The phrase "absence of evidence is not evidence of absence" has always bugged me. We throw it around all the time but I'm not sure we're using it correctly.
If there is no evidence for something where there should be something if the something existed, then that is evidence that the thing does not exist.
In a science context wiki gives the example of drug research
"Evidence of absence and absence of evidence are similar but distinct concepts. This distinction is captured in the aphorism "Absence of evidence is not evidence of absence." [...] Despite what the expression may seem to imply, a lack of evidence can be informative. For example, when testing a new drug, if no harmful effects are observed then this suggests that the drug is safe.[3] This is because, if the drug were harmful, evidence of that fact can be expected to turn up during testing. The expectation of evidence makes its absence significant.[4]"
When it does apply is when we've not actually looked for the evidence or not looked properly, then it's simply an argument from ignorance.
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.[6]
Personally, I find the lack of evidence for an interventionist god where we should find such evidence - such as payer works, miracles happen, god changes the bread and wine into his body and blood at the eucharist, god loves me etc - actually
is evidence founded on absence.
Evidence of absence - Wikipedia.
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"Science adjusts it's views based on what's observed.
Faith is the denial of observation so that Belief can be preserved."
- Tim Minchin, in his beat poem, Storm.