Locke described the argumentum ad ignorantiam as a way that 'men ordinarily use to drive others and force them to submit their judgments and receive the opinion in debate.' Locke defined this type of argument as the kind of move where one party in such a debate requires the other party to admit what the first party alleges as a proof or assign a better. In other words, what the arguer is saying is, 'I offered you what I think constitutes a proof, so we have to tentatively accept it unless you can offer a proof to the contrary.' In other words, the arguer is saying he has a right to put this proposition forward as a judgment that both parties should receive or accept, at least tentatively, until the other party can disprove it, or put some proposition in its place that is proved. (Douglas Walton)That said, the fact is that the expression argumentum ad ignorantiam has morphed to mean something very different from what Locke intended. Some of the various uses of the expression that have sprung from the Latin expression can be seen in the various ways it has been translated into English: argument to ignorance, argument from ignorance, and appeal to ignorance. One will also find closely related discussions regarding the evidence of absence and the absence of evidence. Also, some writers have associated the argumentum ad ignorantiam with the idea of proving, or not being able to prove, a negative.
Many logic texts list the argumentum ad ignorantiam as a fallacy of reasoning. Examples vary, but some of the more popular ones refer to Sen. Joseph McCarthy's justifying a name remaining on a list of suspected Communists because "there is nothing in the files to disprove his Communist connections." I used to call this the "Mike Wallace fallacy" when I was teaching logic courses; I named it after a tactic Mr. Wallace frequently used on "60 Minutes." He would show up unannounced, confront a surprised person with accusations of some sort of wrongdoing, and then the scene would cut to a slamming door or a grainy film of a car driving out of a parking lot. Wallace would then announce something to the effect of: Mr. X refuses to answer our questions and still has not shown any signs that he is innocent of the charges we've made. It should be obvious that not having proof that someone is not a Communist is not proof that he is and not defending yourself against charges is not the same as admitting they are true.
Another common example given in text books is from the Salem witch trials of 1692 where some of those testifying claimed that they could see specters or auras around the accused, but these specters were visible only to the witnesses. Such claims are impossible to disprove. They're in the same class as the claims of mediums who say they are getting messages from the dead. One would assume that a reasonable person would require more evidence than just the word of a witness or medium when judging either the cause of the perception or the veracity of the sensations reported. Furthermore, the fact that an accused witch could not prove that she didn't have a demon's specter around her or that a skeptic cannot prove that John Edward is not getting messages from someone's Aunt Sadie does not imply that the accused is a witch or that Edward is really psychic.
I remember, and hope I am remembering accurately, a televised speech by Ronald Reagan where he defended the notion that a fetus is a person by noting that scientists haven't proved that the fetus isn't a person. It is true that scientists haven't proved that a fetus isn't a person, but being a person in this context is not a matter of discovery but of definition. There is no imaginable discovery any scientist could make that would be proof that a fetus is a person. It is irrelevant to the issue of whether a fetus is a person to point out that scientists haven't proved that fetuses are not persons. The U.S. Supreme Court has ruled that corporations are persons. One day perhaps dolphins and chimpanzees will be declared persons by lawmakers somewhere. You can't turn a corporation into a biological human being by definition, but you can put them both in the class of persons by definition.
Clearly, there are times when not knowing whether something exists does not mean that that something does not exist. The fact that the U.S. or other international agents did not discover weapons of mass destruction in Iraq before President George W. Bush ordered the invasion of that country did not prove that there weren't any such weapons in Iraq. Now, several years after the invasion and having had plenty of time to locate such weapons, it seems highly unlikely that Iraq possessed such weapons.
Given all the time to find evidence for Biblical stories like the universal flood that the god of the Hebrews allegedly inflicted on creation, it is reasonable to reject the story as a myth. Scientists know what kinds of evidence there should be on the planet had such a universal flood ever occurred. The lack of such evidence and the appeal to such things as the Grand Canyon as evidence of The Flood make belief in this story rather absurd. To appeal to miracles or divine intervention to clean up the evidence just makes the belief even less defensible. Such appeals are clearly ad hoc and have no basis in reality. Nobody can prove that The Flood didn't occur, but no reasonable person can believe that it did without giving up the basis of reasonable belief: considering all the available evidence rather than speculating about miracles and question-begging interventions from supernatural forces. Often, a tactic of Bible defenders who are challenged to provide positive evidence for some story or belief respond by trying to shift the burden of proof by challenging an opponent to prove the supernatural speculations are wrong. Here it seems appropriate to bring in Bertrand Russell's famous illustration involving a celestial teapot:
If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes.
But if I were to go on to say that, since my assertion cannot be disproved, it is an intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense.
If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.
Religion isn't the only arena where these kinds of battles take place, of course. The fact that Nessie has not been discovered in Loch Ness, even though there have been numerous expeditions that have scoured that lake from one end to the other, does not prove that Nessie doesn't exist but it makes it highly unlikely. No solid evidence for her existence has ever been found, but something should have been found by now if the creature exists. I can't prove Bigfoot doesn't exist, but I can demonstrate that it is very unlikely given all the time and energy spent trying to establishing its existence but with nothing to show for it except a few questionable sightings, photos, and imprints. I can't prove psi doesn't exist, but I can demonstrate that the evidence for psi is below the threshold any reasonable person should demand given the time, effort, intelligence, and energy put into finding proof of psychic phenomena. I can't prove that the experience you had yesterday was not clairvoyance, but my inability to prove it wasn't clairvoyance is irrelevant to whether it was. I can't prove that what you saw in the sky last night was not an alien spacecraft, but that doesn't mean it was. Furthermore, there is an abundance of independent evidence that makes it more probable that whatever you saw was something besides a spacecraft from another planet. So, if we're interested in what is reasonable to believe rather than what is possibly true, we take into account all the evidence.
You may find some logic texts referring to the following kind of argument as an argument to ignorance: I'm justified in believing that a god exists because neither you nor anyone else can prove she doesn't. It is irrelevant to the truth-value of the statement "A god does not exist" to note that nobody has proven or can prove that a god exists. I may not have the knowledge needed to prove that a god exists, but my ignorance is irrelevant to the truth-value of the claim "A god does not exist." Likewise, we don't know whether there is life on other planets, but our ignorance is irrelevant to whether such life exists. In any case, for many of the existence claims brought up regarding arguments from ignorance, the issue isn't so much whether something like a god, a plesiosaur, or something psychic certainly exists or not but whether the evidence is sufficient to warrant tentative assent that gods do or don't exist, that Nessie does or doesn't exist, or that psychic phenomena do or don't exist. The fact that it is possible that a god, a sprite, a leprechaun, a Nessie, or a Bigfoot exists isn't of much interest in the quest to determine whether the evidence warrants assent one way or the other. The fact that it is possible that a subtle energy (qi) exists and that we can't prove that qi does not permeate all things isn't much of a reason for believing such an energy exists and affects our well-being. Maybe we can't prove that it was the chemotherapy rather than the reiki and the coffee enemas that led to your cancer's remission, but we can point to a strong body of scientific evidence supporting the effectiveness of the chemotherapy you received. You may be able to point to several anecdotes in favor of reiki and coffee enemas, but the mere fact that we can't prove they had any effect does not mean that they did. Furthermore, the anecdotal evidence pales in comparison to the scientific evidence regarding cancer treatment.
Creationists such as Michael Behe have defended their belief in what they call intelligent design by claiming that scientists don't know how certain biochemical processes at the cellular level evolved. They then assume scientists will never discover a natural explanation for the evolution of these processes and conclude that an intelligent designer is responsible for them. Rupert Sheldrake and other dualists seem to think that an argument in favor of believing that consciousness is immaterial is the fact that scientists do not know how or what brain processes bring about consciousness. The fact that scientists do not yet have a consensus explanation for the nature of consciousness in terms of physical and chemical brain processes does not provide support for the immaterialist hypothesis any more than the fact that dualists cannot explain how an immaterial reality could have material effects provides support for the brain=consciousness hypothesis.
Nobody can prove that a god didn't kill thousands by directly creating an earthquake and tsunami, but people who make such claims should know that not being able to disprove this claim doesn't provide any support for it. Also, if one is willing to accept such claims there is no end to the list of unsupported speculations one could make about the invisible causes of things happening in the visible world. Some readers will be aware of the mockery made of this kind of religious presumption by the ideas of the Flying Spaghetti Monster and the Invisible Pink Unicorn.
Another topic frequently visited by those writing about the argument to ignorance is the idea of proving a negative. I can prove I have no money in my pocket by turning the pocket inside out and showing you. I can prove nobody else is in the other room by taking you to the room and inspecting it with you. I can't prove that there's not an invisible dragon in your garage, but I can prove that -5 plus +3 equals -2. Is that proving a negative? When I make plans for tomorrow, I am being directed by what does not exist...yet. Is this absence of what I plan to do a negative that I can prove by doing it?
More to the point, a lab test that comes back negative for signs of cancer does not mean you do not have some cancer cells still active in your body. An X-ray may show no sign of fracture, but you might still have a fracture.
Finally, there are examples involving ignorance, lack of evidence or proof, and argumentation in our courtrooms. Just because a prosecutor can't prove to a jury that the accused is guilty beyond a reasonable doubt does not mean that the accused didn't do what he is accused of. The absence of the victim's blood on the arrested's clothes does not mean he didn't commit the bloody crime. The inability of the accused to prove that he didn't commit the crime he is accused of does not mean that he probably committed the crime. The fact that the accused can't account for his whereabouts at the time a crime was committed does not mean that it is reasonable to assume he was at the scene of the crime. On the other hand, the inability to locate the body of a suspected victim is not sufficient to prevent a person from being charged and convicted of murder.