A more accurate way to state this is “Absence of evidence is not always evidence of absence”. Or “Absence of evidence may or may not be evidence of absence depending on the circumstance”, though it doesn’t roll off the tongue as easily.
"When trying to reject the null hypothesis (sensu Fisher), absence of evidence (for its invalidity) is not evidence of absence (of its invalidity)."
Fisher wasn't the only statistician, he wasn't the only confused one either. We can go with Neyman and Pearson instead:
"When trying to reject the null hypothesis with a sufficiently powerful experiment, absence of evidence is evidence of absence."
This requires a power analysis, which is something most studies lack. Constantly reminding everyone of these details and misguided philosophical differences is tiresome, so I prefer to go with Laplace, Bayes, and Jaynes:
A more accurate way to state this is “Absence of evidence is not always evidence of absence”. Or “Absence of evidence may or may not be evidence of absence depending on the circumstance”, though it doesn’t roll off the tongue as easily.