Can a negative ever be proven?

Someone has recently told me several times that a negative cannot be proven, ever, and that it is ‘a logical theorem’ whatever that means. Is this actually true? To me it seems rather easy to ‘prove’ to any reasonable person that the negative statement “That duck does not have two heads” is true. You just count ‘em. Similarly the negative statement “The moon does not contain any material that isn’t green cheese” seems rather easy to disprove. What is he going on about? Is there any such restriction on logical argument, since I know there isn’t in scientific argument where disproving a patently false theorem is the whole basis for logical thought.

Generally, when I’ve heard the comment, “You can’t prove a negative”, in context it was taken to mean, “You can’t prove that x does not exist.” In other words, you can’t prove God doesn’t exist, or the Loch Ness Monster, or Bigfoot, for examples.

This is true to a certain extent, but it’s not really because of a restriction to logical arguments.

For example, mathematically speaking, you can prove the non-existence of things. You can prove there is no number that is both even and odd, for example, or that there are no nontrivial integral solutions to the equation x[sup]n[/sup] + y[sup]n[/sup] = z[sup]n[/sup] when n is greater than 2.

However, as I said, in practice, this is true to a certain extent. How can you prove Bigfoot does not exist? (And, of course, I don’t mean provide evidence that gives a 99.9% probability of the nonexistence of Bigfoot; I mean actually prove that it doesn’t exist, so that even the most credulous of us would be convinced by the argument). I really don’t think it can be done, logically. To do so would be tantamount to requiring all spots on (and inside) earth to be monitored at the same moment, confirming whether he is here, there, or nowhere.

And even then, some ass would claim he just took off in his UFO.

In formal research-methodology research, you make a falsifiable hypothesis from your theory. Ideally it should be such that if your theory holds water, any success at falsifying your hypothesis trashes it thoroughly.

Example Theory: All forms of chlorine are deadly even in small quantities.

Example Hypothesis: A concentration of 1000 parts per million of every known chlorine combination will be inhaled, ingested, and injected into the bloodstream of volunteers from the 101 freshman class, and none of them will survive to pass the course.

Falsifying Finding: You get a survivor. If this happens, your theory hurls big-time. (Incidentally, the chick who got the sodium chloride looks OK…)

Because the hypothesis must be falsifiable, you get into trouble if you start off with a negative construct as your theory, e.g., “NOT all forms of chlorine are deadly even in small quantities”.

What is your falsifiable hypothesis? What possible falsifying finding could you come up with that could shoot this theory out of the water? Your findings won’t be considered to be worth a damn (in theory at least) if it were not possible to come up with a finding that ruins your theory. Note also that you can never prove your theory, you can only support it by trying rigorously to ruin it, yet failing.

You publish your attempts to shoot yourself down.


In the example given by AHunter3above would it be equally true to re-state the original theory as “no form of chlorine will fail to be deadly, even in small doses”? This should then make a negative statement that can be disproven by the same finding.

Read the article I am not a giraffe and I can prove it at the link below:

The giraffe link, for whatever reason, was blank (it flashed Acrobat and quit). Anyway, I guess Gaspode is right that his duck statement is correct. I do not think that it needs any “prove”. As long as he is talking of the given duck. But in generalized form: “(All) ducks do not have two heads” it cannot be proven by the laws (theorems, principles) of formal logic.
My question (if Gaspode and mods permit): in scientific community, would my argument be accepted: that two headed ducks do not exsist becase two-headedness was never observed, is contradictory to common knowledge, etc. Or will I be made to make the same argument again and again about 3 heads, 4 heads, n heads? And it is easier to prove that the number of duck heads is limited to one?

You can prove a negative by asking yourself “if this theory were true what would be some of the inevitable consequences of it?” If the consequences do not exist, then the theory must be false.

The link worked fine for me. It is to the last double issue of Swift. Discusses concept of “proving a negative” and how misuse of the statement can destroy skeptical arguments.
Essentially what is being described here.

The link works OK for me, but it’s about mystics and Christmas mostly. Kyberneticist what page is the article supposed to be on?
And I’d rather you didn’t hijack this thread just yet Peice. You’ve already had adequate opportunity to answer this question here.

Gaspode: The article is on pages 14 and 15 of Kyberneticist’s link.

Thank you Cabbage and thank you especially Kyberneticist
This will do me nicely.
And now to summarize that one can indeed prove a negative: I can prove that the world is not flat, that there cannot be an undiscovered continent on Earth larger than North America, that there is not an elephant in my living room, that I am not a woman, that I am not a gi-raffe, and that two parts of hydro-gen plus one part of oxygen do not produce sulfuric acid.
Now dor the next part: Does anyone here actually support the statement “A negative can never be proven”?

This .PDF file opened today. I agree with the author; his aguments make sense to me.

Sure, definitely, no problem.

I remember many times when I was a kid and had my own photo lab and darkroom in the basement. There were many times that I would take a negative and make a proof of it.

As alluded to by Little Nemo, you can prove a negative through reductio ad absurdum (apologies if I screwed up the Latin). That is, assume the positive. If that assumption leads to an impossible situation, than the negative must be true.

IIRC, this leads us into a particular fight within the philosophy of mathematics. Most mathematicians have no problems with reductio arguments, and big parts of mathematics are built on them. The alternative approach, constructivist mathematics, rejects reductio arguments. You can do lots (but not nearly all) math without reductio.

In my new profession, history, the phrase “you can’t prove a negative” generally means that you cannot prove something did NOT happen. It’s impossible to PROVE that Monica Lewinski was NOT a secret agent of the Republican National Committee–after all, maybe the coverup was SO good that all the evidence was shredded.

That’s why things like blaming Roosevelt for Pearl Harbor are so difficult to deal with–it simply can never be proven that he did NOT know.




Sure, okay, I’ll bite. Depends a bit on what you mean by the word “prove.” Do you mean “empirically prove?” If that’s the case, then I submit to you that you can’t even prove a positive.

So we’ll start with something simple: please prove that you exist.

Extra credit if you can do this in 25 words or less.


Boo, Keeve. Boo.

To me it seems rather easy to ‘prove’ to any reasonable person that the negative statement “That duck does not have two heads” is true. **
If you can produce the ‘reasonable man’ that hangs around all those courtrooms I reckon I could prove I exist.
I actually agree with you totally. It is almost impossible to prove some positives. My question was whether it was always impossible to prove a negative, which was what was put to me. It appears the answer is ‘No’. it is sometimes very easy to prove a negative.

OK. I’ll jump in with my WAG.

It seems to me that “you can’t prove a negative” is more scientific rule-of-thumb than scientific law, sort of like Occam’s Razor. It also appears that it applies to open systems, or ones where all data are unavailable, and/or impossible to gather. I would humbly posit that the relevant negative is “No chickens have two heads” rather than “That chicken has two heads”.

In my field (planetary astronomy), there are lots of questions for which this seems to work-- among the “unprovable negatives” are

  1. There are no Earth-like planets outside our solar system
  2. There have never been any comets which have come from outside our solar system
  3. There has never been life on Mars
  4. There is no intelligent life outside the Earth

Each of these statements can easily be proven wrong by a single couterexample-- look for #1 to be proven wrong sometime in the next decade or two. Scientific inquiries to prove one of these statements to be correct will find it difficult since there could always be an undiscovered counterexample. Pick up a single fossil on Mars, on the other hand, and you’ve disproven #3 in a moment.

Abstract this argument (or disregard it) as necessary. :slight_smile:


In my line of work (environmental engineering), I’m often dealing with attempts to prove a negative (such as “prove that you cleaned 100% of the contamination”…i.e., prove that you did not miss a small bit).

It often comes down to a weight-of-evidence proof (100% proof is impossible).

Your examples are good/simple ones, but here’s an example of the mindset you may find…

Can you prove that this is ALWAYS true? (Maybe you blinked and missed it.)
Maybe the second head is just really small and you can’t find it.

Are you sure NASA’s moon rock samples were accurately analyzed?
Are you sure NASA landed on the Earth’s moon and not some asteroid?

Of course, I’m taking this to the extreme, but real-life examples of being asked to prove a negative are in a much grayer area than your examples.

‘Proving’ something means different things according to context.

In the domain of empirical science, the maxim that ‘one cannot prove a negative’ is generally put forward when students are learning about falsifiability and experimental observation. If I say ‘Black dogs exist’ I only need to find one black dog to prove my theory empirically. If I say ‘No purple dogs with seven legs exist’ then it may seem, to some beginners, that by observing lots and lots of dogs, none of which fit this description, I am going some way to ‘proving’ my statement. This is, of course, flawed reasoning. Even if I survey a million different dogs, none of which fit the description, I have not proved anything at all about whether the purple seven-legged ones exist or not. While this may seem a rather elementary point to the wise folk of the SDMB, it IS a point worth making to beginners in the subject.

In other domains, such as pure logic and mathematical argument, it is perfectly possible to prove a statement which is framed so as to include a negative.

When sceptics and rationalists debate with believers in assorted pseudo-sciences, the ‘cannot prove a negative’ mantra tends to be hauled into the debate, sometimes by people who know what they are talking about and sometimes not. In these cases, it tends to refer to the empirical science principle mentioned above. For example, sceptics might challenge believers in astrology to ‘prove’ that it really works. Adherents to the astrological cause might (if they are rather ignorant) retort by saying ‘prove it does NOT work’, at which point the sage old sceptics will say ‘Ah, but it is impossible to prove a negative’. What they mean is that even if one can produce a zillion instances of astrology failing to work, this does not, in and of itself, ‘prove’ whether or not astrology works.

And finally… about a century ago here in England the word ‘prove’ simply meant ‘to put to the test, to examine’. A ‘proving house’ would perform such tasks as testing whether anchor chain was as strong as its manufcaturer claimed. Hence ‘the exception that proves the rule’ means ‘since we have found an exception to the rule, it TESTS the rule and makes us think again about how good that rule is’.