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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-existen

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 11y ago
The paper very specifically addresses that problem at length.
- jsprogrammer 11y 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 11y 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!
Retra 11y ago
Then how would you prove that such a proof cannot exist?
- jsprogrammer 11y 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?
  - Retra 11y ago
    The more relevant question you should be asking is "what is a proof?" 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.
    - jsprogrammer 11y ago
      I do think 'proof' is mostly meaningless in this circumstance (non-existence). 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.
shkkmo 11y 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.
- jsprogrammer 11y 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.
  - shkkmo 11y ago
    First off, it depends on the assumption that infinity exists. Secondly, just because something is a tautology, does not mean it is not a statement about non-existence.
    - Dylan16807 11y ago
      I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? Also don't forget to say 'nonzero'.
      - shkkmo 11y ago
        > I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? 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

ginolomelino 11y ago

The paper very specifically addresses that problem at length.

jsprogrammer 11y 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 11y 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!

Retra 11y ago

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

jsprogrammer 11y 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?
- Retra 11y ago
  The more relevant question you should be asking is "what is a proof?" 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.
  - jsprogrammer 11y ago
    I do think 'proof' is mostly meaningless in this circumstance (non-existence). 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.

shkkmo 11y 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.

jsprogrammer 11y 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.
- shkkmo 11y ago
  First off, it depends on the assumption that infinity exists. Secondly, just because something is a tautology, does not mean it is not a statement about non-existence.
  - Dylan16807 11y ago
    I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? Also don't forget to say 'nonzero'.
    - shkkmo 11y ago
      > I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? 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