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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

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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.

Answer 6

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.