The author of that paper makes some good points, although it’s full of razzle-dazzle. At the end he pretty much acknowledges that you can’t, in fact, prove a negative with a degree of certainty equal to 1. His argument is basically that if the chances of occurrence X are so minute and there is no observable evidence to support X as real, we should accept it as “good enough” proof that X does not exist. And for practical purposes, I agree. But from the purely intellectual standpoint, it’s not quite accurate.
And yes, the statement “you can’t prove a negative” is itself a negative. Which if you could prove it, would make it untrue. But you can’t. Which would seem to support the theory that you can’t prove a negative.
Short answer: It depends on how anal you feel like being.
Practically speaking, yes, for certain things you can show a thing to be so unlikely that it’s not worth quibbling over and simply better to call proved.
Only to the extent that you can’t prove anything with a degree of certainty equal to 1. His argument is that as soon as we accept an inductive logic, negatives are every bit as provable as any other universal claim.
Again, this is the same standard that we accept in any other proof, including the ones mentioned (for example, the sun will rise tomorrow - resisting the temptation to phrase that as an “Annie” lyric :))
These problems completely disappear if we adopt an axiomatic scheme. For this, proving negatives becomes commonplace - every math major learns that there are no rational numbers r such that r^2 = 2.
Razzle dazzle? I thought the article was very clear (up to the point I read) especially compared to others.
Now first, to deal with your sentence, say you can prove a negative does not imply that you can prove all negatives, which is trivially false. (I can’t prove that it is not true that 2 + 2 = 4).
Second, he implies but doesn’t make clear that there are two types of negatives - logical negatives and existential negatives. Logical negatives are proved all the time. To prove that a statement is false you can use proof by contradiction, where you assume it to be true, show that this leads to a contradiction, and thus demonstrate that it is false. You can do this in sudoku - if you want to show that a 3 does not go into a certain square when you are almost done, assume it does, fill in the squares implied by the original square being a 3, and if you get to an impossible state you have proved that a 3 does not go there.
He mostly covers existential proofs, such as there are no unicorns. The confusion maybe comes from a missing premise on page 2
2a. The complete fossil record that might contain a unicorn as been discovered.
Since this is not true, we can’t actually prove this negative. In fact, our confidence in the negative is directly proportional to the percentage of the fossil record which has been uncovered, in the sense of areas excavated. Clearly you are not as confident about not finding something when you start digging as you are when you finish.
So you can’t logically prove an existential negative, but if you expand proof as in burden of proof, giving it its legal meaning, you can do it reasonably well. I can’t totally prove that there are no Roman ruins buried in by backyard (I live in California) but I can come pretty close.
Of course you can prove a negative - proving a negative is just proving a positive of the negated statement. “It is not the case that Socrates was not a man” is precisely as provable as the statement “It is the case that Socrates was a man.” The difference between a positive and a negative is essentially semantic.
It’s just a lot more difficult to prove negatives in the limited situation where the proof of the positive just requires you to find one example of the place or thing you’re looking for, and the proof of the negative requires you to examine every possible example of the place or thing you’re looking for.
Well, that’s the point. “There’s nothing in this box”, provable.
“Socrates never thought about a purple elephant”, unprovable.
Depending on its scope, a negative can be anywhere from easy to prove to impossible to prove, and the way that most people use the phrase “asking me to prove a negative”, they mean the impossible, not the easy kind.
No, because proving a negative would mean knowing the answer to all the potential positives, and that by necessity includes the group of things we aren’t aware we don’t know, which forces us to not fully accept even the most accurate list. Even a situation which limits potential positives is problematic since we can’t necessarily assume that the limits are accurate.
But it’s impossible to really prove a positive, either. Generally it’s just easier and takes less work to get to a more reasonable likelihood point with a positive than a negative.
Not true. You can use reductio ad absurdum, for example.
Indeed, if one cannot prove a negative, then one cannot prove that one cannot prove a negative. After all, the claim in question is logically equivalent to saying, “There is no way to prove a negative” – in other words, it is itself a negative statement.
It’s not about proving a negative – nor a positive. It’s about proving “existential” (or in other words, “all-encompassing”) statements.
I can no more prove “all panthers are black” than I can prove “no panthers are green with yellow polka-dots.”
However, I can easily prove both “not all panthers are yellow,” and “some panthers are black.” (well, for some values of “easy,” in this particular case…:))
Meh. You can prove a positive, by evidence, you can only infer a negative from lack of evidence. This is of course different from falsification which is similar to proving a negative but which is really just finding evidence of a contradiction. And also difference from proving a logic negative, which is something else entirely. Much ado about… literally… nothing…
In 1994, Andrew Wiles proved Fermat’s Last Theorem, which is a negative theorem (No three positive integers can satisfy the equation a[sup]n[/sup]+b[sup]n[/sup]+c[sup]n[/sup], when n > 2).
My first thought was that statements like “all panthers are black” or “no panthers are green with yellow polka-dots” are statements about what does not exist, while statements like Fermat’s Last Theorem or the irrationality of the square root of 2 are statements about what cannot exist. The latter are proved by logical reasoning, by showing that the definitions and axioms we’re working with are incompatible with such a thing existing. “No panther is a halibut” would be this sort of statement. The former could be proved by observation, but only by observing all places where such a thing could possibly exist.
But wait, I thought. Surely a statement like “there are no counterexamples to Goldbach’s Conjecture” is the kind of mathematical, logical statement, that can be proved (if it can at all) only by showing that such counterexamples cannot, logically exist. But what about a statement like “there are no counterexamples to Goldbach’s conjecture less than one million.” We can check that by looking to see whether there are any in all of the places one might exist—but in doing so, are we engaging in observation, or logical reasoning, or is there no clear distinction between the two?
I think this hinges on a negative proposition being one that produces no evidence. If Socrates never thought of a purple elephant, there would be no evidence. If he did, he may have written of the fact, or told a friend who recorded it. The positive case can leave an evidentiary trail. Finding one instance of that documentation would prove the case, where the opposite, not finding any instances of the documentation, can’t prove that they don’t actually exist somewhere. (edit: not talking about all negative propositions, just the ones that fit the common mantra.)
