Evidence of absence

Not to be confused with Absence of evidence.

Evidence of absence is evidence of any kind that suggests something is missing or that it does not exist.

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. However, when interpreted as a claim about conditional probabilities, absence of evidence generally is evidence of absence.

Overview

The difference between evidence that something is absent (e.g., an observation that suggests there were no dragons here today) and 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 regards whether the experiment would have detected the phenomenon of interest if it were there.

The argument from ignorance for "absence of evidence" is not necessarily fallacious, for example, that a potentially life-saving new drug poses no long-term health risk unless proved 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" with cosmologist Martin Rees' maxim, "Absence of evidence is not evidence of absence".

An exhaustive inspection of the attic for vermin can provide evidence of absence, but any sign of mice will always suffice to the contrary.

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, the confidence of the inference, as well as confirmation bias within the community.

Proof and evidence

Main article: problem of induction

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, …, a_n 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, …, a_n are all Fs that are also G.

2. a_n+1 is also F... [therefore]

3. a_n+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

Conditional probability

Suppose a hypothesis $H$ and evidence $E$ are events in a probability space. If $\Pr(H\mid E)>\Pr(H)$ (meaning that observing $E$ increases our confidence in $H$), then the absence of $E$ should reduce our confidence in $H$, as shown by the following proof:

${\begin{aligned}\Pr(H)&=\Pr(H\mid E)\Pr(E)+\Pr(H\mid \neg E)\Pr(\neg E)\\&>\Pr(H)\Pr(E)+\Pr(H\mid \neg E)\Pr(\neg E)\\\Pr(H){\bigl (}1-\Pr(E){\bigr )}&>\Pr(H\mid \neg E)\Pr(\neg E)\\\Pr(H)\Pr(\neg E)&>\Pr(H\mid \neg E)\Pr(\neg E)\\\Pr(H)&>\Pr(H\mid \neg E)\end{aligned}}$

In order for this proof to work, the probability of $E$ must be strictly between 0 and 1.

Probabilistic reasoning of this sort is particularly important in legal disputes. For example, a jury can be swayed by "evidentiary lacunae", or a lack of evidence they expect to hear.

Proving a negative

A negative claim is a colloquialism for an affirmative claim that asserts the non-existence or exclusion of something. Claiming that it is impossible to prove a negative is a pseudologic, because there are many proofs that substantiate negative claims in mathematics, science, and economics, including Euclid's theorem, which proves that there is no largest prime number, and Arrow's impossibility theorem. There can be multiple claims within a debate, nevertheless, whoever makes a claim carries the burden of proof regardless of positive or negative content in the claim.

A negative claim may or may not exist as a counterpoint to a previous claim. A proof of impossibility or an evidence of absence argument are typical methods to fulfill the burden of proof for a negative claim.

Philosopher Steven Hales argues 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.

See also

• Argument from ignorance
• Argument from silence
• Contraposition
• Contraposition (traditional logic)
• Pitfalls
• Probatio diabolica
• Proof by exhaustion
• Transposition (logic)