Evidence of absence
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Evidence of absence is evidence of any kind that suggests something is missing or that it does not exist. A simple example of evidence of absence: A baker never fails to put finished pies on her windowsill, so if there is no pie on the windowsill, then no finished pies exist. This can be formulated as modus tollens in propositional logic: P implies Q, but Q is false, therefore P is false.
Per the traditional aphorism, "absence of evidence is not evidence of absence", positive evidence of this kind is distinct from a lack of evidence or ignorance  of that which should have been found already, had it existed. In this regard Irving Copi writes:
In some circumstances it can be safely assumed that if a certain event had occurred, evidence of it could be discovered by qualified investigators. In such circumstances it is perfectly reasonable to take the absence of proof of its occurrence as positive proof of its non-occurrence.
The difference between evidence that something is absent (e.g. an observation that suggests there were no dragons here today) and a simple absence of evidence (e.g. no careful research has been done) can be nuanced. Indeed, scientists will often debate whether an experiment's result should be considered evidence of absence, or if it remains absence of evidence. The debate is whether the experiment would have detected the phenomenon of interest if it was there.
The argument from ignorance for "absence of evidence" isn't necessarily fallacious, for example, that a potentially life saving new drug poses no long term health risk unless proven otherwise. On the other hand, were such an argument to rely imprudently on the lack of research to promote its conclusion, it would be considered an informal fallacy whereas the former can be a persuasive way to shift the burden of proof in an argument or debate. Carl Sagan criticized such "impatience with ambiguity" in cosmologist Martin Rees' maxim, "Absence of evidence is not evidence of absence".
In carefully designed scientific experiments, even null results can be evidence of absence. For instance, a hypothesis may be falsified if a vital predicted observation is not found empirically. (At this point, the underlying hypothesis may be rejected or revised and sometimes, additional ad hoc explanations may even be warranted.) Whether the scientific community will accept a null result as evidence of absence depends on many factors, including the detection power of the applied methods, and the confidence of the inference.
Absence of evidence
If someone were to assert that there is an elephant on the quad, then the failure to observe an elephant there would be good reason to think that there is no elephant there. But if someone were to assert that there is a flea on the quad, then one's failure to observe it there would not constitute good evidence that there is no flea on the quad. The salient difference between these two cases is that in the one, but not the other, we should expect to see some evidence of the entity if in fact it existed. Moreover, the justification conferred in such cases will be proportional to the ratio between the amount of evidence that we do have and the amount that we should expect to have if the entity existed. If the ratio is small, then little justification is conferred on the belief that the entity does not exist. [For example] in the absence of evidence rendering the existence of some entity probable, we are justified in believing that it does not exist, provided that (1) it is not something that might leave no traces and (2) we have comprehensively surveyed the area where the evidence would be found if the entity existed...—J.P. Moreland and W.L. Craig, Philosophical Foundations for a Christian Worldview
Proof and evidence
The Pyrrhonian skeptic, Sextus Empiricus, questioned the apodicticity of inductive reasoning because a universal rule cannot be established from an incomplete set of particular instances: "When they propose to establish the universal from the particulars by means of induction, they will effect this by a review of either all or some of the particulars. But if they review some, the induction will be insecure, since some of the particulars omitted in the induction may contravene the universal; while if they are to review all, they will be toiling at the impossible, since the particulars are infinite and indefinite".
Until about the middle of the previous century induction was treated as a quite specific method of inference: inference of a universal affirmative proposition (All swans are white) from its instances (a is a white swan, b is a white swan, etc.) The method had also a probabilistic form, in which the conclusion stated a probabilistic connection between the properties in question... The Oxford English Dictionary defines “induction”, in the sense relevant here, as follows: "[The] process of inferring a general law or principle from the observation of particular instances..."
[Much] of what contemporary epistemology, logic, and the philosophy of science count as induction infers neither from observation nor from particulars and does not lead to general laws or principles. [Induction] was understood to be what we now know as enumerative induction or universal inference; inference from particular instances:
- a1, a2, …, an are all Fs that are also G... [to a general law or principle] All Fs are G.
A weaker form of enumerative induction, singular predictive inference, leads not to a generalization but to a singular prediction:
- 1. a1, a2, …, an are all Fs that are also G.
- 2. an+1 is also F... [therefore]
- 3. an+1 is also G.
Singular predictive inference also has a more general probabilistic form:
- 1. The proportion p of observed Fs have also been Gs.
- 2. a, not yet observed, is an F... [therefore]
- 3. The probability that a is G is p.—John Vickers, "The Problem of Induction" in The Stanford Encyclopedia of Philosophy
"You can't prove a negative"
In 1992 during a presentation at Caltech, skeptic James Randi uses the phrase "you can't prove a negative". He claims that he cannot prove a negative (such that telepathy does not exist), but he also argues that an individual who claims telepathy exists must prove so. He discusses that induction is often used as a mode of proving a thesis, but if an individual assumes that something is or is not, then the person must prove so. Further, as he says, he does not take an advocacy position, as a lawyer would. He says that he cannot prove that a negative is true, but he could attempt to use evidence and induction to support a claim that he is biased toward, such as a claim that something does not exist (ex. flying reindeer). 
Philosopher Steven Hales points out that typically one can logically be as confident with the negation of an affirmation.
Hales says that if one's standards of certainty leads them to say "there is never 'proof' of non-existence", then they must also say that "there is never 'proof' of existence either". Hales argues that there are many cases where we may be able to prove something does not exist with as much certainty as proving something does exist.
Existence of God
A deity's existence can mean different things to different people. Some claims about the existence of gods or about their actions can be falsified, such as the claim that a god Helios pulls the sun across the sky. Some related claims can be empirically tested: There is evidence of absence for the power of faith healing (which the American Cancer Society also calls potentially dangerous if it replaces proper medical care).
- Argument from ignorance
- Argument from silence
- Contraposition (traditional logic)
- Transposition (logic)
- Turvey, B.E. (2008). Criminal Profiling: An Introduction to Behavioral Evidence Analysis. Elsevier. p. 267. ISBN 9780123741004. LCCN 2008274380.
- Martin, M. (2007). The Cambridge Companion to Atheism. Cambridge Companions to Philosophy. Cambridge University Press. p. 70. ISBN 9780521842709. LCCN 2006005949. "[Advocates] of the presumption of atheism... insist that it is precisely the absence of evidence for theism that justifies their claim that God des not exist. The problem with such a position is captured neatly by the aphorism, beloved of forensic scientists, that "absence of evidence is not evidence of absence." The absence of evidence is evidence of absence only in case in which, were the postulated entity to exist, we should expect to have more evidence of its existence than we do."
- Walton, Douglas (1992). Nonfallacious arguments from ignorance. American Philosophical Quarterly. pp. 381–387.
- Sagan, Carl (1997). The Demon-Haunted World: Science as a Candle in the Dark (1st ed.). New York: Ballantine. p. 213. ISBN 0-345-40946-9. OCLC 32855551. "Appeal to ignorance—the claim that whatever has not been proved false must be true, and vice versa (e.g., There is no compelling evidence that UFOs are not visiting the Earth; therefore UFOs exist—and there is intelligent life elsewhere in the Universe. Or: There may be seventy kazillion other worlds, but not one is known to have the moral advancement of the Earth, so we're still central to the Universe.) This impatience with ambiguity can be criticized in the phrase: absence of evidence is not evidence of absence."
- Moreland, J.P.; Craig, W.L. (2003). Philosophical Foundations for a Christian Worldview. InterVarsity Press. pp. 155–156. ISBN 9780830826940. LCCN 2002154307.
- Sextus Empiricus. Outlines of Pyrrhonism trans. R.G. Bury (Loeb edn) (London: W. Heinemann, 1933), p. 283.
- Vickers, John (2011). "The Problem of Induction". In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Fall 2011 ed.).
- "James Randi giving one of many talks where he uses the phrase". Youtube.com. 2009-11-05. Retrieved 2012-02-20.
- Hales, Steven D. (2005). "Thinking Tools: You can Prove a Negative". Think 4 (4): 109–112. doi:10.1017/S1477175600001287.
- "Faith Healing." Making Treatment Decisions. American Cancer Society. June 15, 2009.http://www.cancer.org/docroot/ETO/content/ETO_5_3X_Faith_Healing.asp