You Can Prove a Negative [pdf]

http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf

by miralabs • 11 years ago

39 76 11 years ago

departments.bloomu.edu

awptimus5• 11 years ago

This is an unnecessary. "You can't prove a negative" isn't a statement about logic, it's a statement statistical evidence (incorporating logic). You need to observe the entire population in order to reduce a probability to zero.

_delirium• 11 years ago

Yeah, the first half of this seems to be mostly for color rather than really debunking the slogan. Not even the most naive users of the slogan "you can't prove a negative" literally think that you cannot prove negatively phrased mathematical or logical claims, such as "1 + 2 != 5".

awptimus5• 11 years ago

It's just missing the better alternative. It's much more correct and useful to say "given all evidence the probability there are yeti is almost zero" correctly doesn't rule it impossible and correctly asserts that we shouldn't behave as if they do. I mean, if we're fine when the entire physics community go nuts with excitement when we make a mistaken measurement of a neutrino going faster than light, why worry about a person who thinks they saw a yeti try to find a yeti?

coldtea2• 11 years ago

Not even the most naive users of the slogan "you can't prove a negative" literally think that you cannot prove negatively phrased mathematical or logical claims, such as "1 + 2 != 5".

What happens is that the most "naive users of the slogan" don't take their thinking that far, that's why they believe it. It's not that don't think you cannot prove negatively phrased mathematical or logical claims, it's they don't understand that that would be a counter-example to their "can't prove a negative" claim.

TelmoMenezes3• 11 years ago

And likewise you can't prove a positive. In fact, you can't prove anything with statistics, and science is not about proof. Mathematics is about proof. Science is doubt. All theories are forever on probation -- or until they are falsified by new experiments.

awptimus5• 11 years ago

Not true. Given a hypothesis "there exists" the first observed instance is proof of the positive. The inverse however, again, requires you to sample the entire population and collect all data.

stefantalpalaru• 11 years ago

[...] science is not about proof. Mathematics is about proof.

Mathematics is a science: https://en.wikipedia.org/wiki/Mathematics#Mathematics_as_sci...

eli_gottlieb2• 11 years ago

Claiming that the word "proof" can only be used in the formal sciences is just vocabulary nitpicking.

danielam3• 11 years ago

I know you're being terse, but I hate the way the word "doubt" is used here so I'll take the opportunity to pontificate.

Doubt, too, must be justified as much as certainty. We tend not to doubt something until there is a reason to doubt it and certainty grows with corroborating evidence. Furthermore, if I were to say that I have a coin in my right hand, you would have neither doubt nor certainty about the claim because given your knowledge there is no reason to doubt that I had such a coin but likewise no evidence that I do.

Doubt that does not obey such a process becomes very unproductive and indeed science is most productive when it abides by a falliblistic principle of economy.

animefan• 11 years ago

Similarly in frequentist statistics "you can only disprove a hypothesis" is false.

If you think you can disprove x > y, (where x and y are parameters like the mean of some quantity for two different populations) then it follows that you can prove x <= y.

On the other hand you cannot ever prove x = y since there will always be values of x and y that are so close that your test has no power. but you can still put bounds on how close x and y must be in this case

darkmighty• 11 years ago

It really depends on the statistical model of the quantities you're working with. If somehow your model dictates that x and y must be discrete, you can get certainty of measurement (i.e. prove x=y) even with some noise.

In this case I think both sides are right, so the argument should be used in context. You can only prove things about the real world (and not an axiomatic system for example) assuming there are no unknowns. In some (perhaps most) cases those unknowns are not there, but in some important cases unexpected behavior may occur in spite of "overwhelming evidence", because the assumptions made by the conclusions failed. It would be a much simpler discussion to state "When you want to prove something, make your assumptions clear, and test them well. Everything that hopes for rigor is proved within a well defined set of assumptions, so it's not like nothing can be proved either.".

coldtea2• 11 years ago

On the other hand you cannot ever prove x = y since there will always be values of x and y that are so close that your test has no power.

If they are so close that your test has no power, then they might as well be the same for the purposes of your test.

jsprogrammer7• 11 years ago

I think, "not being able to prove a negative" is more related to the idea that you cannot prove/disprove (ie. Assign an absolute truth value to) the non-existence of something. For example, you cannot prove that a God does not exist.

ginolomelino• 11 years ago

The paper very specifically addresses that problem at length.

jsprogrammer7• 11 years ago

It's more that it avoided the issue by deflecting to a different example, while assuming unproven and fallacious (to the original point) premises to construct an inductive argument, which the author later admits is not any kind of proof according to the original question.

themodelplumber• 11 years ago

I wouldn't say it addresses the problem at length. At best it goes one level deeper than "you can't prove a negative" to say "you _can_ actually offer some evidence or probability for said negative" but even then it hands off the threshold of [what's been proven vs. not proven at that point] to epistemology.

After that, the author concludes that you should never dismiss an inductive argument just because you think any probability it provides is immaterial. After all, we rely on induction all the time in day-to-day life! Well, duh. So let's hear it for induction, everybody. At least it gives us some probability.

I dunno. It's just not as intriguing a line of thinking as I hoped it would be based on the title of the paper. Despite all the excitement about proving a negative, the author goes completely silent on the issue of how much probability proves what!

Retra9• 11 years ago

Then how would you prove that such a proof cannot exist?

jsprogrammer7• 11 years ago

What proof is valid that a thing cannot exist?

The only one I can think of is:

1) Exhaustively compare all things that exist, to your objective

2) Terminate when identity is detected

This brings up all kinds of problems, such as the fundamental: "what is a 'thing'?". Can you actually enumerate all of them? How do you validate identity?

shkkmo14• 11 years ago

You can prove that a thing does not exist, you can even do it without induction.

For, example you can prove "there is no integer that is not a factor of another integer".

Or another example, my first example disproves your negative statement "you cannot prove the non-existence of something".

I do agree that you can't disprove the existence of God, but that has nothing to do with induction or the rules of logic. I think that has much more to do with the semantic and cultural difficulty of adequately defining "God" in a way or ways that satisfy everyone and contain sufficient referential specificity to be disprovable.

jsprogrammer7• 11 years ago

"there is no integer that is not a factor of another integer"

I believe this is a tautology. It is true by the definition of integers (which you provide in the statement itself). It therefore doesn't make a statement about non-existence, but rather, the quality of integers.

scoofy6• 11 years ago

Induction allows us to use the existential quantifier, deductions allows us to add the universal quantifier. If we want to say something about the world, we are bound to empiricism, thus our deductive claims rest upon our inductive evidence. This is a framework. Any use of the universal quantifier (necessary for a negative claim), is unsupported by our empirical data. This is essentially Popper/Hume.

Retra9• 11 years ago

That's one model of the process. Another is that the distinction between negative and positive claims is entirely artificial, and that there are really only "claims" and the truth of them is determined by their effectiveness at constructing reliable models of reality. And thus any demonstration of self-contradiction is a disprove of a claim, whether negative or not.

scoofy6• 11 years ago

Negative and positive claims are not artificial or equivalent, one is making a claim about positive existence, the other is making claims about lack of existence.

Of course we can manufacture the universal quantifier insofar as it's a mapping of the existential quantifier. That is to say, proving something is "not nonexistent" is not a statement that requires the universal quantifier, it's a claim that has simply mapped the equivalency of the existential quantifier, using the universal one.

This is still only a statement about positive knowledge.

ThePhysicist2• 11 years ago

In physics (and other natural sciences) you can ONLY prove negatives in fact, so to a physicist this statement is not surprising at all.

Example: If we observe an apple falling to the ground with the same acceleration many times we will try to generalize this observation and create a theory that explains this phenomenon and similar ones (such as Newton's laws and his theory of gravity). Unfortunately we can never prove such a theory in the mathematical sense since we can never be sure that the next time we perform a given experiment it will still yield the same result (it could be that the laws of physics change over time or work differently in other parts of the universe). However, we can easily falsify a given theory if we can produce even one single observation where the theory does not fit reality (which is precisely why general relativity was needed to replace the classical theory of gravity, which could not explain all experimental data [actually the new theory came even before the data in that case, which is amazing in itself]). So, in a sense, proving that something is not true is the only thing we can do with absolute certainty.

That said, theories that are proven wrong by experiment and replaced by other theories are still valid in their domain of applicability, so to say as an approxomation of the larger theory.

scoofy6• 11 years ago

You're reference for negative here is wrong. When you prove that something is not the case, then you do so by demonstrating an there exist something contrary to your hypothesis. This is positive knowledge, not negative knowledge.

This is a very convoluted philosophical concept, but it's essential Karl Popper's argument. We don't, and can't prove negatives, but what we mean by that, as i discuss below, is that we cannot apply the universal quantifier with any certainty.

Retra9• 11 years ago

What exactly do you mean by "positive knowledge" or "negative knowledge?"

ThePhysicist2• 11 years ago

I see, so you talk about the case where we not observe any events at all? In that case I would indeed be surprised if it was possible to (mathematically) prove a theory using this negative knowledge.

We can make use of negative knowledge to draw conclusions about a given system though. For example, if we do not know whether an electron bound to an atom is in its ground state or not we might observe the system to see if it emits a photon or not. The longer we observe the system not emitting a photon, the more likely it is that it is in its ground state. So here we actually gain quantifiable knowledge about a system from the observation of a "non-event".

danielam3• 11 years ago

However, we can easily falsify a given theory if we can produce even one single observation where the theory does not fit reality [...]

Actually, strictly speaking, it's not that easy. The Duhem-Quine Thesis quite nicely shows how critical experiments aren't quite as decisive as some like to think, especially as theoretical depth increases. A theory can depend on numerous other theories being true. The reason a theory in question is the subject of a critical experiment is because of the principle of economy. Once supporting theories have been sufficiently corroborated, we justifiably tend to trust them more than the theory that we are currently considering.

wfo• 11 years ago

I think the article dances around what people really mean when they say "you can't prove a negative". The writer touches on it but somehow skirts the real issue. When we say "you can't prove a negative" you're really saying that in order to prove a statement of the form "not exists x P(x)", it's equivalent to proving "for all x not P(x)" and so you're really making a statement about EVERYTHING that has or does or could ever exist somehow. These proofs are either analytic and trivial (i.e. there doesn't exist a rectangular circle) or not proofs.

He suggests you get around this by providing an argument about unicorns and then goes on to talk about how premises don't always need to be justified, but the problem is proving the nonexistence of a thing -- and he provides a formal deductive example of a proof of nonexistence of a thing by instead substituting as a premise nonexistence of another thing (evidence). Sure, if you can prove no evidence for unicorns exists and that for unicorns to exist they MUST have left evidence you can prove they don't exist. But the first one is a reduction to the original problem -- in order to show something doesn't exist (be it unicorns or evidence for unicorns) you need complete and total knowledge about existence, which no human up to this point has had enough hubris to claim, and the second requires a level of certainty that doesn't really exist.

If we're discussing the existence of a thing that doesn't leave evidence (god) especially it's very fair to say you can't prove a negative and I think his attack on this tactic is a non-starter.

There has never been even the tiniest bit of evidence against the existence of god. There is certainly evidence against a god with specific qualities; a benevolent god, a god that wants this to happen, etc. But if you don't require god to have any specific qualities there isn't a shred.

Kenji2• 11 years ago

About the part with god, there's a specific expression for that: God of the gaps. As our (scientific) understanding increases, a proposed god people want to believe in has to lose one characteristic after another because they would not hold up against reason and evidence.

cristianpascu2• 11 years ago

True, my father told me that the day science came up with the tunneling effect, he stoped believing in a loving God. How can one believe in God when we have computers so powerful that we can post pictures of cats on the internet?

shkkmo14• 11 years ago

Do you have an example of one such characteristic that has been disproved?

I would posit the opposite: As the capability of our language to describe characteristics of god expands, it becomes harder to disprove god because the number of possible descriptions increases.

cristianpascu2• 11 years ago

I think the problem of proving the (non) existence of something is as simple as understanding that (most of the time) it can not be deductive. I'd say that when it comes to existence, it is always inductive. What kind of proof can anyone give me that she/he exists? Or the rest of the world? There is no such thing. Even in mathematics, if one doubts basic entities as numbers and their properties, the whole establishment of mathematics starts to crack, at least for that particualr one human being.

All of our reasoning about the world is inference based, hence probabilistic. That's in the nature of science, religion and everything else in between. There is strong evidence for the existence of God, but it's inherently inductive. Not to mention various conceputal complication which follow from in different kind of reasoning about God (in whatever view).

shkkmo14• 11 years ago

You can, in fact, prove non-existence deductively through proof by contradiction.

Proofs can be valid even if their premises (assumptions) are not true. Mathematics does not rest on the truth of it's assumptions, but on the validity of it's proofs.

If you can create an interesting system of proofs off of an a different sent of premises, then that can also be very useful (see non-euclidean geometry). If you can't create an interesting system, then those premises have no value.

I also disagree that all our reasoning about the world is probabilistic. By saying that all reasoning is inductive, you make you earlier assertion that "I'd say that when it comes to existence, it is always inductive." empty because you are claiming that all reasoning is inductive.

Dylan168076• 11 years ago

Even in mathematics, if one doubts basic entities as numbers and their properties, the whole establishment of mathematics starts to crack, at least for that particualr one human being.

That's not how it works. Numbers are their properties. You are free to use any properties you want when you commit mathematics, and you will get different sorts of behavior from your numbers depending on your choices. There is nothing to doubt about a set of properties being itself. No 'establishment' tells you how numbers work. Use whatever you like.

You Can Prove a Negative [pdf]

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