A Mythical Axiom

When someone gets charged with committing a crime, the police prosecutor believes there is sufficient evidence to arrest the suspect. The law tells us what we must not do. We are permitted to do anything, so long as it is not prohibited by the law.

The allegation is that we did something which is prohibited by law. The suspect and his lawyer must now try to prove a negative, i.e. the suspect did not commit the said offence.Those who believe that it is impossible to prove a negative are saying there's no point in denying it because, "You can't prove a negative".

So there's no point in having courts of law, judges, lawyers, jurors etc. because the suspect can't prove a negative? He can't prove that he did not commit the crime? Of course, those who assert that you can't prove a negative don't apply their generalization (their axiom) in these cases. They will argue that every suspect has the right to attempt to prove that they did not commit the crime, yet still they assert that one can't prove a negative.

They are selective as to the application of the axiom. They get around this contradiction by re-wording the case for the defence as, "The defendant may attempt to prove his innocence". That doesn't sound like a negative. So everything is ok, right? So, the axiom is not challenged.

The problem with this word manipulation is that innocent means not guilty.
"The defendant is guilty" means the same as "The defendant is not innocent".

Fortunately in law, the onus of proof remains on the prosecution at all times.

In science however, researchers write papers describing how they arrived at their conclusions. Let's imagine that a researcher has concluded that cyclones are caused by the level of carbon dioxide in the atmosphere. Other researchers are sceptical of this conclusion and must set out to prove that the level of carbon dioxide does not cause cyclones. Oh no! You cannot prove a negative. Is there any point in trying?

Fortunately in science, researchers must provide the evidence that supports their conclusions, and consider any evidence which may contradict their conclusions.

Whether a statement contains the words no or not does not mean that the statement cannot be proven or disproven. Make any statement you like, you must then be able to give evidence to support that statement.
Quod gratis asseritur, gratis negatur – What is asserted without reason (or evidence) may be denied without reason (or evidence).

Here's a statement that some consider to be a fundamental, self-evident truth (an axiom):
"You cannot prove a negative". The statement itself is a negative.
Can you prove that you cannot prove a negative?
If you can, then you have just proven a negative and contradicted your statement.
If you cannot, then the statement is not an axiom.


The statement, "The absence of evidence is not evidence of absence", is a negative.
If "You can't prove a negative" is true, then "The absence of evidence is not evidence of absence" cannot be proven to be true. Both cannot be true.

If "The absence of evidence is not evidence of absence" can be proven to be true, then "You can't prove a negative" is false.  ...{1}

1) If evidence has not been sought or is impossible to obtain then the absence of evidence is not evidence of absence. For example, I am unable to obtain evidence that life exists on Neptune's moon, Triton. This absence of evidence is not evidence of the absence of life on Triton. As such, I cannot say that there is no life on Triton. In this case, "The absence of evidence is not evidence of absence", is a true statement. Therefore, from {1} above, "You can't prove a negative" is false.

2) If evidence has been sought then the absence of evidence is evidence of absence.
For example, the Michelson-Morley experiment produced no evidence to support the idea that light requires aether to travel through, i.e., there is no evidence of aether.
"In addition, recent resonator experiments have confirmed the absence of any aether wind at the 10−17 level". That sounds like proof of a negative to me, i.e., there is no aether wind. http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment

When a doctor conducts exhaustive tests for malignant cancer cells in a patient but finds the results are negative then the doctor is correct to say that there is no evidence of malignant cancer in the patient. Of course the doctor cannot conclude that the patient does not have cancer, merely that there is no evidence of cancer. So in these cases, the absence of evidence is evidence of absence.

Strangely, "The absence of evidence is not evidence of absence" is true, and "The absence of evidence is evidence of absence" is also true.
The obvious contradiction is caused by equivocation (ambiguity) over the meaning of "the absence of evidence". Both sentences are a clever play on words, however they lead to misinterpretation and misunderstanding.

I should point out that proof, is not the same as developing a mathematical proof which is an absolute truth. The meaning of proof as used in this context is the same as "Evidence sufficing or helping to establish a fact".

What one can or can't prove does not depend on whether the statement is worded as a negative. If it is true that one cannot prove a negative, then — Any statement that can be worded as a negative cannot be proven, including this statement; Any statement that can be worded as a negative cannot be proven.
Here are four more negative statements which are provable:

4 ≠ 5
Sir Isaac Newton did not watch television.
Kangaroos are not native to Japan.
I am not the smartest person in the world.

Is it even necessary to prove these statements? They are self-evident truths.


Ducks are animals.
Ducks only have two legs.
Proposition: If ducks only have two legs then ducks don't have four legs.

The negative consequent (ducks don't have four legs) is just as provable as the positive antecedent (ducks only have two legs).

Conclusion: Any animal that has four legs is not a duck.
By starting with a positive assertion (ducks only have two legs), I derived a negative consequent (ducks don't have four legs) and from those I drew a negative conclusion (Any animal that has four legs is not a duck).

2 + 2 = 4       (If n is any number other than 4, then 2 + 2 ≠ n)
5 ≠ 4
Therefore 2 + 2 ≠ 5

The good thing about proving a positive is that it disproves a whole lot of negatives. For example, by proving 2 + 2 = 4, we have proven that 2 + 2 ≠ 5 and 2 + 2 ≠ 6, etc. Proving a negative can disprove a whole bunch of positives, e.g. any animal that has four legs is not a duck.

For any generalization, or axiom, only one exception is required to disprove it.

How many times have I proven a negative?