poincaré conjecture is true.
Evidence: Nobody has found a counterexample.
Reasonable doubt: Human thought does not cover all of the possible 3-manifolds all of the time.
Other conjectures have gone for a century without a counterexample, only to have some mathematician find a disproof.
The Poincaré Conjecture: Second try
Claim: The Poincaré conjecture is unprovable.
Evidence: It has gone unproven for nearly a century.
Reasonable doubt: Other conjectures have gone unproven for a century and were either proven, disproven, or proven to be Gödel-complete.
And now, to brainwave's example:
Fairies: First try
Claim: Fairies do not exist.
Evidence: No respected human biologist has ever seen a fairy directly.
Reasonable doubt: Human reconnaissance does not cover all of the universe all of the time. Specifically, human recon is bad at peering into the underground structures where some fairies are said to make their homes.
Fairies: Second try
According to my measurement of an artist's conception drawn by Myrea Pettit and shown on the web at http://www.fairiesworld.com/homepage.htm, a fairy is about eight times as tall as an acorn; this puts the height of a fairy roughly in the same order of magnitude as the height of a Lilliputian. But a fairy is supposed to be both small and intelligent enough to use language reminiscent of that of a human being. These are conflicting goals because brain-case size limits number of neurons, which in turn limits intelligence (Alex the Parrot notwithstanding).
Claim: Fairies cannot exist because beings with the size and intelligence of a fairy do not exist.
Evidence: No respected human biologist has ever seen a being with the size and intelligence of a fairy.
Reasonable doubt: Human recon does not cover all of the universe all of the time. There may be other creatures and other forms of intelligent life that we just don't know about.