How to prove a negative can't be proven?

How do you prove that a negative can't be proven, since it’s negative?

 

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What?

 

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The statement “a negative can't be proven” is negative. So how do you prove that the statement is true?

 

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Sorry, I should have been more specific. What I mean to ask is what does the statement "you can't prove a negative" mean? I have never understood that statement.

 

Do you mean negative or negation!

Here's an example. It originally involved old greeks, but I cant't remember their names so I'll change it.

Arnold makes a statemant in a speech whilst running for the position of governer of California that All Austrians are liars. However, Arnold himself is Austrian, and so this statement must be a lie, therefore the negation of this statement is "All Austrians are not liars". But Arnold lied in his original statement.

It goes something like that.

 

Maybe Arnold no longer considers himself an Austrian?

I'm talking about negative statements like "pigs can’t fly." An explanation I found: A negative cannot be proven because if something is absent it would not be there to prove its absence.

 

rather than a proof that a negation cannot be proven, i give you a counterexample, of a negation that i know to be provable: i can prove that 1 does not equal 0.

 

Yes of course this is true, mathematical negations are usually trivial due to the rigourous logical foundations of definitions of mathematics. However spoken language tends to be full of logical inconsistancies. The error in the statement given above is I think the fact that it is said that ALL Austrians are liars, not most or some. Elements of a set tend to have common features or characteristics, and so when we compose the set of all elements that are Austrians, we tend to think that other characteristics are also shared by all elements of the set, which is not true. It is an error in human logic and perception.

 

Perhaps there are more conditions that need be met for a negative statement to be unprovable. Whereas numbers are defined as the same value always, a pig could fly off the moment you look away. We need some logicians in here.

(Just read ryans but I'll post this anyway.)

 

clearly you do need more conditions, i have given you a counterexample to your statement.

well, now you are talking about experimental facts. these can never be proven, whether they are negative or positive, they can only be disproven.

i can never prove that all pigs (in the universe) can fly. i can also never prove that all pigs cannot fly.

this is because there are conceivably an infinite number of pigs in the universe, and i only have a finite lifetime.

logical statements are different: they assume certain axioms, things which can never be proved by observation.

so the mathematical notion of proof is quite different from the scientific notion of proof.

if one of your axioms is "No mammals can fly" and another is "Pigs are mammals", then you can construct a mathematical proof that pigs cannot fly, based on those axioms. but the axioms might not be physical, right? (hint: bats are mammals)

 

In mathematics, there are existence and nonexistence proofs. Some differential equations can be proven to have a solution although it might be impossible to find it (other than numerically). Some equations can be proven to have no solution (EG: Fermat’s Last Theorem).

Statements in natural languages are notoriously difficult to deal with logically. People often have difficulty in agreeing on the semantics of natural language statements.

In general, self referential statements are considered invalid and meaningless for purposes of logic. They generally cause at least some confusion.

• Every statement I make is a lie (no truth value can be assigned). Every statement I make is true (can logically be true or false). The latter would be considered false (good opinion) by most, but it might be very difficult to prove if I do not say much, or you cannot list everything I ever said.
• This English sentence is difficult to translate into French. A literal translation is easy, but seems like nonsense. Cette sentence francois est difficule transxxx au anglais. (Excuse my bad French) The given translation captures the spirit, but is it a valid translation?
• Either you are the person reading this sentence or the person who wrote it. Which are you? This is easy for you to answer, but I have a problem with it.

Self referential sentences and subtle circular definitions are considered verbotten for purposes of formal logic. They should similarly be ignored except for informal discussions.

Statements like pigs cannot fly are not generally made out of a context involving somebody who claims that they can fly. The proper approach to somebody making an unsupported claim is not a mere negative statment. A better response is:Your statement lacks credibility in the absence of evidence, proof, or at least a reasonable argument. The burden of proof is on you, the person making such a claim, not on me, the person not willing to believe.

 

In general, proving or disproving a negative revolves around the ability to gain facts on the total set of possible outcomes.

If one establishes that a coin must land on either heads or tails, but not both then it would be possible to make provable negative statements about the coin. A coin with its heads side facing up does not also have its tails side facing up.

In general, proving a negative is (impossible)difficult for cases when the total set of outcomes is unknown. I would think that given a specific negative assertion one wanted to prove one could establish its provability based on the known information.

Cheers!

 

Wittgenstein used in his writings an example "cat is not on the mat" proposing that "not" is a logical operator which doesn't form a part of what is being described. An example is a board in the park with a picture of a dog crossed with a red line. Red line in the meaning of "not" is not a part of the picture (it operates outside of the picture) W. believed that the world could be understood from the description of this world in language (the description is thus a mirror <i> a picture</i> of the existing reality) You cannot paint "nothing" (you cannot draw a cat not being on the mat - you can draw the cat only where it is ...thousands of other cats and amimals are not on the mat too.. ) Negative is then a logical operator (stands outside the reality - reality is never negative)

 

I am still not sure what a negative really is. And google is not my friend

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OK, I think I now understand better why a negative can't be proven and also why "a negative can't be proven" is a true statement despite being negative. Thanks all for your help.

 

i still believe that it is, in fact, not a true statement at all. see my counterexample above.

 

I meant a true statement in some, not all, cases.

 

because it is not positive, if it was positive, you would be able to prove it!

 

Well, there is a standard proof that the square root of 2 is not rational. This is a negative sentence that has been proven.

 

Off the subject of mathematics, the anti-trust laws as of the 1940's (and perhaps to this day) required an impossible proof of a negative.

At that time my father ran a business which bought forged steel products and sold them under his registered trademark. He had competitors who sold essentially the same product, differing only in the trademark stamped in by the forging dies.

The prices charged by my father and his competitors seldom differed. This was not surprising. They both bought from similar (some times the same) large manufacturers who had huge forging equipment. They both sold through what were called mill supply houses (hardware stores for heavy industry). A significant difference in price would result in most potential customers buying from the competitor.

The anti-trust laws stated that if essentially the same products were sold with little price differences by all those supplying the products, then there was prima facie evidence of a conspiracy to fix prices. Therefore, the burden of proof would be on my father and his competitors. They would have to prove that they never met and conspired to fix prices.

How could you ever prove that you never met or corresponded with a particular person in order to fix prices? You might have a chance if you lived in different cities, were illiterate, and could prove that you never left your home town.

It never actually happened, but my father often discussed the possibility with his competitors at trade shows. He was friendly with them. The potential fines were enough to bankrupt my father. He believed that those provisions of the anti-trust laws would never hold up in court, but who knows?