Today I learned the adage "you can't prove a negative" is false because you can prove a negative. It's very easy. We need to stop saying that. A better statement to use is "he who asserts a positive has the burden of proof."
Here are some of the reasons why this adage is false:

By stating that you cannot prove a negative, you are making a negative statement that, if true, contradicts your statement. In other words, it's selfrefuting.

If you cannot prove the statement, it's an arbitrary statement that has no basis and is not useful in any logical sense (i.e., it's nonsense).

There are many examples where this is not true. You can prove that 1 does not equal 0. You can prove that 2 is not greater than 3. You can prove that an equilateral triangle doesn't have any right angles. You can prove you don't have a million dollars in your pocket right now. You can prove that a flipped coin that lands on heads is not tails. You can prove that there isn't a rainbowcolored, firebreathing unicorn sitting on your shoulder. You can prove you are not reading this article while having sex on the moon. You can prove you are not dead.

To have identity is, by definition, to have a single/the same identity. Therefore, if you can't prove a negative, you also can't prove a positive. (Or, another way to say it: every claim of positive knowledge is a claim that disproves all alternatives.) This is also referred to as the Aristotle's Law of Identity (aka A is A). For example, to prove that 1 = 1 is to disprove that 1 = 0.

Related to the law of identity and another one of the three classic laws of thought, the law of noncontradiction states that "contradictory propositions are not true simultaneously." Reality conforms to the law of noncontradiction (i.e., you can't have a certain attribute and not have the same attribute at the same time). Yet the law of noncontradiction is a negative statement and relies on negation. (There are controversial theories in quantum mechanics such as the Copenhagen interpretation which imply reality violates the law of noncontradiction, but they currently remain inconclusive and people can't seem to even precisely agree on "any concise statement which defines the full Copenhagen interpretation.")

You can't reasonably test every proposition. For example, to test the proposition that there aren't any such things as rainbowcolored unicorns, you would have to check every part of the universe at the same time. The best we can do is infer using inductive reasoning to make generalizations from what we do and can know.

Science is all about proving negatives, which can be done in as few as one experiment.
People make negative statements when they are unable to provide evidence of their beliefs (e.g., "can you prove that God doesn't exist?"). So, instead of retorting that you can't prove a negative, focus on the lack of evidence to test the proposition. Focus on the reasonability of doubt. The inability to invalidate a proposition does not make it true. That's the key.
James Randi Lecture @ Caltech  Cant Prove a Negative
What did you learn today?
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Original posting by Braincrave Second Life staff on Sep 27, 2011 at http://www.braincrave.com/viewblog.php?id=652