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Negative evidence denotes a failed necessary prediction of a theory, which thus counts as evidence against said theory.
Note that this negative evidence does not support any rival theories, per se — it just means that the theory being tested failed to line up with reality.
Negative evidence is produced when any given theory by necessity predicts something specific which, upon closer inspection, fails to materialize.
Absence of something predicted by the theory in question (the "negative" evidence) thus suggests that the theory is mistaken.
Negative evidence should not be confused with the discovery of novel evidence not predicted (or even considered) by an existing theory.
- P1: If X, then always Y.
- P2: Not Y.
- C1: Not X.
 In science
In order for a prospective theory to be falsifiable, and thus qualify as sound science, hypotheses must propose testable predictions in the observable world. Then, you do the experiment and see if the prediction comes true.
Essentially, negative evidence is considered found when no positive evidence could be located when and where the theory being tested predicts that such positive evidence would need to be.
For example, evolutionary theory predicts that a dynamic environment causes new species to evolve via a natural process of selection and adaptation, branching them out from their respective ancestor species into various new forms of life.
If the ground had contained billion-year old fossils of anatomically modern humans right next to fossils of triceratopses, trilobites and golden retrievers of the same age and strata, Darwinism would be in trouble.
Thankfully, this turned out not to be the case — every adaption appears at the right time and place in the geological timeline — and so, another test was passed by evolutionary theory.
An absence of something can be proved in various ways, e.g., by a reductio ad absurdum or by proving something else that is inconsistent with the presence of that something (a very useful approach known in mathematics as proof by contradiction). For example, in law, a party may have the burden of proving nonreceipt of certain correspondence and may bear that burden of proof (at least by a preponderance of the evidence) by introducing into evidence a docket record in which the correspondence would have been noted. In mathematics, there are plenty of proofs of negative propositions, such as "there is no largest prime number" or "there is no rational square root of 2".
Proving a negative is often harder than proving a positive.
If your neighbor claimed that teenagers were racing their cars outside of his house, the absence of teenagers, cars, and tire marks on the street would be negative evidence against his claim.
The problem of evil convinces Richard Carrier that omniscient, omnipotent, and omnibenevolent (OOO) god is extremely unlikely. This is because, if an OOO god existed, then evil should not exist. Evil does exist. Ergo (thinks Carrier) an OOO god likely does not.
Taken too far, negative evidence becomes an appeal to ignorance, the mistake of believing that because we have not found evidence, it must not exist.
Negative evidence occurs when an absence of evidence is actively found when actively looking for something. (Don't see your car in the garage? It's probably not in the garage.) An appeal to ignorance occurs when an absence of evidence is found when nobody has done any looking! (Maybe you should actually walk to the garage and check, before assuming your car isn't there.)
 See also