Proving A Negative

[Regina Minx]

An often-heard line of argument goes that you can’t prove a negative. It’s a popular notion, but unfortunately it is quite false. Negatives can, in fact, be proven, and it’s quite easy to do so. First of all, the three fundamental laws of logic consists of one negative among their number, the law of non-contradiction. That law states that a statement cannot (note the negative) be both true and not true. Furthermore, this law is not just asserted, but is proveable. It can be formally proven using basic set theory and valid rules of inference, the full details of which are beyond this fluffy little piece, but can be found here, under ‘For Advanced Students’, for those truly interested.

One of the laws of logic is a provable negative…but that would mean that we have just proven that it is not the case that you can’t prove a negative. So we’ve proven two negatives, in just 173 words!

In fact, ‘you can’t prove a negative’ is itself a negative. So even if that statement wasn’t demonstrable untrue, even if that statement was 100% true, you couldn’t prove it! And if you could prove it was true, it wouldn’t be true! Egads!

Furthermore, the rule of double negation states that any claim can be expressed as a negative. In logic, it is equally valid to prove P true as it is to prove not-not-P true. So pick anything you can prove. You own existence, you say? Great! Give me that proof. OK, I accept it. Now, putting your existence through the filter of double negation, that means that if you can prove your own existence, you can prove that you aren’t non-existant. Congratulations, you’ve proven a negative. For any P that is proven true, it is simultaneously and necessarily proven that P is not false.

Some people use ‘you can’t prove a negative’ only in the sense of proving an existential negative claim, such as that Santa Clause or Bigfoot don’t exist. Of course, as it should be clear by now, I’m going to dispute this. Here is a formal syllogistic argument using modus tollens:

P1: If Bigfoot existed, there would be fossil evidence of Bigfoot remains.
P2: There is no fossil evidence of Bigfoot remains.
C: Therefore, Bigfoot don’t exist.

Now someone could note that I have not demonstrated that P1 or P2 is true. I have just asserted it. That is correct. However, it is a mistake to insist that someone prove to 100% deductive certainty all of the premises of any argument. The only way to prove P2, for instance, is to set up another argument that has other premises, and reaches P2 as a conclusion. Of course, then I would have to prove those OTHER premises, and so on ad infinitum. Which premises we should grant and which require demonstration is a subject for another discussion on epistemology, but one thing is certain. If proving things requires an infinite chain of proven premises, then nothing can be proven, either positive things or negative.

Sometimes people will criticize induction and state that a negative statement cannot be proven inductively beyond any shadow of a doubt. For example, suppose we scour the world for evidence of living or fossilized Bigfoot, found zero evidence, and conclude that there is no Bigfoot. This is basic induction.

Someone could nevertheless invent an ad hoc explanation for why we can’t find evidence of Bigfoot, such as that Bigfoot has magical powers of concealment, or more credibly that they have superior abilities at self-concealment than any other recorded species. And then they could insist that you can’t prove that wrong!
First of all, from a mathematician, attempting to rescue a theory with ad hoc assertions that cannot be proven independently only lowers the prior probability of the theory on a proper inductive analysis.. But from a broader picture response, inductive arguments can’t give us certainty about negative claims…but also they can’t give us 100% certainty about anything at all, even positive claims. Inductive arguments make conclusions probable, not certain.

Don’t be in such a hurry to chuck induction out the window. Why do you think the sun will rise tomorrow? Why do you think that turning the spigot in our kitchen will give you water? Why do you think that the chocolate bar you’re about to eat is sweet? Why do you think the coffee in your cup is energizing and not poisonous? The short answer is that you believe all of those things because the sun has risen every day (outside the arctic circles, naturally), because you’ve always gotten water from the tap, because you’ve always had sweet chocolate bars before, and because coffee has never poisoned you in the past. In other words, you’re using a model of induction described by the Rule of Succession.

So to sum up. You can prove negatives deductively, and to the extent that you can’t prove them inductively the same holds true for any claim, be it positive or negative. People tend to insist that negatives can’t be proven, however, not because this is a bedrock law of logic, but as a deflection in an attempt to keep believing what they want to believe, rather than what the evidence suggests.

[FreeElk]

I think "you can't prove a negative" would have come, mostly, from discussions about the supernatural. As rebuttals to questions like:

"Ah, but can you prove unicorns do not exist?"

Which, if you could check everywhere in the universe, you could say that yes you could prove it (or you could prove they do if you found one). The proof that they don't exist requires checking every place in the universe, the proof they do exist only requires checking until you've found one. So, you can prove the negative if you can check everywhere but then not many people have that ability (or none, depending on your position on those supernaturals ;) ).

Unfortunately people quite often shift to false arguments in attempts to keep the perceived upper hand. If the person were to admit that, no, they couldn't prove unicorns do not exist they would probably feel they had lost a little in their argument despite it being the correct answer. These things come back to bite you, however, when someone later finds you've been spouting nonsense as fact to back up your argument. That is then used against a person and used to dismiss the argument (however solid) along with the person who attempted to argue for it.

[Regina Minx]

I think "you can't prove a negative" would have come, mostly, from discussions about the supernatural...

There's actually two things going on here. The first is proving a negative, which is not really a thing. The second is an unjust shifting of the burden of proof. In essence, the person making a claim is the one that has the obligation to offer sufficient demonstration of its truthfulness. If you and Alicia were having a conversation and the topic of unicorns came up, and you made the claim that unicorns did not exist, the burden of proof would be on you then.

Of course, given my above discussion on induction, most people would consider it sufficient to say, "We have no evidence of them and we would expect there to be evidence. Therefore they probably don't...".

If Alicia was the one who brought it up and said "Oh absolutely unicorns exist," the burden of proof is on her. But the [url=https://en.wikipedia.org/wiki/Prior_probability]prior probability of 'unicorns exist' and 'unicorns do not exit' are not equal, so Alicia would have a higher burden of proof in the same discussion.

[FreeElk]

The second is an unjust shifting of the burden of proof. In essence, the person making a claim is the one that has the obligation to offer sufficient demonstration of its truthfulness. If you and Alicia were having a conversation and the topic of unicorns came up, and you made the claim that unicorns did not exist, the burden of proof would be on you then.

I guess this is an interesting point. My first instinct was to say that the person claiming something beyond the accepted holds the burden of proof but, when I thought about this, it doesn't really hold up. If there were two people today with differing opinions about the supernatural - lets say magic - the one who believed would then have the burden of proof but take that same pair back a few hundred years and the non-believer would hold the burden of proof if we're going on public opinion.

The claims, however, would come this way around. In a world where few people believe in magic or unicorns then to make the claim "Magic/The unicorn doesn't exist" without first receiving some indication that the other person believed otherwise.

[Dreaming Space]

Well, keep in mind that there's a distinction between logical proofs and empirical ones.

Proving that unicorns exist empirically requires producing one.

Proving that they don't exist, or never existed, is more of a challenge.

[Regina Minx]

I'm almost 98% positive that I addressed the use of inductive logic. But to put it in other terms:

The absence of evidence IS evidence of absence to the precise and mathematical degree to which the absence of evidence is less expected on the theory of something existing.