You Can Prove a Negative [pdf]
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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.
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.
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".
Negative knowledge is: there is not an instance of x (where not existing is not a predicate of x)
It's not just the word "not" it's an odd number of nots (double negatives and all that).
It is a statement about an abstraction, not existence. Like saying "the cup is above the table" is a statement about the existence of the "above-ness with respect to the table" of the cup. One needn't survey all of reality to know if it is possible for some abstraction to exist. It need only be consistent with the rest of your language, so that's all you need to survey.
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.
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.
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).
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.
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.
I'm reminded of Duhem's rather humorous "German Science" where he characterizes German mathematics as being disproportionately oriented toward rigorously deducing nonsense from arbitrary axioms as opposed to the French tendency to rely on intuition to arrive at sensible axioms but neglecting systematic rigor afterwards.
Math is a tool to help you make logical conclusions.
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.
If I believe John knocked the book off the table, but then I learn it was the wind, that doesn't change the characteristics of John.
Most certainly, guilty as charged.
> It does if you believed that John knocked the book off the table to punish you.
I don't see how my belief about the reason John took that action makes whether John took that action (or not) a defining characteristic of John.
I suggest Hume and the writings of the logical positivists. This will wipe the idea of being able to "prove" with science or observation right out of your brain. No matter how much information you have, you don't have enough to predict the future. If our planet collides with some huge energy burst, all the people who said "I know for a fact that the earth will still be moving around the sun at the approximately same rate and mass" will have been PROVEN false. But there is no such trick to prove anything about tomorrow.
So, if you want to lie to yourself, I guess you could "prove" something to yourself. But you know you can't trust your observations for about a billion different reasons.
Why is this important? Because you can always know if a bridge has failed, but you never know if it's not going to fail. Never. No matter how certain you are, unless you can form a deductive proof showing that the structure will NEVER fail (which I suspect is impossible for a computer, let alone a human), you will always have doubt. If you don't recognize the doubt, you are lying to yourself.
Furthermore, once you realize how.... loosely science binds together, you realize that the "laws" we think we "know" are broadly accurate but fall to pieces in the details.
So no, proving a negative is not possible. It's not a proof unless you can show the state of the universe, which is again impossible without consensus, which is itself probabilistic and flawed. If you are 100% certain that an event has happened in any way that you could deductively prove something, that gamma ray burst could easily mess with your plans.
A non-deterministic universe is one of increasing, but always <100%, certainty.
1. If Bigfoot were in Times Square, someone would notice.
2. No one has noticed.
Therefore it cannot be the case that Bigfoot is in Times Square.
That proof works because of the particular properties of Times Square (e.g. that it's full of people 24x7) and doesn't apply to the forests of the Pacific Northwest.
If your assumptions are false, then so is your conclusion, in the end.
Besides, Bigfoot is Jewish.
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.
I defend that using the word "proof" in the context of scientific theories is misleading. It invites people to suspend doubt, which is precisely the opposite of the scientific stance. Religions deal in absolute truth. Science is something much better than that, precisely because it is rationally humble.
Science is fundamentally subject to abductive, statistical reasoning. As such, the whole "doubt" and "humility" shtick wears out once you get to well-tested ideas.
We do not have "some evidence" that "supports", for instance, our conclusion that the Earth orbits the sun. We can simply say that all available evidence supports such a statement. We can imagine some chance, some probability, that the statement is wrong, but that probability involves revoking so very much of our established corpus of scientific observations that it makes no odds. In well-established matters like these, underconfidence in colloquial speech is every bit as misleading as overconfidence in bleeding-edge research.
Note that when you start trying to talk about things anti-scientific people wish to dispute, such as evolution and global warming, all of a sudden it's the enemies of science yacking on about the humility of mere empiricism, as though empiricism is mere.
Mathematics is a science: https://en.wikipedia.org/wiki/Mathematics#Mathematics_as_sci...
Still, if this bothers you, replace "science" by "natural science" in my claim and the relevant aspect of the argument remains unchanged.
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.
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
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.".
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.
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.
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.
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.
Secondly, just because something is a tautology, does not mean it is not a statement about non-existence.
Also don't forget to say 'nonzero'.
That proof relies on induction which in turn relies on the existence of infinity. To see how infinity is necessary, assume there are not infinite integers. Let X be the highest positive integer and multiply it by 2, the result does not exist.
> Also don't forget to say 'nonzero'.
I did forget :), thanks
>Let X be the highest positive integer
That contradicts the definition of integer, so it doesn't really show anything.
Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers.
What do you mean by unbounded? If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true.
> That contradicts the definition of integer, so it doesn't really show anything.
Which definition does it contradict?
> Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers.
You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity.
There is no largest integer.
> If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true.
Sort of, it depends on how you define factor. What stops me from factoring -4 into 4 * 63?
> Which definition does it contradict?
Where you take counting numbers and add 0 and negatives. If you reach a point where you can't count more, you screwed something up.
> You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity.
When you give me an integer n, I only need to apply the successor function n times to provide your factoring example. No problem there.
Or you ran out of time, or atoms.
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?
A proof, essentially, is that which compels an unbiased rational mind into agreement. Thus we have to ask what compels a rational mind. (And what is a rational mind?)
The answer, roughly, is "information." And if your claim requires information behavior that hasn't been observed, one cannot accept the claim.
So all you have to do is identify a structure internal to the claim that contradicts what is known about information processing rules. Which induces a kind of language dependence, but this is not a problem because you cannot make language-independent claims in the first place. And more importantly, language cannot be meaningful unless it obeys some laws, so naturally any claim that violates them cannot be true. (E.g., Conservation of Probability prevents the existence of true claims of supernatural phenomena or self-contradiction.)
In this manner, truth is a 'stack' of verifiable justifications. Something is true if:
* you are justified in believing it, and
* you are justified in asserting your justification, and
* you are justified in asserting the previous assertion, and
* Et cetera., ad infinitum, or for as far as the flow of entropy in your environment allows.
You don't have to enumerate things. You just have to properly understand the claim. You have to accurately process the information encoded in its expression. To ask for a full enumeration is to ask for infinite information -- something that doesn't exist -- and thus cannot be what proof is about, or else the word 'proof' would be meaningless.
You've touched on the issue of language, which I think is key to the confusion people have in this discussion.
My current thinking is that the proposition of "X does not exist" has no answer/proof because the statement is meaningless.
To consider that proposition, at some level you must assume that X does not exist, so the most obvious question is: What is X? If it doesn't exist, how can we even refer to it, or know its properties? If we can actually define X, then it must exist (in the most technical sense of a 'proof': X is an element of the current system you are working in).
It's a contradiction, likely caused by malformed propositions.
I would agree with that, given that I would take existence to be a tautological primitive if left to my own devices. But normally, I interpret it as "X does not exist independently of mental constructs and their derivative effects," which is a workable claim that can be false. And that's how most people mean it when making those kinds of claims.
So there are two meanings to existence: existence 'proper', and existence as a name for the extra-mental nature of a thing.
a) The thing itself b) A name for the thing itself
The question is then, can there be a real b) if there is no corresponding "a)", the thing that is being named.
It would seem the obvious answer is, no. However, I'm sure you could try to construct some twisted logic in an attempt to allow it. Such a logic would probably have flaws (incompleteness theorems likely have something to do with it) and are likely near useless.
I would agree that there are multiple definitions of existence that can be considered fruitfully. However, as a matter of formal logic, attempting to disprove/prove the non-existence of a 'thing' [0] is a trap.
[0] I use quotes here, because I'm not sure anyone here has yet defined what exactly a 'thing' is. :)
If you can describe a thing that exists, in sufficient detail to check the description, then you can describe a thing that doesn't exist too. For the answer to be 'no', you have to reject the entire idea of descriptions. You have to reject the entire idea of categories, leaving only sets with predefined member lists.
I think that line of thought is a waste. If you reject language as too imprecise to use on reality, you end up concluding that proofs cannot exist, so you must not actually be talking about proofs.
Plus, it's easy to name something that doesn't exist.
There is no piece of paper on Earth that has printed on it the string T5hYEQdla6ZLTjxQnA3D. This is something that could exist. You could even make one yourself! But as I make this comment, it does not exist.
>a) The thing itself b) A name for the thing itself
Yes, that is generally accepted. Generally philosophers talk about the name, 'A', the thing, A.
> It would seem the obvious answer is, no.
If you think that is obvious, you need to spend more time thinking about it. For example, you seem to be implying that we don't know if 'alien' is a real word or not since we don't know if aliens exist. (Some people may claim to know, but most of us have not seen proof.)
However, there is a level where it is not about the words, but the actual things. In your example, the word 'alien' is real, but there is no 'real' definition of it, in the sense that there is no definition that can be given that could select an actual instance of 'alien' (I may have confused the quotes by this point) precisely (perhaps) because 'aliens' don't actually exist (even though there is a word that can be mapped to some vague concept of what an alien might actually be like).
The concept 'alien' is actually pretty well defined and specific, the least specific part of it is probably the definition of 'life'.
I would argue that the statement does indeed have meaning, because you have some understanding of logic. If the statement had no meaning it would be impossible to disprove.
It is quite possible to create a meaningful definition that has no referent: "My first born child" is a statement that can have at most 1 referent but can also have 0 referents.
Just the fact that it is possible that a definition has 0 referents does not make that definition "a contradiction" or "meaningless".
1) An inductive proof that everything that does exist is not that thing.
2) A proof by contradiction where you start by assuming the thing does exist and then derive a contradiction.
#1 only works if you accept induction is valid for the set of items you are working on.
#2 actually only proves that one of your assumptions is not valid, not which one. Presumably all your other assumptions are without reproach
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!