Hardly a mathematical statement but it can be a useful shortcut to explaining why an argument isn’t well-structured…

From ScepticWiki:

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 Law

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.**

are inadmissible.

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.

### Other cases

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.

http://www.skepticwiki.org/wiki/index.php/%22You_Can’t_Prove_a_Negative%22