What practical methods can be used to prove a negative claim?



I realize that the burden of proof falls squarely on the shoulders of the person making the claim, but they often seem unwilling to do the footwork.

It is difficult to prove a negative case except by pure logic, so once that avenue is exhausted would it be worthwhile to attempt to make the best positive case to show that it still fails to pass muster? This seems to be the way Mythbusters tackles many difficult claims, however 'true-believers' still seem unlikely to respect their methods.

Example (for the sake of argument): If another individual claims "Bigfoot is an extant creature living in North America," the natural way to argue against this is to counter with "Bigfoot is not an extant creature living in North America." This calls for one to prove a negative claim.

Are there any other alternative methods to prove a negative claim?


Posted 2015-07-23T14:52:54.973

Reputation: 213


If you opponent is unwilling to do the footwork, then its a complete non starter. You cannot prove that Russell's teapot is not orbiting the sun. Its up to the person making the positive claim to give proof.

– crthompson – 2015-07-23T17:55:03.737

@paqogomez I agree completely, but I feel an obligation to help rid people of their unreasonable assumptions about unprovable things. Not all unprovable things, of course, just the ones that I am continuously bashed over the head with.

As such, I am willing to do extra legwork to help disprove it because I feel like they deserve my help in any way that I can give it. – Gavin42 – 2015-07-23T18:09:10.590


Due to the source restrictions on Skeptics.SE, they sometimes have a problem with this for some claims. For example, see here - which had "original resource" (reasoning) answers which were deleted, or the linked meta question about it.

– DoubleDouble – 2015-07-23T19:04:16.320

I deleted several comments that were afield of the question in question. If anyone really wants one or more of them back, please open a question on meta. – virmaior – 2015-07-24T03:11:48.443

Proof lies in the assertion. If an opponent is convinced of the opposite, the best tactic is to ask them to explain their reasons for their assertion and poke some holes in their assertions. – Swami Vishwananda – 2015-07-24T04:46:11.530



As noted earlier, it is probably impossible to firmly logically prove there is no Bigfoot, in the same way as Russell's Teapot. However, and this come from the Mathematician in me, there are a few strategies you can try to either disprove or weaken a claim.

Proof by Contradiction

Basically, assume that the claim is true, and show that it being true causes some logical impossibility. This is more or less the only way to actually prove something false. Things to help support this kind of argument:

  • Include requirements. I.E If Bigfoot is 8 feet tall and a mammal, then his heart has to be some size, food intake this, skeletal structure that...
  • Include other, given truths. I.E If we assume Bigfoot exists and the Sun is hot, then ..{argument}.. which is a contradiction.
  • Take an extreme. I.E if Bigfoots exists, there must be a smallest Bigfoot..
  • Generalize. I.E If Bigfoot exists, then at least one undocumented mammal exists...

Proof by Contrapositive

A => B implies not B => not A.

This one is weird, and probably not applicably outside of rigidly defined areas like math. But essentially, you take the contrapositive and prove it must be true. In a single statement case this is redundant, and becomes proving "Bigfoot does not exist", which is where we started. But apply it to a specific implication used, and it can be useful. I.E "Most photographs are not fake => Bigfoot exists" would also imply "If BigFoot does not exist => Most photographs are fake" which is just a silly implication.

I know the example for this one is weak, I'm trying to come up with a better one, but these cases are usually relatively subtle and require a lot of context

Probabilistic Proof

Instead of absolutely proving a claim wrong, you can prove that a claim is not likely to be true, therefore the inverse is more likely to be true, therefore the claim is more likely to be false than not. Not an absolute proof, but certainly weakens a claim if it's most likely to be false. Tips:

  • Look at implications. Bigfoot existing need X acres of land never surveyed, there have been Y acres surveyed out of Z, so Bigfoot's X being within those Y acres is only ~Y/Z*X
  • Look for correlations. I.E Number of recorded sightings of Bigfoot decreased in proportion to availability of cameras, a regression or even common sense says that's very unlikely if Bigfoot exists.


Posted 2015-07-23T14:52:54.973

Reputation: 634

I personally think, your first case, proof by contradiction could fail since according to you it requires requirements, which instantly and simultaneously calls for a certain claim. In that case, you already presumed by that claim, prior to prove the negative claim. I thought. – None – 2015-07-25T21:17:34.583

One might think easily by presenting the contradiction, he or she could prove the non-existence, but to me actually not. Since the -- contradiction -- of 2 or more existing objects or -- claim -- whatever, just contradicts. I can not call it the proof of negative, although that tactic can come to one's mind very easily. – None – 2015-07-25T21:28:55.997

There is actually a firm logical base to Proof by Contradiction. As with any type of argument, it's only as strong as the premises it is based on. But if two premises assumed together cause a logical impossibility, then one must be wrong. And if you have one that you are certain is correct, then the other has been proven to be false. – Cain – 2015-07-25T22:44:16.143

You are talking as if one of the claim is 100% certain..., which is in my opinion complete "opposite" side of the "false or wrong.... – None – 2015-07-25T22:54:59.387

Do you take into consideration even the --positive-- existence could not be 100% sure to us? – None – 2015-07-25T22:58:15.993

It really just depends on the premise. If you can't take any premise as certain, you can't really prove anything. You have to start with some paradigm of given premises, whether those are axioms in math or just assumptions in other fields. If you're talking about % certainty in things again, that becomes the Probabilistic Proof, not one of the stronger ones. – Cain – 2015-07-26T01:26:26.707

Ugh, we are talking same thing....sorry I read ( really ) and forgot!. Upvote + deleting mine. – None – 2015-07-26T01:30:52.500

Actually, this is not a good saying, I don't think I am fond of Russel so much. – None – 2015-07-26T01:36:39.447


Assume that you and your opponent agree that the positive claim, e.g. "Bigfoot exists", is unprovable. And assume that you consider it a false claim, but you do not know how to disprove it.

First, I would ask my opponent: Tell me the arguments which support your claim. I would try to convince him of the weakness of his arguments.

Secondly, I would ask him: What argument would you consider a disproof of your claim? If he cannot imagine any argument for a possible disproof then we have an example of immunization.

If the opponent retorts in the end "Nevertheless, it could be possible!", I would reply: It is also possible that little green men exist - but we just did not detect them. At least, that's my pragmatic method.

Jo Wehler

Posted 2015-07-23T14:52:54.973

Reputation: 17 204

"Bigfoot exists", if it were true, would be very easily provable - just catch a Bigfoot. It's only hard to prove because it isn't true. – gnasher729 – 2015-07-23T16:48:34.707

I think the questioner is asking about the metaphysically aka from the religious side claim which is at the ultimate end negative. ( Whatever that word could mean. ) So I don't think it is not a good idea for you to say because it isn't true --> gnasher729. Thank you. – None – 2015-07-23T16:57:01.373

@KentaroTomono you've got a double negative in that last sentence, but I think your point is that gnasher shouldnt say its not true. I agree with gnasher but in reality, he should probably say he's agnostic about it. It cannot be proven, but cannot be disproved. In the same way Russel's teapot cannot be proven or disproved.

– crthompson – 2015-07-23T18:11:38.980

@paqogomez No, I didn't try to mean it. Once you say, -- because it isn't true, then instantly and simultaneously you gave some definition to it what kind of context the message is in. ( I am sorry, here it is deep night, so let me hit the sack. ) – None – 2015-07-23T18:30:53.117

@gnasher729 - The claim "Bigfoot exists" is provable. One must only capture a Bigfoot. It is wholly impractical to prove it isn't true, however it could be proven true at any moment. Much like, if God appeared to the world simultaneously and performed whatever 'magic' were requested of Him, some more-than-current number of people would likely believe in God afterward. Some would likely still reject the evidence because it doesn't suit their mental model of the universe, so there would have to be another explanation. People can just reject truth by choosing to. – MrWonderful – 2015-07-23T18:52:10.197

@paqogomez I learned a bit of Russel's teapot. But the answer to you, could be "then so what?" with sorry. The problem of could be or could not be, I think was annulled when once gnashar729 said if it were... thing. Or more correctly saying, I am not sure what Mr. gnashar729 is trying to mean by his first statement. I meant by the phrase at the ultimate end, somewhere at the endpoint which I don't know, as after that I wrote Whatever that could mean. I think further discussion might kill my nerve with smile and apology........ – None – 2015-07-23T21:49:25.830

@KentaroTomono perhaps tired has gotten to you a bit. I tried really hard to follow your point in that comment, but could not. Your previous comment however was clear. I understood you to mean that once a point is stated it gives direction to the argument. With that I agree, but its still the positive aspect of the point that needs to be proven. Not the negative. – crthompson – 2015-07-23T21:57:29.060

@paqogomez Yes, anything could take some kind of --direction-- thing, of course, I started feel like this is the matter of the definition of the words of positive and negative. I am afraid let me have personal fun or rest for a while.......thank you. – None – 2015-07-23T22:03:33.143

Please rest, I certainly dont require an immediate answer. I apologize if I miss your point. By positive, i mean tangible. Bigfoot is the claim. So if you say he exists then show proof of it. The negative is the opposite. I couldnt possibly test every square inch of the planet and say with certainty that there is no bigfoot on the earth. Its as if I told you that there were an english speaking colony of pink buffalo on saturn. Carl Sagan popularized the phrase "Extraordinary claims require extraordinary evidence." I better have some extraordinary evidence if i'm going to make that claim. – crthompson – 2015-07-23T22:14:35.587


Proving the negative is impossible in many paradigms, especially in science. In particular, any philosophy which derives from Empiricism runs into trouble proving a negative because one would have to experience everything in the universe to prove that something is not there. Consider that it was "impossible" to violate Bell's inequalities until Quantum Physics experimentally proved that you could.* If one wishes to prove the negative, it often calls for ontological truths which go beyond empiricism, such as those religions often contain. Just off hand, "there is no way to the father except through me" from Christianity comes to mind, but there's examples in every religion.

The best success I have had is taking a slightly different tack. Instead of arguing "Bigfoot does not exist," I argue that acting on the belief that Bigfoot exists is counterproductive. By adding a layer of action on top of the belief, there is room to make arguments that would be impossible if I simply targeted the belief itself.

Best of all, never once do I need to argue against the belief, which is often a deeply held belief. Instead, they are free to keep the belief as long as they please, but I seek to gain agreement on what behaviors should or should not come from it. It is remarkably hard to argue that I should care about anything beyond that. From there, I let them wrestle with the belief themselves.

  • Disclaimer: I know I've got the order of events backwards. However, the statement is still true. Bell's inequalities were valid under classical physics models, so could have arguably been penned before QM.

Cort Ammon

Posted 2015-07-23T14:52:54.973

Reputation: 16 681

3You don't prove anything in science, so the first sentence is misleading. Also, you can gather evidence for the absence of something by checking everywhere it could be. See the Periodic Table of Elements for an example. You don't have to examine every atom in the universe to check for Bigfoot, only all habitats for large land mammals in North America. – Rex Kerr – 2015-07-24T08:05:41.853

I apologize for having been a bit offensive. That said, still, I must say to Rex that you are misunderstanding regarding your first You don't prove anything in science, so the first sentence is misleading. Better be gone for a while.... – None – 2015-07-24T13:01:45.390

@RexKerr I agree, but the OP's question indicates that they are dealing with people who do not. It is not uncommon for people who believe in unpopular concepts like "big foot" to validate their actions with the doubt that comes from failing to execute a complete check of the space. I have seen lines along "Bigfoot doesn't exist because we've mapped North America with satellites and haven't seen him." / "But maybe he was just in his cave when the satellites were going overhead." – Cort Ammon – 2015-07-24T14:10:37.540

2You are certainly addressing the readers of SE.Philosophy, but not necessarily the people the OP is thinking about. No reason not to be accurate unless you're quoting yourself--but you don't (more or less) do that until the second paragraph. When someone asks "how do you prove a negative?" a misleading statement about the capabilities of scientific inquiry isn't a good way to answer. As always, save for things resolved via logic alone, science is about as good a way as any to obtain reliable knowledge of the presence or absence of things. – Rex Kerr – 2015-07-24T19:26:18.967

I would like to ask why Rex thinks taking the science is not appropriate. Then we have to escape to the inductive method, whose premise is obvious, the meaning of which could be whatever you can name. This is obvious, so... can you elaborate more so that we can clarify the difference between the ontological call or the belief from the inductive method? – None – 2015-07-25T12:02:16.590


I see two problems here:

The first one is that the difficulty of proving "Bigfoot is not an extant creature living in North America" has nothing to do with the logical form of the statement. Consider the statement "Elephants are not an extant creature living in my room", which has the same logical form, but can be proved without effort.

The difficulties of proving statements of the form ¬(]x)(Fx) stem from the fact that ¬(]x)(Fx) is equivalent to (Vx)¬(Fx) which is an universal statement. Universal statements, if true, are true of everything in the universe, and, ignoring logical truths, this implies that proving them true requires one to make sure that every single object in the universe satisfies the conditions.

Adding a second condition to the universal statement can make this task manageable. "Elephants are not an extant creature living in my room" is easily provable because it's trivially true of almost every object in the universe that they are not in my room, and I can easily check whether any object in my room is an elephant; however, if there's a lot of objects that satisfy the second condition, proving that they lack the first one can be unmanageable.

The second problem I see is that the notion of 'burden of proof' is not really related with the positivity or negativity of statements. For instance, it's generally accepted that that lack of vitamin C causes scurvy. You can deny this claim, but the burden of proof is on you. Also, answering 'who has the burden of proof' seems to be context dependent. It sometimes depend on shared assumptions about whats true, sometimes on power relations, sometimes on the consequences of accepting that a statement is true. Saying 'Bigfoot exists' at a coffee table is, after all, not the same as asking for a research grant to search for Bigfoot.


Posted 2015-07-23T14:52:54.973

Reputation: 161


In certain scientific fields, negative proofs are possible. For example, in Theoretical Computer Science there's something called "the Halting Problem," which describes a program that can't possibly exist due to a contradiction. From there, there's a set of proofs that one can do to show that other sort of programs are impossible by reducing it to the Halting Problem.

In Pure Mathematics, there are a couple techniques that come to mind. For example, the Cantor Diagonalization argument was used to show there is no function from the integers to the real numbers that is both one-to-one and onto. In dealing with finite sets, one can use the Pigeon Hole principle to do similar sorts of arguments.

In the real world, as Cain raised, one needs to take a probabilistic approach to many problems. One useful way of thinking about this is using a Bayesian framework: for the "bigfoot" example, even starting with a weak prior (Bigfoot is equally likely to exist as not), looking at the data makes it extremely unlikely that Bigfoot is a real thing.

The example of Russell's Teapot has been brought up: the thought experiment is of course a great example of how (1) the burden of proof of a claim ought to be on the one proving it and not the one making the negative claim and (2) given that one side has offered an argument, it is then begging the question to dismiss it because the claim is impossible.

James Kingsbery

Posted 2015-07-23T14:52:54.973

Reputation: 5 868


Negative claims are difficult; to give one example Von Neumann claimed to have proved that hidden-variable theories weren't possible in QM.

However de Broglie and then later Bohm were able to show that QM could be reinterpreted to do so; however at the cost of non-locality.

A justified negative claim offers a critique; a limit that may be in dialogue with another; and that might be possible to shunt around or move over.

An unjustified negative claim is probably best ignored.

Mozibur Ullah

Posted 2015-07-23T14:52:54.973

Reputation: 1