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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 8 '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 8 '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. –  YequalsX Jul 8 '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 8 '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? –  YequalsX Jul 8 '11 at 17:54
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6 Answers

up vote 8 down vote accepted

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.

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+1 excellent way of explaining it I think. –  Chad Jul 8 '11 at 19:27
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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. –  YequalsX Jul 8 '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 8 '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 Ericson Jul 8 '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? –  YequalsX Jul 8 '11 at 20:32
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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.

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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. –  YequalsX Jul 8 '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 8 '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 9 '11 at 6:51
    
@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. –  YequalsX Jul 9 '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? –  mixedmath Jul 9 '11 at 15:27
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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.

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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. –  YequalsX Jul 9 '11 at 15:11
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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. –  YequalsX Jul 9 '11 at 21:06
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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 1:33
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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 Ericson Jul 11 '11 at 19:55
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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 Ericson Jul 13 '11 at 20:51
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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)

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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.

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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.

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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. –  YequalsX Jul 8 '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 Ericson Jul 8 '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 8 '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 Ericson Jul 8 '11 at 19:56
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@YequalsX: Fine. This deserves to be downvoted (on math.stackexchange.com). Sorry I can't help with your question. :-( –  Jon Ericson Jul 8 '11 at 21:39
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