Hardly a mathematical statement but it can be a useful shortcut to explaining why an argument isn’t well-structured…
The claim “you can’t prove a negative” is often used as a shorthand in discussions to refer to the difficulty of gathering experimental evidence to “prove” that something does not exist. Proving that a phenomenon isn’t real takes a lot more time and effort than it takes to demonstrate it. This is especially true when the definition of the phenomenon can be changed at will by its believers.
In criminal law, the maxim of “You can’t prove a negative” is reflected in the “presumption of innocence”. That is, arguments of the form
- You can’t prove that the defendant didn’t commit the crime.
Since it is perfectly possible to not-commit a crime and at the same time, have no evidence of that non-commission, the lack of proof does not imply anything.
Often, of course, one can prove that a defendant didn’t commit a crime, which again demonstrates that “You can’t prove a negative” is of limited value.
It has been claimed that the United Nations demanded that Saddam Hussein prove the non-existence of weapons of mass-destruction under his control. This may be historically inaccurate, but had such a demand been made, it would have required evidence that every square inch of Iraq did not contain WMD.