You can prove a negative


Well, no it isn't philosophically impossible... read on:

It is commonly thought that one cannot prove a negative, but of course I can. If I say "there are no weasels in my right pocket", all I need to do is enumerate the objects in my right pocket and find a dearth of weasels among them to prove that negative claim. So why do people think one can't prove a negative?

Negative claims are of the form


Or, in English, "NOT Thereis an x such that x Fs"... oh, OK, it asserts that no x is F.

Now to prove this claim, you need something that logicians call "the Universe of Discourse", or "the Domain". That is, the totality of the world or worlds that the claim ranges across. In the pocket example, that is my right pocket. If the domain or universe is small enough, and all the objects in it accessible in a reasonable time, we certainly do think that we can make proof claims. Consider the extinction of the Yellow River dolphin. Pretty well all areas in which that animal can exist are under constant observation by a very large population that has the means to report its existence. So we can safely say that it no longer exists.

So why is it a common claim? This has to do with the development of the medieval logics, and ambiguity (errors in logic are nearly always due to ambiguity in some way or another). The medievals had what they called "the Square of Opposition". It went like this:

Propositions of the form E are negative claims. But if the universe is not defined, as it wasn't (the medievals thought that logical possibility ranged across all the universe and possible universes unrestrictedly), one cannot find out if something is false until one encounters an existing contradiction to the claim (an x that is F). Because they had an unrestricted domain, they could never prove that negative if none were ever encountered.

I suspect, though, that it is easier to prove that Obama is not a Muslim or a terrorist than to prove that no gods are green, for example. The domain is smaller, and more manageable.

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I can prove that 4 is not greater than 5 without much effort.
1 + 1 + 1 + 1 = 4 + 1 = 5.............
I love it when someone says that it's impossible to prove a negative. Even an ignorans like me can do that....

By Brian English (not verified) on 24 Jun 2008 #permalink

I wonder if my dissertation committee will buy that. They can't prove I'm not done with it, amirite?

I never thought I'd see John Wilkins run a negative campaign...!

I had just assumed that "It can be difficult to prove a negative" had degraded into "You canna do it, man" over time.

But now that I know that all those morons who say this out of convenience are well informed by medieval philosophy, it all makes sense somehow...

Ian: you have to prove I approved of this negative.

Greg: It's sort of the common heritage of the educated world c1850, so it still gets played out in public even though they aren't sure where. But notice that the homeschool movement has its own logic based on the classical medieval logic.

Incidentally, the medievals are well worth reading. They were not the fools of the mythologising of the renaissance, nor were they the mindless idiots of the latter nineteenth century.

I'm curious about where that idea came from. It's so obvious that "one can prove a negative", why would anybody ever think otherwise? After all, it's even self-defeating:
"One can't prove a negative" is itself a negative.

False statements about Obama aren't even of the form ~(∃x)F(x), except in some redundant way, like ~(∃x)(x is Obama, x is a muslim), which reduces to ~(Obama is a muslim), if Obama's existence and uniqueness are taken for granted.

I guess the phrase 'you can't prove a negative' falls into the same category of supposedly self evident profound philosophical discoveries as 'everything has a cause'.

Legally, at least in one state, the burden of 'proving a negative dependant fact or right' falls directly upon the party asserting it (even if it requires 'burden shifting' from plaintiff to defendant) because the courts deem such a burden "impossible" or "particularly difficult".

The medievals were brilliant. They were just ... so ... I don't know, ... medieval, I guess.

I always thought it degraded from "You cannot prove a universal negative" which is more or less true. Here universal is defined as "Everywhere and always", i.e. the domain is infinite. Disproving this would then require to be able to see all time, etc. Of course, you can still disprove this using a reductio ad absurdum or some method which does not rely on simply searching through the domain.

I am not a logician, so there may be glaring errors I do not see.

I think why many people like to use them as a counterclaim is that they are often insanely difficult to disprove. What is more, artificially increase your domain and it becomes more and more difficult to disprove...repeat until your opponent tires and walks gloat in your new-found righteousness. ;)

But "you can't prove a negative" isn't about class enumeration; it points to difficulties in delimiting the boundaries of the domain. It's used rhetorically to counter the claim that all the evidence isn't in yet; that somewhere out there (perhaps in the antipodes) we can find the smoking gun, the black swan-- or even something that would force us to revise the very definition of the domain. And you can't prove that we can't.

By Elihu M. Gerson (not verified) on 25 Jun 2008 #permalink

As I understand it, it's impossible to prove a negative statement through logic alone. It can be done empirically but only when the statement is something that can be tested empirically. That's why statements like "There are no UFOs" or "There is no God" can't be proven based on logic alone.

V. useful debate started, and it's good to remind us that in some cases it is possible to prove the negative.
But Romeo Vitelli #12, and Elihu #11 get the point that an awful lot of woo is based on the impossibility of disproving (universally and in strict logic) statements like "homeopathy works" or "astrology works".

Eli, I don't see how this is contrary to what I said. If the domain is indefinitely large, or inaccessible, then the possibility of a black swan case always exists. But when we have explored the domain sufficiently, we are entitled to say there are no (for instance) green swans.

It is always possible that we have the wrong domain. Since science is based on a fallibilist epistemology, we might be mistaken, but the likelihood that a thoroughly explored domain contains objects not found diminishes dramatically as the exploration proceeds.

Science of course doesn't do that sort of exhaustive exploration in the main (bird watchers are a class to themselves). It works out that there is no reason to think that some phenomenon exists based on testing, and in those cases there remains most of the time a possibility that phenomenon exists. But that wasn't my point. Sure we can do a modus ponens on any scientific claim if we do happen to find a counterinstance. But in finite and accessible cases, we most certainly can prove a negative. Prove me wrong, I dare you...

Re #15: John, it depends on what you mean by "accessible". For example, consider the beach at 57th st. in Hyde Park in Chicago. Surely the number of grains of sand there is finite, and the locals are used to strange people in the neighborhood doing pointless exercises. Can we then reliably count all of those grains of sand?

My main point is the issue of ensuring stable, reliable, practical definitions of domains-- it is this which is particularly hard to do, especially if there is dedicated opposition. If one doesn't like the play, one can move the goalposts, change the field, change the rules, shoot the umpire, and so on without limit.

Now of course that's contrary to conventional assumption in philosophical argument. But it's also entirely in line with ordinary practice. So, in finite and accessible cases, we can prove a negative. Now all we have to do is decide who decides when a case at issue is finite and accessible. Once we've done that we can go on to prove negatives with respect to that case.

By Elihu M. Gerson (not verified) on 25 Jun 2008 #permalink

I first recall hearing that phrase from the great James Randi, at the first time I saw/heard the great James Randi, in his Nova special. He had just finished showing a class of college students how easy it was to convince people that a generic set of astrology predictions was accurately tied to their specific birth dates. One student then responded passionately, "But that doesn't prove that astrology doesn't work!"

"You're absolutely right," Randi replied, "and I also can't prove to you that Santa Claus doesn't exist. I can't prove a negative. I can only show how the same results can easily be produced by non-magical means." (I probably don't remember the exact words, but those are close.)

I got a levin-bolt shock of understanding from that, so it is a bit depressing to have to agree that the part about proving a negative should have been qualified.

As I am currently reading Grant on medieval cosmology I have a small quibble with your elucidation. One of the fundamental medieval cosmological premises was that the universe is finite! Therefore it follows that all possible domains of discourse in the medieval world are also finite so it is theoretically possible to prove all negatives! Between theory and practice there is, of course, a very, very wide gap ;)

Here's a simplified taxonomy of types of claims or statements: (1) Observational statements, such as "There is a cat on that rug", which can be checked out to find out whether it is true or false, (2) Verifiable statements, such as "There is a cat on some rug somewhere", which can be proved true by a single (or a small number of) observations, but require an infinite number (or a very large number) of observations to prove false (3) Falsifiable statements, such as "Cats never get on rugs" which can be proved false by a single (or a small number of) observations, but require an infinite number (or a very large number) of observations to prove true. There are more complex statements that can never be proved true or false by any finite (small) number of observations. A verifiable statement is the negation of a falsifiable statement.

So when people say "You can't prove a negative", I think they mean that you can't prove the negation of a verifiable statement. If proving a statement requires making infinitely many (or a very large number of) observations, then it isn't provable. Yes, I suppose you could say that a statement such as "There are no unicorns" (the negation of "Unicorns exist") could be proved with a finite number of observations if you split the world up into little 10x10 meter regions and searched each one. But in practice, it's very difficult to know that you've really covered all the possible hiding places.

To me, though, the way around this is to forget about prove and work with likelihood. You can certainly reduce the likelihood that there are unicorns anywhere in the world to a number that is small enough to be negligible.

Consider the extinction of the Yellow River dolphin.

Not the Yellow River, the Long River, most widely known as "Yangtse" (actually Yangzi).

By David Marjanović (not verified) on 25 Jun 2008 #permalink

"If I say "there are no weasels in my right pocket", all I need to do is enumerate the objects in my right pocket and find a dearth of weasels among them to prove that negative claim"

Actually you need to narrow the domain even further, e.g. "there are no visible weasels..." else one could claim that you didn't prove that there were no invisible weasels. Reminds me of some religious claims...

Now John, we are imperfect observers and we can never be absolutely sure if there is a weasel in your pocket.

Is this not the point where Wittgenstein starts waving the poker?

bob h writes: Actually you need to narrow the domain even further, e.g. "there are no visible weasels..." else one could claim that you didn't prove that there were no invisible weasels.

That's a flaw in Popper's philosophy of science, it seems to me. Nothing is really verifiable or falsifiable without making auxiliary assumptions.

I think the comic itself alludes to the falseness of the statement. She stayed up all night doing the "impossible", which is exactly what committed volunteers are good for.

Of course you can prove a negative, but you can't prove an absolute negative. Like God does not exists, or humans do not die, because the enumeration of the positives is theoretically infinite.

Irrespective of the medieval origin, nowadays the expression "you can't prove a negative" is just that - expression, a figure of speech. It's not meant to be taken literally. The suggestion from some commenters that people are morons for using a figure of speech is, I think, ill-considered.

As I said on afp once, "It'll cost you an arm and a leg to stop me from leading you up the garden path, or to put it another way, even if all hell breaks lose, you'll have to pay through the nose to get my tongue out of my cheek. However, if you read between the lines it will be clear that the law is an ass, the world is your oyster, and you can't prove a negative."

So the medievals had no syllogisms that concluded to "E" or "O" statements? That's simply false. So do syllogims not prove statements?

'I can prove that 4 is not greater than 5 without much effort.
1 + 1 + 1 + 1 = 4 + 1 = 5.............
I love it when someone says that it's impossible to prove a negative. Even an ignorans like me can do that....'

More accurately, you proved a mutually exclusive positive- which is the only way to argue a negative