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This question has devolved into a discussion. As I understand the discussion, everything is revolving around the veracity of statement

  1. Nonexistence can never be proven.

and on what exactly constitutes a claim of nonexistence.

In particular, if a statement of the form A does not exist can be reformulated into an equivalent statement of the form B exists does this mean that the former statement is not really a claim of nonexistence?

If a statement of the form A exists can be reformulated into an equivalent statement of the form B does not exist does this mean that the former statement is not really a claim of existence?

EDIT: Consider the following two equivalent statements

  1. There does not exist a largest prime number.
  2. For a given prime number p there exists a prime number q larger than p.
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  • Can you give an example of such a pair "A does not exist" = "B exists"? – Mitch Jul 08 '11 at 15:07
  • I am having a really hard time understanding this question. Examples would be helpful, and maybe try to provide some context for us here -- how does having someone explain this to you advance your study of philosophy? – Joseph Weissman Jul 08 '11 at 17:12
  • Consider the statement: **There does not exist a largest prime number.** Now consider the statement: **For each prime number p there exists a prime number q such that q is larger than p.** These statements are equivalent. One is an assertion, nominally, that something exists and one is an assertion that something does not exist. –  Jul 08 '11 at 17:42
  • @YequalsX: this just makes it sound like mathematics is just really good philosophy, where __all__ the ambiguity is removed. In math, if you make a claim, you have to prove it (otherwise it's just a conjecture). – Mitch Jul 08 '11 at 17:49
  • Maybe I should rephrase the question. What is the definition for a claim being a claim of nonexistence? What criteria must be satisfied. Furthermore, if a claim of form "A doesn't exist" is logically equivalent to a claim of the form "B does exist" then is it still a claim of nonexistence and vice versa. Some claims can be reformulated. Some people believe that nonexistence can never be proven. Does this extend to statements that are equivalent to claims of existence? –  Jul 08 '11 at 17:54
  • @mitch: I'm a mathematician and the easiest example I could think of comes from math. Let me take another stab. **There exists an entity who murdered John.** Now consider the statement, **There does not exist a natural cause of Sally's death.** –  Jul 08 '11 at 17:57
  • We can't draw any conclusions from those two statements. Did you mean "John" in the second statement? And this example really ought to be in the question itself, if it's part of what you are asking. – Jon 'links in bio' Ericson Jul 08 '11 at 18:55
  • Out of curiosity, would the added statement **A or B exists** be enough to reformulate the proof of the existence of **B** given the nonexistence of **A**? Would that added statement mean that if **A does not exist** then **B exists**? In other words, do you interpret `or` as inclusive or exclusive? – Jon 'links in bio' Ericson Jul 08 '11 at 21:49

6 Answers6

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Some claims of existence are mathematical: is a given set of properties consistent? is there a number/object which satisfies a given set of constraints? Whether you set out to prove the positive or the negative, the burden is on the claimant, there's no need to worry about whether it is positive or negative existence or non-existence. There may still be an issue of difficulty (or as your example shows, issues of constructibility and reverse mathematical -logical- axioms (like "p or not p") are allowed).

Other claims are scientific: is there a an instance in the 'real' world? Here the properties are not inconsistent, but not necessary either. Is there a unicorn dancing on my head? (evidence shows not). Is there an atom of atomic number 120? (theoretically its possible, but we can't scan the entire universe, and our current technology only gets us so far).

So for your primes example, existence or non-existence, it doesn't matter (any quantification can be converted from existential to universal or back again with a couple extra negations).

For your 'murderer' vs 'not natural cause' example, you're still playing with the properties of the concepts, which is...mathematical.

Mitch
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    Are you saying that **there is no unicorn dancing on my head** is a fundamentally different concept (construction?) than **there is no prime larger than all others**? What fundamentally makes the latter provable but not the former? Or makes one statement a claim of nonexistence but not the other? Do you agree (or disagree) with the view that claims of nonexistence can never be proven? Thanks for your response. –  Jul 08 '11 at 20:15
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    @YequalsX: 1 - yes, the 'dancing on head' question I think is fundamentally different from the 'primes' question and also the 'atomic #' question (all three). 2 - all questions are provable by different methods, the 'unicorn' one by looking (and disprovable because the looking will also tell you not), the 'primes' one by mathematical proof either way, the 'atomic number' question we can only prove by finding one or creating one, but we can't prove non-existence unless we show the properties are inconsistent. The last qn should be answered by those answers (sometimes yes, sometimes no). – Mitch Jul 08 '11 at 20:25
  • @YequalsX: not to put words in Mitch's mouth, but the largest prime statement is proved with deductive logic and the unicorn statement is proved with inductive logic. (But you could construct a deductive proof: If a unicorn is dancing on my head, I'd see it. I don't see a unicorn dancing on my head, therefore it does not exist.) – Jon 'links in bio' Ericson Jul 08 '11 at 20:28
  • @Mitch: You would say that all three examples are examples of nonexistence. The nature of the objects (or concept of existence of the objects in question) are what makes the questions fundamentally different from each other. Sometimes one can prove nonexistence and sometimes you can't. It depends on the situation. Is this an accurate interpretation of what you are saying? –  Jul 08 '11 at 20:32
  • @YequalsX: yes, it depend on the situation. I doubt my categorization is the best, but as a very first approximation, analytic questions can have proofs for existence or non-existence, but experiential questions (where the properties are not inconsistent and the search space is not finite), I don't think you can prove a negative. E.g. "Is it that there are no more than 8 planets (presuming the current fixed definition)?", "I don't know, I can't know, I haven't looked everywhere and I can't be sure I've looked everywhere." – Mitch Jul 08 '11 at 21:10
  • There is no unicorn dancing on your head is also a fundamentally different concept than "No Unicorns exist." It the former you are acknowledging that unicorns could exist just that there is not one on your head. – Chad Jul 11 '11 at 13:23
  • @Chad: I'd say they are different, yes, but not fundamentally. The search space is much larger (it is unwieldy and possibly infinite), so I agree that proving "No unicorns exist" is difficult or impossible. (formalized though, there is little difference other than the additional constraint that 'x' is dancing on my head. Which is all to say that -some- nonexistence proofs are possible. – Mitch Jul 11 '11 at 14:09
  • They are different as the first acknowledges the the existance of unicorns but says there is not one right here. the other takes the scope out of the equation. The second is the only real claim of non existance. – Chad Jul 11 '11 at 14:31
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This is, at its root, a question of set theory.

The statement "X does not exist" can be easily translated to, "X is not a member of the set of things with the property of existence." Existence is the same: "X is a member of the set of things with the property existence."

Very simple, right? So where does the problem come from?

The problem comes from the fact that we haven't enumerated the set of things that have the property of existence. If we had, it would be trivial to prove non-existence.

Most people feel that the set of things that exist can never be enumerated, as the universe is big enough to make this effectively impossible. Therefore it is effectively impossible to prove non-existence.

Satanicpuppy
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  • As a generally principle I agree with what you have written. It is nicely put. I don't think it is correct to have a blanket belief that nonexistence can never be proven. For instance, I think I can proof that there does not (at this time) exist on my head an object with a mass of 200 million kilograms. –  Jul 08 '11 at 20:24
  • @YequalsX: Yes, you can prove that non-existence of a heavy weight in two ways, either by looking (something that heavy must have certain visible signs of its appearance, and those signs are not tangible), or by noting logically that such a weight on ones head is inconsistent with being alive. Hm..maybe those are the same thing? – Mitch Jul 08 '11 at 21:13
  • @YequalsX: There is an object on your head that has that mass, but it's invisible and not affected by gravity nor inertia. – Lennart Regebro Jul 09 '11 at 06:51
  • @Lennert: You are now making a claim of existence. I gather from your previous writings that the burden of proof rests with you. So prove it. If you are going to engage in this type of thinking then your true position is that nothing can be proven. In which case you should state this and not the weaker claim that nonexistence can't be proven. What ever you believe to be a valid proof of something I can always retort, "Prove it isn't all an illusion." You can't. But then we've delved into the abyss of unreasonableness and nothing further can or ought to be said. –  Jul 09 '11 at 15:15
  • I have a much harder time accepting this set theoretic explanation. It takes for granted the definition of existence. Does that set, for instance, exist? Does a concept exist? A number or a color, unattached to an object in particular? – davidlowryduda Jul 09 '11 at 15:27
  • @mixedmath: Were you responding to me in particular? I believe the natural numbers exist. I don't believe that natural numbers exist in a different way than apples existing. This is how I perceive the universe. It is my belief that concepts do exist. –  Jul 09 '11 at 18:50
  • @YequalsX: Yes, I am making a claim of existence. You ask me to prove my claim of existence. Well, I can't, of course, because it isn't true. Can you disprove it? No you can't. That *should* be enough for one penny to drop. "Prove it isn't all an illusion." - That's just wordgames. As you say, you have entered into the abyss of unreasonableness. So don't do that then. – Lennart Regebro Jul 09 '11 at 20:37
  • Actually you have entered that abyss when you said, **"There is an object on your head that has that mass, but it's invisible and not affected by gravity nor inertia."** Your position essentially is equivalent to nothing can be proven. Anyone can play the game I quoted in bold. Such a game renders any proof invalid if one considers it valid reasoning. I'm merely pointing out that in the interest of consistency you ought not reject the game when it is applied to proofs of existence when you play it for proofs of nonexistence. –  Jul 09 '11 at 21:02
  • @YequalsX: No, that position is not equivalent to that nothing can be proven. It is equivalent to that factual claims of non-existence can't be proven. The point of my statement is that you can't prove it false. You now require of *me* to prove it. Well, that should enable me to say "QED!", but you don't seem to understand the implication. You are now asking *me* for proof of *existence*. I agree that I have the burden of proof. You do *not* agree. Therefore I require you to prove the statement to be false. If you can not, then QED, the burden of proof is on *existence*. – Lennart Regebro Jul 10 '11 at 08:32
  • @Lennart: You appear to believe that one can prove there is an apple in their hand. But that one can't prove that there is no apple in their hand. You appear to accept that it is valid to say, "Maybe the apple is invisible, not affected by gravity, or inertia." You are free to engage in this type of reasoning. It is inconsistent for you not to accept this sort of reasoning when it comes to proofs of existence. One can say, "There's not really an apple in your hand, it's an illusion." I think if you were consistent then you would believe that nothing can be proven. –  Jul 10 '11 at 14:31
  • @YequalsX: You have some non-existence to prove. If you can't, you should, to be intellectually honest, admit that you can't. Every comment from you that fails to prove the non-existence of the object on your head just lends further support to my standpoint. And that goes for that magical weightless invisible apple I have in my hand to. And the invisible Unicorn in my garden. You said you can prove they don't exist. **So do it.** Claiming that they are "invalid" or "unreasonable" doesn't help, that is just pleading for Ockhams razor. Asking me to prove they exist is admitting that I am right. – Lennart Regebro Jul 10 '11 at 14:44
  • @YequalsX: "There's not really an apple in your hand, it's an illusion." - Well, that is a positive claim of existence, namely a claim that what appears to be an apple isn't an apple. So then prove it. All your arguments against my standpoint, including this one, just pushes the burden of evidence from non-existence to existence, hence again proving that I am right: You can't prove non-existence. You are continually arguing for my standpoint, you just fail to realize it. – Lennart Regebro Jul 10 '11 at 14:49
  • @Lennart: You have claimed that one can prove that an apple in one's hand. I'm just pointing out that one can retort, "You can't prove you have an apple in your hand because it could all be an illusion." It's the same sort of game that you played with the **There is an object on your head that has that mass, but it's invisible and not affected by gravity nor inertia.** comment. I'm just pointing out that this sort of reasoning renders all statements unprovable. Rejecting this type of rebuttal for claims of existence but not for nonexistence is inconsistent. –  Jul 10 '11 at 20:53
  • @YequalsX: And I have pointed out that when you say "The apple is an illusion" then you acknowledge the existence of the apple. It then becomes a discussion on the nature of the apple, but you acknowledged its *existence*. And no, that statement does not render everything unprovable. It only renders **non-existence** unprovable. Which is the whole point. – Lennart Regebro Jul 11 '11 at 06:53
  • @Lennart: I thought the gist was clear. You claim it is possible to prove an apple is in your hand. Using your reasoning **There is an object on your head that has that mass, but it's invisible and not affected by gravity nor inertia.** I can apply a similar objection to your proof about the apple. I can say, **Your senses deceive you and there is nothing in your hand.** You appear to think countering a proof of nonexistence with a counter statement that can conceivably be true is valid but not when it comes to proofs of existence. It's inconsistent. –  Jul 12 '11 at 20:11
  • @YequalsX: That is not a similar objection at all. You are now only reformulating "The apple is an illusion" argument, which is not valid. I see and feel the apple. You see and feel the apple. Machines we build measure the apple. If our senses deceive us, ie if the apple is an illusion, that is just a statement about the nature of apple. And it will continue to be so. This line of argument is already dealt with, and is fruitless. You have reached a dead-end, repeating it is pointless, you are wrong and I have explained why. – Lennart Regebro Jul 12 '11 at 20:58
  • My objection when you say "The apple does not exist" is "It exists, you just did not detect it". Examples of this kind of apples are for example the top quark, which was assumed to exist, but wasn't discovered until 1995. According to you, up until 1995, the top quark was proven not to exist (even though everyone knew it did, it just hadn't been found). Your objection is "Although you can detect it, it doesn't exist *really*" which is just silly word games and disintegrates into childish claims that reality is not real and other nonsense. So new question: Does the top quark exist or not? – Lennart Regebro Jul 12 '11 at 21:04
  • @Lennart: You wrote _Your objection is "Although you can detect it, it doesn't exist really" which is just silly word games and disintegrates into childish claims that reality is not real and other nonsense._ I was thinking just this when you said, **There is an object on your head that has that mass, but it's invisible and not affected by gravity nor inertia.** in response to my claim that there is no object with a mass of 200 million kilograms sitting on my head. –  Jul 14 '11 at 12:00
  • @YequalsX: Well, maybe now I have been able to explain myself more clearly? Or do you claim that the top quark doesn't exist? – Lennart Regebro Jul 14 '11 at 12:08
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Claims of non-existence are claims that X does not exist. These are indeed not provable. As you yourself point out, your first claim can be reformulated as the second claim. So is it a claim of existence or non-existence?

Well, neither.

"There does not exist a largest prime number" can not be proven as a fact, since that would require you to calculate all prime numbers, and since they are according to the statement itself infinite, you can't do that if it is true. You can't prove it false either, as this would require you to show that all numbers above X is not primes, which again requires infinite calculations.

"For a given prime number p there exists a prime number q larger than p" becomes a provable fact once you substitute "a given prime number p" with a specific number, such as 7, and you get "There exists a prime number larger than 7". This is easily provable by finding it, say, 11. But you can not prove the general statement, because it would require you to test if every number is a prime or not, which requires infinite calculations.

The claim "there does not exist a largest prime number" is therefore not a factual claim at all, but a theoretical claim, and can only be proven true or false within its own theoretical framework.

A real factual claim of non-existence are such as "There are no black swans". Famous for being proven false, by encountering black swans.

Lennart Regebro
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  • I will be more precise. It is a provable fact that within the set of natural numbers there is no largest prime. It is a provable fact that within the set of natural numbers, given a prime p there exists a prime q that is larger than p. These are factual claims. They are provable. One is a claim of nonexistence and one is a claim of existence. These claims are not different types of factual claims than saying Washington D.C. is the capitol of the United States. –  Jul 09 '11 at 15:11
  • @YequalsX: Rephrasing the question will not yield a different answer. You should instead tell me what needs clarifying in the answer. They are not factual claims, and I tried to explain why. Just stating that they are factual in that situation is not constructive. – Lennart Regebro Jul 09 '11 at 18:36
  • Your belief that statements about black swans are more real (and/or factual) than statements about the natural numbers doesn't make sense to me. I don't agree with it. All proofs, no matter what the proof is about, requires a theoretical framework with which to work with. Specifically, it requires a theory of logic and axioms about truth and a calculus on which to work with that logic. A statement about black swans is no more or less real and/or factual than a statement about natural numbers. At least, in my opinion. –  Jul 09 '11 at 18:48
  • @YequalsX: No, proofs of existence does not require a theoretical framework. If I have an apple in my hand, I have an apple in my hand. There is no theory, no framework, no logic, no calculus, just existence. – Lennart Regebro Jul 09 '11 at 20:24
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    Of course there is a framework. There's an implicit assumption that your eyesight is accurate. The logic is that your eyesight is accurate in this case. Your eyes see an apple in your hand. Therefore there is an apple in your hand. Your reasoning follows a famous syllogism. By definition of the word proof it requires a logical framework under which valid conclusions can be made. –  Jul 09 '11 at 21:06
  • One can prove a nonexistence statement for either math or physical reality if the search space is finite. I have three oranges in frnot of me. I can prove that none of these are apples by checking each one. – Mitch Jul 10 '11 at 01:05
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    @lennart: there are no proofs of the infinitude of primes that rely on testing each number to infinity. All existing proofs deal with that infinity by the natural numbers (or subsets of them) as a single entity (with many members) or by not invoking infinity at all and mimicking an infinite process (given any -finite- set, produce a new prime; this allows you can always get another prime, which is what an infinite process is). – Mitch Jul 10 '11 at 01:33
  • @YequalsX: That is neither a framework nor logic. Your definition of proof is a mathematical one which excludes proofs of existence. That is very narrow and limiting. – Lennart Regebro Jul 10 '11 at 05:20
  • @Mitch: Yes, you are correct, you can prove non-existence if the search-space is finite, which it only can be within abstract theoretical frameworks, or with the addition of things like Ockhams razor. All existing proofs of the infinitude of primes rely on an abstract mathematical framework, and are not factual proofs, but theoretical proofs. Se my answer above. A *factual* proof as much as that wording makes sense within maths (ie not much) would require testing infinite numbers. (See my answer above). – Lennart Regebro Jul 10 '11 at 05:25
  • @lennart: I think we can agree that if the search space is 'unmanageable' (like the set of swans) then one cannot prove a 'it is not the case that there exists an x such that P(x) is true' if P is consistent. But as to 'testing infinite numbers', that is not needed (and not doable anyway). See any proof of the infinitude of primes: none do such an infinite list of tests. – Mitch Jul 10 '11 at 15:41
  • @Mitch: All existing proofs of the infinitude of primes rely on an abstract mathematical framework, and are not factual proofs, but **theoretical proofs**. A **factual proof** as much as that wording makes sense within maths (ie not much) would require testing infinite numbers. (See my answer above) – Lennart Regebro Jul 11 '11 at 06:58
  • @Mitch: Or let me reformulate this: How would you prove the infinitude of primes **without using any form of algebra**? – Lennart Regebro Jul 11 '11 at 07:00
  • @lennart: sure it would be difficult since you couldn't define primality to begin with, let alone some proof. You seem to be stuck on "theoretical framework", when we've already established that there's more than one kind of existence (or nonexistence) proof, that 'factual' (experiential) proofs still need a theoretical framework, and nonexistence proofs based on 'factual' information can still be made (when the search space is finite. We've already granted that you can't prove infinitude of primes by testing the infinite set. – Mitch Jul 11 '11 at 13:27
  • The existence of and infinite number prime numbers does not preclude the existence of a largest prime number. In fact using Calculus I can show that a Prime Infinty(p) = a Prime Infinty(p + 1). Therefore a prime infinity is the largest prime number. – Chad Jul 11 '11 at 18:02
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    I would suggest that even the claim "there are no black swans" is a theoretical one to an avian taxonomist. The scientific name of the black swan (_Cygnus atratus_) captures that theoretical interest. In the framework of modern biologist, they are still swans, but in the framework of, say, romantic poetry the probably are not. – Jon 'links in bio' Ericson Jul 11 '11 at 19:55
  • @Mitch: We have established none of these things. You claim them, but they are wrong. @Jon: Poetry maybe can be called a framework. Reality is not and can not be called a framework. Although black swans may be non-existent in fiction, in reality, they exist. The difference is self-evident. – Lennart Regebro Jul 12 '11 at 07:42
  • So yet again: Whether you can define primality without calculus or not, the fact remains is that you can't prove the non-existence (or existence) of a largest prime without using the theoretical abstract frameworks we all know as mathematics. And when existence/non-existence is discussed it is factual existence, not theoretical existence. It is the factual non-existence that can not be proven, but factual existence *can* be proven. This is why the person who claims a positive factual claim has the burden of proof. – Lennart Regebro Jul 12 '11 at 07:52
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    [Just so you know, notifications don't get sent to the second or subsequent users called out by `at` symbols.] Reality may not be a framework (it's hard to know what you mean by it), but taxonomy, i.e. the way biological reality is categorized, most certainly is. It is not self-evident that color should not be a distinguishing characteristic of a species, which would mean that all swans are white _by definition_. We sadly must always view reality through fallible frameworks. – Jon 'links in bio' Ericson Jul 13 '11 at 20:51
  • @Jon: But this is still then a matter of the **nature** or **classification** of the thing, which means the **existence** of it has been acknowledged. QED, again. :-) – Lennart Regebro Jul 14 '11 at 09:14
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You just use existance elimination. Assume ∃x, derive a contradiction and you're done. For instance assuming that there exists a Barber(x) and Shave(x,y) = x shaves y leads to the conclusion ¬∃x (Barber(x) ∧ ∀y (Shave(x, y) ↔ ¬Shave(y, y))) since it's impossible that a barber exists who neither shaves himself nor doesn't shave himself according the law of excluded middle the statement Shave(x, y) ↔ ¬Shave(y, y) can't be true for Shave(c, c) ↔ ¬Shave(c, c)

Niklas Rosencrantz
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The nonexistence of something can be proven if by proven we mean logically deduced, or if mean that it cannot be conceived. We can, for instance, prove that a particular kind of thing cannot exist if, given a set of properties, we show that they lead, taken together, to a contradiction e.g. a square circle. Some proofs for the nonexistence of God, for instance, are proofs for the nonexistence of a particular kind or conception of God.

Now as far as a negative claim having a burden of proof: of course it does, because the "negativeness" of the claim is not in the making-of-a-claim (you can only make "positive" claims, that is, about a state of affairs being such and such, making it a redundant adjective). For instance, when someone makes any claim, they face a burden of proof. The popular example today among armchair intellectuals is the existence of God. In this context, some claim that they don't need to prove the nonexistence of God in order to make the claim that God doesn't exist. This is wrong. They can claim that they don't see any reason to believe in God or that they don't find the proofs convincing without burden of proof, but to claim God does not exist is logically equivalent to making a claim about a state of affairs in which God is not only unnecessary but necessarily absent. In other words, a proof of nonexistence must show that a thing cannot exist. Empirical claims of nonexistence are merely special cases constrained by time and space.

danielm
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You need to add more propositions, which may not be accepted. You suggested:

(1) A does not exist

and

(2) B exists

But (1) has nothing to say about (2) and vice versa, so you need to add another proposition. Perhaps:

(3) A or B exists

If you could show (1) is correct, then (2) is also correct via (3). But you can't prove (1) if (2) is correct unless you assert something like:

(4) Either A or B exists, but not both

This is the exclusive "or", which is much harder to show than the usual inclusive "or" found in (3). Binary choices are common in artificial environments (such as computers), but are more difficult to assert in cases where binary choices are not common. In the real world, it's harder to assert something like: either God exists or evil exists, but not both. It's not immediately clear that propositions in the form (4) are to be preferred over propositions of the form (3). Intuitively, we'd assume the reverse.

You also brought up the statement: "Nonexistence can never be proven." That can trivially be shown to be false. A standard counterexample would be the existence of a married bachelor, which is false by definition. Another example: I don't have at least a million dollars in my bank account and I can prove it. Or: I don't have a best selling book that I've written or a tattoo that says "Mom" on my arm. So you'd need to add some qualifications to that statement to make it true.

If you buy into inductive logic, flying horses do not exist because their is no evidence for them. We can never be 100% sure of that statement because a single counterexample would invalidate all other evidence, but we can be mostly certain which is good enough for most purposes.

In summary, if you convert a claim of nonexistence into a claim of existence, you must take on the burden of proving the premises you used to do the conversion in addition to proving the new claim. In some cases, the extra burden in not worth the effort.

Jon 'links in bio' Ericson
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  • I'm talking about statements that are equivalent to each other but one is of the form that something exists and one is of the form that something does not exist. Overall the question is about what criteria must a statement meet in order to be considered a claim about nonexistence. –  Jul 08 '11 at 17:48
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    @YequalsX: Then you need to provide us with a premise like (4) above that connects (1) and (2). As it stands, these propositions are unrelated. – Jon 'links in bio' Ericson Jul 08 '11 at 18:53
  • -1 You can not prove something by claiming something also unprovable. Proving something requires existance. At least until you can prove that it does not. A claim of non existance is not proof of non existance. – Chad Jul 08 '11 at 19:47
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    @Chad: I really don't see my answer saying those sort of things. It's really dealing with the need to provide more premises to make (1) and (2) equivalent. (7) is false, but we aren't really talking about whether (5) and (6) are true, but whether they are equivalent. (And now I'm confused about which part of my answer caused the -1 vote. Can you clarify your objection, please?) – Jon 'links in bio' Ericson Jul 08 '11 at 19:56
  • Your logic that 7 proves 5. You must first be able to prove 5 which would defeat is the whole premise that non existance is unprovable. Throwing science and logic as we know it into complete chaos. – Chad Jul 08 '11 at 19:59
  • @Chad: I've update my answer to address your concern. It's a _non sequitur_ to say that you must first prove (5) to prove (7). (5) is true and (7) is explicitly false! It was a hypothetical example. I hope the introduction of a truly nonexistent "crime" number, will help you see past your mathematical intuition. Of course showing (7) to be true would throw lots of things into chaos, but that wasn't in any way what I was attempting to do. – Jon 'links in bio' Ericson Jul 08 '11 at 20:22
  • @Jon: (5) and (6) are equivalent. One can not logically be true without the other being true. If there are no prime numbers then it is vacuously true that **if p is a prime number then there exists a prime number q larger than p**. The bold statement is true because the premise is false in the hypothetical situation you mentioned. –  Jul 08 '11 at 20:37
  • If **crime** numbers don't exist then any conditional whose premise involves a property of a crime number is true. –  Jul 08 '11 at 20:40
  • @YequalsX: (5) and (6) are only known to be equivalent because someone went through all the time and effort of proving the equivalence. (And I'd bet my million dollars that the equivalence was proven in order to make one easier to prove after a proof for the other was formulated.) I feel you are still trapped by your excellent mathematical intuition. What I'm trying to suggest is that you need to show your work in order to convince a lesser mathematician that (5) and (6) are equivalent. It's not obvious to me. – Jon 'links in bio' Ericson Jul 08 '11 at 20:49
  • @YequalsX: My intuition would say that any conditional whose premise involves a property of a crime number is false if the truth value can be determined at all. I find (9) to have exactly the same truth value as `1/0 = 0`. Right? – Jon 'links in bio' Ericson Jul 08 '11 at 20:52
  • @Jon: In mathematical logic any statement of the form **If A then B** is true if A is false. (5) and (6) are equivalent because there are two possibilities. 1. Primes exist and 2. Primes don't exist. If primes don't exist then it is true that there is no largest prime and it it would also be true that if p is a prime then there is a larger prime. In the case that primes do exist then both statements are false or both true. There is no situation in which one statement is true but the other is false. –  Jul 08 '11 at 21:04
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    @YequalsX: Fine. This deserves to be downvoted (on `math.stackexchange.com`). Sorry I can't help with your question. :-( – Jon 'links in bio' Ericson Jul 08 '11 at 21:39
  • @Jon the premise is thate you can not prove a claim of non existance. You are trying make a claim of non existance to prove a claim of non existance. I understand that if you prove non existance of a larger set then it also would include the limited definition. However before we can accept this as possible you must overcome the original premise that you can not prove a claim of non existance. Your claim is similar (but opposite) to a claim of god exists because he wrote the bible. You would first need to prove that he actually wrote the bible before we could accept this claim. – Chad Jul 10 '11 at 22:27
  • @Chad: My answer actually asserts that claims of nonexistence are easily provable in some situations. I gave several examples of this, so it seems somewhat unfair to say that the answer is wrong because it disagrees with the very premise the question would like to have addressed. The vast majority of this answer as it stands is trying to address the edited addition, which asserts that two statements are **obviously** equivalent, which they are not. I should never have been distracted by that tangent, it seems. – Jon 'links in bio' Ericson Jul 11 '11 at 17:45
  • @Jon I see what you are saying. And for the most part I agree with you. 1 and 2 are not neccessarily diometrically opposed. And proving 2 does not disprove or prove 1. And while I agree that proving 7 would disprove 5 (5 would be a subset of 7 so if the superset does not exist then there can be no subset). Your assumption that you can prove 7 is what I was challenging. Until you can prove 7 or even that 7 can be proven for this case it must be assumed that 7 can not be proven. And I have seen nothing that would logically allow for me to concede that it can be proven. – Chad Jul 11 '11 at 17:57
  • @Chad: The fact that (7) can not be proven prompted me to introduce (8), (9), and (10). (10) can be proven and is a much simpler path to prove (8) than (9). In fact, I assert that (9) is utterly meaningless. Note that the "crime" number example exactly duplicates the prime number example with the trivial difference that (10) is true by design. Perhaps it would have been better to introduce another premise that proves (5) but does not assert non-existence, but I'm not mathematically clever enough to do that. (And it's utterly beside the point of the answer anyway.) – Jon 'links in bio' Ericson Jul 11 '11 at 18:16
  • I reject that you have proven 10. With your example 0 crime numbers are known to exist. This does not mean that there are no crime numbers. And this is the preposition that the first (1) in the example eludes to. – Chad Jul 11 '11 at 18:59
  • @Chad: I designed the definition of "crime" numbers to be nonexistent, therefore I've utterly failed to answer your objection. I'll remove the entire distraction. [But I assert that "0 crime numbers are known to exist" means that "crime numbers are known to not exist". I'm an optimist when it comes to inductive reasoning. What's more, any proof of (10) would prove (8), but not (9), which is the whole point.] – Jon 'links in bio' Ericson Jul 11 '11 at 19:29
  • @Jon actually that is totally against the point. You can only prove that there is no known objects existance. Not that none exist. You can assume nonexistance. But that does not prove non-existance. So you can not prove crime number non existance. So if you cant prove 10 then you can not use it to prove 8. – Chad Jul 11 '11 at 19:36
  • @Chad: I think you owe us an answer to this question, therefore. – Jon 'links in bio' Ericson Jul 11 '11 at 19:40
  • @Jon I do not have a good answer. My only issue with your example is that you use an example of a claim of non existance to prove a claim of non existance with out proving the claim of non existance. But there is also a part of me that accepts that either of these are truely claims of non existance as it feels like the claim that would be non existance here is would be "No numbers exist" – Chad Jul 11 '11 at 19:51