If I say, “there are no penguins on Mars”, that is a negative statement. No one can prove 100% that there are no penguins there because I could simply say that we are not looking in the right places.
However, if I say “there on penguins on Mars”, that is a positive statement. Could I not use the same argument as above in my defense, we are just not looking in the right places.
Maybe I’ve got this all wrong but I can’t wrap my head around it.
The point is that you need only find a single example to prove the positive statement, but you must do an exhaustive search to prove the negative.
The aphorism is, however, incorrect on its face. It is trivial to prove that there are no penguins in my living room. It would be better to say that it becomes increasingly difficult to prove a negative if it is highly generalized or poorly specified.
The fallacy is closely related to the fallacious claim that “absence of evidence is not evidence of absence”. As any good Bayesian will tell you, that depends on whether the hypothesis under consideration implies that we should expect to see evidence.
You can’t prove that there are no penguins on mars because, as you rightly say, there are myriad reasons why you may not have seen one* yet*.
However, you can prove that there are penguins on mars by finding just one solitary single penguin.
The person making the positive claim is the one who bears the burden of proof and this is why, for instance, atheists like me do not make the claim that god does not exist. It is a pointless and unproveable statement. Much better to say that the person claiming god does exists has to show evidence for this.
That makes sense. Thanks.
“There is no gas in the car’s gas tank”. That’s a negative, but it can be proved by looking in the tank or using the gas meter. Likewise, you could also use this to prove “There IS has in the car’s gas tank”.
Not sure what you’re trying to get straighten out. But does this help?
Really? Did you check behind the sofa?
Very well. Now, did you check under the table? Even if you did, check again. And don’t look at me like that. The penguin may have run from somewhere else it was hiding, to under the table, while you were checking behind the sofa. Penguins are tricky like that.
Also, there are those tiny penguins that hide between the cushions of your couch and in your carpet. And don’t get me started on the invisible penguins, and the ones made out of dark matter. Those guys are a real pain to find.
But why should such silliness apply only to negative assertions?
Maybe the penguin that you did see on Mars was an optical illusion. Maybe you were drunk. Maybe it was a meerkat in fancy dress on its way to a party.
The fact is that it’s perfectly possible to prove negative assertions that are well-specified - i.e. regular penguins, that are visible and lack the ability to teleport - and limited in scope, i.e. where an exhaustive search is logistically feasible.
If you doubt that, put your money where your mouth is. I have $1000 that says there are no penguins in my living room.
Of course, in the real world everything is best addressed on Bayesian principles anyway. We don’t need absolute proofs of negative assertions, we update the probabilities as more data becomes available, based on what evidence we should expect to see under the competing hypotheses.
Actually, the point of the maxim is a bit different than outlined. It’s best example comes from The Amazing Kreskin: “Can you prove it didn’t happen?”
That’s the negative statement. You can’t prove it didn’t happen because you can’t come up with positive evidence to show it didn’t happen. Any evidence you can give is to say, "It didn’t happen in examples A, B, C, D . . . " But the retort is that “you didn’t show it couldn’t have happened somewhere else.” There is an infinite number of events and you’ll never be able to disprove all of them.
Did you forget about that “Bloom County” book over there on the bookshelf?
As in, specifying what counts as a penguin. 
“What’s that on the television then?”
“Looks like a penguin.”
[bold added]
[bold added]
Broadening the definition to include representations of penguins, invisible penguins, or perhaps penguins that exist only in the imagination of a person sitting in the room really does not speak to the principle of the difficulty of proving a negative.
In politics “prove a negative” is often mixed up in arguments over causation.
In other words the claim is “Government action X caused (or prevented) result Y”.
It’s at least plausible that we could gather statistics to see whether Y became more or less common after action X occurred.
Saying for certain that the increase or decrease in Y happened because of X is a very much taller order. Almost everything in society happens for multiple overlapping reasons in complex nth order feedback loops. As such, assigning strict causation is often a fool’s errand.
So it becomes a handy rhetorical device to throw sand in the Other Side’s gears.
Very well. Define “penguin”.
Define “define”.
We could play this game forever. Why do you think this game is only applicable to negative assertions?
Honkus Magnificentus
I think RealityChuck has outlined the real point of the statement about not proving a negative.
And we might add that other claims, such as proving there is no Santa Claus, can only be tested by examining positive claims: that if there is a Santa Claus, he lives at the North Pole 364 days a year, so let’s go take a look, for example. But without such positive claims, it is impossible to prove he doesn’t exist.
Was it the Amazing Kreskin who asked, “can you prove it didn’t happen?” I know the Amazing Crisswell asked it, most notably in Plan Nine from Outer Space.
Rather than arguing penguins, consider for example the assertion:
“No prime number contains within it the sequence of digits …123456789…”
Again, I need only find one such prime to prove the positive, such a prime does exist.
I have to enumerate every prime in existence to prove the negative. Or come up with a general theory in primes that mathematically shows the negative. Good luck with that.
BTW - is it true? I don’t know, I pulled that theorem out of thin air. Given the plethora of big primes, I bet it is not true. But… to prove it is?
I take it you never met Wittgenstein*. I doubt that finding a hippo in a room is easier than finding a penguin.
- Although being a beery swine might have been an issue in his case.