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.
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!