Proving a Negative

1. Double negatives
    Just as with someone who says, "I haven't done no crime!". Do they mean:

    a.  There is no crime that they haven't committed?
    b.  That they have done no crime? I.e. They have committed no crime.
    c.  That they haven't done a crime? I.e. They haven't committed a crime.
    d.  That they have done crime? I.e. What they have done, was a crime.

    So too with "You can't prove a negative". Do they mean (for example):

    a.  You can't prove that pigs can't fly. (Or even; You can't prove that flying pigs don't exist.)
    b.  You can prove that pigs can't fly.
    c.  You can't prove that pigs can fly.
    d.  You can prove that pigs can fly.

    In the first case (the suspected criminal), it's generally interpretations b or c, which the claimant means with their double negative.

    Claims b and c (in each example), mean the same thing, as do a and d.

    When people use a double negative, It is important to get them to clarify which of the three interpretations (b, c or d) do they mean. Double negatives are ambiguous.

2. Unfalsifiable
    If something is unprovable, then it is untestable and unscientific.

    When someone says, "You cannot prove ..." then they are warning me that what follows is probably unfalsifiable and unscientific.

    E.g. You can't prove that God exists. Or, you can't prove that God doesn't exist.

    Questions of religion or the supernatural are not scientific. Science is about the natural, the provable, the repeatable, the testable, the evidence.

3. Examples of provable negatives
    Here are five negative claims which are provable.
    I.e. You can prove that...:

        2 + 2 ≠ 5
        Sir Isaac Newton did not watch television.
        Kangaroos do not exist on the moon.
        I am not the smartest person in the world. (That should be self-evident)
        Automobiles did not exist in the 16th century.

    These are just a few exceptions to the "rule" that you can't prove a negative.

"The exception proves that the rule is wrong. That is the principle of science. If there is an exception to any rule, and if it can be proved by observation, that rule is wrong." – Richard Feynman, according to The Meaning of it All, 1999 ­­­

4. Can you prove that?
    The claim, "You can't prove a negative" is itself, a negative.

    Can you prove that you can't prove a negative?

    If you can, then you've contradicted your claim and the claim is false.
    If you can't, then the claim is meaningless and unfalsifiable.

    A negative claim which is not provable; "You cannot prove a negative".


5. "Universal, existential negative"
 If it is impossible to prove an existential universal negative then it is equally impossible to prove an existential universal affirmative. I.e. If it is impossible to prove that something does not exist somewhere in the universe, then it is equally impossible to prove that the same something does exist somewhere in the universe.

    For every negative claim which is impossible to prove then the opposite claim is also impossible to prove.
    If it is impossible to prove that pigs can't fly then it is impossible to prove that pigs can fly.
    If it is impossible to prove that God doesn't exist then it is impossible to prove that God does exist.

To highlight the lengths that some people will go to hold onto their "you can't prove a negative" myth, I came across this: (my bolding)

"You can't prove a negative" This Error is propagating across the internet, because of this article. People are beginning to say "ha you can prove SOME negatives so this rule doesn't hold". Actually "You can't prove a negative" is actually common short hand for “you can’t prove a universal negative,” or better "existential negative". In other words, you cannot prove that some hypothetical does not exist, anywhere in the universe, because that would require that you be able to look everywhere at the same moment. And, of course, if the hypothetical something, in question is claimed to be invisible and undetectable by any means, in principle, it gets even sillier to attempt to disprove that hypothetical's existence. e.g. God or telepathy etc.Stewgreen2 (talk) 10:11, 7 May 2013 (UTC)

http://en.wikipedia.org/wiki/Talk:Evidence_of_absence

If Stewgreen2 thinks this proves that one can't prove a universal,existential negative then he should consider the following:
The obvious response to this is; you cannot prove that some hypothetical does  not  exist, anywhere in the universe, because that would require that you be able to look everywhere at the same moment. And, of course, if the hypothetical something, in question is claimed to be invisible and undetectable by any means, in principle, it gets even sillier to attempt to dis prove that hypothetical's existence. e.g. God or telepathy etc.


6. A real life proof

The Michaelson/Morley experiment showed how a claim that luminiferous aether, “an invisible and infinite material with no interaction with physical objects”, was proven to not exist. The generalisations that one can't prove a negative, or that one can't prove universal or absolute non-existence [of something], have been disproven. They are false statements. 

I need only one example to disprove the generalisations, just as a single black swan refutes the assumption that all swans are white.

 

James Randi Lecture @ CalTech in 1992.

http://www.youtube.com/watch?v=qWJTUAezxAI

James Randi describes, in a humourous way, a thought experiment, intended to demonstrate that one cannot prove a negative.  He imagines taking 1000 reindeer to the top of the World Trade Centre to push them, one at a time, off the roof. "You're going to test whether or not reindeer can fly."
As expected, they all fall to their deaths. His comedic way is quite entertaining, but tapers off as he gets closer to making his point.

"Now, what have we proven with this experiment?
Have we proven that reindeer cannot fly? No! Of course not.
Think about it now. We've only shown that; on this occasion, under these conditions of atmospheric pressure, temperature, radiation, all these sort of things, at this position geographically, on this season etc. that these 1000 reindeer either could not fly or, chose not to..."

These 1000 reindeer did not fly, however it may have been the "atmospheric pressure" etc, that prevented them from flying? By that same reasoning, you can't prove that dead sparrows or rocks, can't fly because, it may be the atmospheric pressure etc, preventing them from flying.

"Now, if the second, then we know something certainly about the IQ of the average reindeer...
However, we have not proven a negative. We cannot prove a negative, technically, rationally and philosophically speaking.
However, people will often look at this example and say, 'Well, how many reindeer would you have to test?' Now I'm not going to get into the statistics of the thing... I'm not going to get into the arguments on the matter, I will only tell you that you cannot prove a negative and I think that's a rather good example that you might wish to use in illustrating your point.
The other folks who claim that it is so are required to prove it. And, if it's so, it's very easy. Just show me one flying reindeer!
Of course then they come up with the rationalizations and they say, 'Oh no it's only the 8 tiny reindeer who live at the north pole who can and will, on the evening of December the 24th, fly to do that specific job'. In that case you have to throw up your hands and say, 'Well, I don't think that your hypothesis is very testable."

Let's start with his first question; "Now, what have we proven with this experiment?".
Have we proven that reindeer cannot fly? Mr Randi's answer : No!
Have we proven that reindeer can fly? Answer : No!
So what have we proven with this experiment?  Answer : If you throw reindeer off a tall building, they will probably die.
So the rest of his performance is simply to reiterate his assertion that you cannot prove a negative.

What was the point of his thought experiment?
"We cannot prove a negative, technically, rationally and philosophically speaking". Where is the evidence?

Note the association; those who would argue against his "You can't prove a negative" assertion, would also argue that reindeer can fly. And of course, people who would argue that reindeer can fly are stupid, just like those who would argue against his statement.
Like all good comedians, the punchline is stated as a quick one-liner: "The other folks who claim that it is so, are required to prove it. And, if it's so, it's very easy. Just show me one flying reindeer!".

This is a common misdirection technique used by con-men and magicians.
He plays his audience as any professional magician or comedian would. He distracts his audience with the preposterous concept of throwing 1000 reindeer off the (then) world's tallest buildings. While the audience is still visualizing such a ghastly, ridiculous scenario, he quickly slips in his message that he wants the audience to believe.
Now comes the switch. Let everyone know that only those people who believe that reindeer can fly, would also believe that it is possible to prove a negative. So, after watching his show, if you believe that he has proved that "You cannot prove a negative", then you have been conned.

Personally, I'm not sure whether he was there to give a lecture or to be the comic relief. Unfortunately, so many people believe his joke.

Here's the statement that Mr Randi considers 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.

From Wikipedia - Evidence of absence. As of 23 Sept 2013 :

"In 1992 during a presentation at Caltech, skeptic James Randi uses the phrase "you can't prove a negative". He claims that he cannot prove a negative (such that telepathy does not exist), but he also argues that an individual who claims telepathy exists must prove so. He discusses that induction is often used as a mode of proving a thesis, but if an individual assumes that something is or is not, then the person must prove so. Further, as he says, he does not take an advocacy position, as a lawyer would. He says that he cannot prove that a negative is true, but he could attempt to use evidence and induction to support a claim that he is biased toward, such as a claim that something does not exist (ex. flying reindeer)."

http://en.wikipedia.org/wiki/Evidence_of_absence

Of course Mr Randi should use evidence and induction to support his claim, that reindeer cannot fly, but what he hasn't proven, nor given any evidence to support, is his assertion that one cannot prove a negative.
All in all, Mr Randi has just dismissed his 1000 dead reindeer evidence as inconclusive, simply because he wants to believe that you can't prove a negative.

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?