DAN: Hi, I’m Dan the illogical scientist. That program you’re writing will never work and I can prove it.
ASOK: I don’t mean to be rude, but it’s impossible to prove something can’t be done.
DAN: It’s impossible for most people, but I’m a trained scientist.
ASOK: Did the training involve electric shocks?
-Dilbert
Libertarian will probably be able to say what I’m about to say more articulately and accurately, but here goes:
A proposition has whatever meaning is assigned to it. After all, A=A could mean “this apple is this apple” or “George W. Bush is George W. Bush.” A=A is not “falsifiable” as I understand the term; it is an axiom of logic. It is true no matter what meaning is assigned to its parts.
A proposition which may be true under some interpretation but false under another interpretation (e.g. (A or B) => B could be true, but might not be depending on the truth values of A and B) is often called satisfiable. This is the only thing I could think of that might be justifiably called a “falsifiable” proposition, although I’ve never heard it called that.
Unfortunately, it has been proved that there are some statements that are true within a system but cannot be proved to be true within that system. (VERY rough description of Godel’s famous Incompleteness Theorem; someone please elaborate if you feel like it) This would seem to imply that there are also false statements that cannot be proved to be false (just the negation of the true proposition that can’t be proven). Yet these propositions whose truth value cannot be proved within a system do have meaning within the system.
Not sure how this relates to not being able to prove a negative though.
So, er, has this helped clarify anything in any way?
Yeah, DEDUCTIVE negatives are often proven, under the “Reductio ad absurdam” technique. Mainly, (A & ~A) or any contradiction at all cannot arrive from a correct proof. Correct me if I’m wrong, but if both (the apple is red) and (it is not true that the apple is red) are true at the same time, then all logic and indeed, all language, has gone to hell.
If I want to prove a negative (~A), I just have to show that A results in a contradiction. Then it is obvious that A cannot exist, and that A’s opposite, ~A, exists.
Arguments, in order to be accepted, must be both VALID & SOUND.
For example:
Conclusion: the world is square
This is a VALID argument. If the premise is true, then the conclusion MUST be true. In an argument that is valid, the truth of the premises FORCES the truth of the conclusion. We know that the premise is not true, and so we don’t have to accept the conclusion. However, it’s still a valid argument.
c: An apple is not a pear
This argument is both SOUND and VALID. A sound argument is an argument which is valid and which also has true premises. It is valid because the premises force the conclusion, and also because the premises are incontravertably true.
Therefore, SOUND arguments are VAlID arguments in which the premises are true. If an argument is valid and the premises are true, then the argument MUST be true.
Conclusion: The pres of NK is not my neighbor
This is a valid argument. However, it is only sound if its premises are true. If one doesn’t believe the premises, one can’t believe the conclusion. This is one problem with logical arguments: if you refuse to believe any premises, then no one can prove anything. However, if you refuse to believe that A = A is true, then you’re going to have a tough time with the world in general.
However, those are for deductive arguments. INDUCTIVE arguments are arguments that would be IMPROBABLE that the conclusion was false if all the premises are true.
C: The speaker is a male
You can concede that the argument is PROBABLY true, but is definitely not waterproof, and is not sound or valid because those terms only apply to deductive arguments. A COGENT inductive argument is one that is inductively strong with all its premises true.
C: The daughter is biologically mine
Let’s assume that thge premises are true on this one. This is COGENT because it’s probably true, all the premises are true, but there is still room for error. You could claim that the neighbor/NK thing is inductive/cogent rather than deductive/valid.
Moral of the story: make sure people believe your premises, and make your arguments deductive, not inductive. Then you should be able to prove anything 
P.S.
A statement in logic can be as follows: A & D
Unless there is an English meaning assigned to that statement, it means nothing.
However, let’s say that it means: Apples exist and Dragons exist.
This is easy to disprove. If you believe that “Apples exist” and that “Dragons don’t exist” then you can easily say that they both don’t exist.
Therefore, since (A & D)is false, then ~(A & D) must be true. It is an axiom on logic that if A is false, then ~A must be true, and vice versa. In formal logic, it is possible to prove things false.
However, without an interpreation assigned to it, A & D is neither true nor false.
I hope that’s what you were looking for.
Ugh, just when I thought I was getting it.
When you say “both”, do you mean that I can prove that:
apples don’t exist AND dragons don’t exist
-or-
apples and dragons, as a combination, don’t both exist?
Now this is particulary painful to think about. This seems to say that if I believed (even mistakenly) that both dragons and apples existed, that the statement would be true for me. Even though I’m wrong, you would logically have to disprove my assumptions. Doesn’t this put us on a course of circular logic and/or infinte tangents?
Sorry if it was confusing.
~(A & B) means that is it false that both apples and dragons exist-basically, only one can exist out of the 2. (You can take that to mean that apples don’t exist and dragons do, but to prevent that you’d need a more complicated sentence which would be useless for a simple example of negation.) Obviously then, the simple (A & B) just means that both exist, while
~A & ~B means that neither apples nor dragons exist.
And in response to your philosophical question, in formal logic there are absolute truths, absolute falsities, and indeterminate statements, but nothing wishy washy.
“A” would be indeterminate.
“Susie has red hair” is indeterminate. Unless you know something about Susie, you can’t confirm or deny this statement. It all depends on the interpretation. If you interpret “susie” to mean the susie you know that has blond hair, then it’s false. If you only know a susie with red hair though, then it’s true.
“This is a statement” is true. It cannot be false, no matter what your interpretation is. Likewise “an apple is an apple.”
Anyway, the philosophical question of “perception” has no place in formal logic. There is no point in proving anything based on a premise that half the world is going to disagree with. Therefore, most arguments are originally based on the most simple truths, like identity, contrapositives (if A is true then ~A is false), conjunction extraction (if A&B is true then A is true and B is true), etc. and built up from there.
However, if you really want a headache, think about one of the fundamental building blocks of formal logic: modus ponens. The argument basically looks like
c: q
and sounds like “If it rains, the grass will be wet. It is raining. Therefore, the grass is wet.” Assuming the premises are true, then the conclusion has to be true. right?
It sounds simple and reasonable, yet THERE IS NO LOGICAL OR FORMAL MATHEMATICAL PROOF FOR THIS STATEMENT. It is used in basically every proof for complex logical notions, yet can’t be proved by the system that it holds intact.
Isn’t logic wonderful?
If => symbolizes implication, then (A or B) => B is true if B is true, regardless of the truth value of A. This is called a disjunctive propostion.
At its root, logic (and any other epistemological system) is a house of cards, predicated on the arbitrary assumption of its validity. Godel understood intuitively, and proved formally, that you cannot escape a petitio principii (begging the question) using any logical system. This is actually something that most of us understand intuitively, and we will be quick to point it out in arguments against people whose views we find to be primative. For example, consider this argument:
[li] The Bible is the inerrant word of God. This is true because the Bible says so.[/li]
People will likely jump all over this, pointing out correctly that it is begging the question. However, they might fail to see the same, though more subtle, error in this argument:
[li] The statement I made is true because I proved it by deductive logic.[/li]
Implicit in this argument is that deductive logic is valid. The arguer is relying on logic being valid because it is logical. Thus, the validity of deductive logic rests on one of its own fallacies. To accept the validity of logic requires a willingness to “overlook” this flaw.
Or the other way around, where we assume the opposite to be true and then contradict our assumption. A classic example is proof that the square root of two is irrational.
Given a square with unit sides, the diagonal of the square is x, such that x = 1[sup]2[/sup] + 1[sup]2[/sup]. Therefore, x[sup]2[/sup] = 2, or x = 2[sup]1/2[/sup].
Hypothesis: Where x = 2[sup]1/2[/sup], x is rational.
Let x = p/q, where p and q have no factors in common, thus giving us a rational number reduced to its lowest form.
x[sup]2[/sup] = p[sup]2[/sup]/q[sup]2[/sup].
p[sup]2[/sup] = x[sup]2/sup.
By substitution (x[sup]2[/sup] = 2), p[sup]2[/sup] = 2(q[sup]2[/sup]).
Taking as axiomatic that 2a must always be an even number for every value of a, i.e., 2a must always have 2 as a prime factor, we note that a[sup]2[/sup] must always be even, since it must always have a prime factor of 2[sup]2[/sup]; therefore, let p = 2a.
Thus, (2a)[sup]2[/sup] = 2(q[sup]2[/sup]).
4a[sup]2[/sup] = 2(q[sup]2[/sup]).
2a[sup]2[/sup] = q[sup]2[/sup].
q[sup]2[/sup] therefore must always be even, for the reasons cited previously.
But if p and q are both even, then each must have at least one factor of 2 in common. Because this contradicts our assumption that p/q is reduced to its lowest form, and because all ratios can be reduced to their lowest forms, our assumption must have been incorrect.
Therefore, the square root of 2 cannot be rational.
It is important to note that not all schools of logic accept this method of proof, that is, they do not allow that Not(Not A) = A.
For the record, that is not an inductive argument, but a deductive one with a non sequitur conclusion.
Not A is not the contrapositive of A, but the negation of it. A contrapositive is an implication that is both negated and permuted, thus, if A => B, then its contrapositive is Not B => Not A. Its converse is B => A, and its inverse is Not A => Not B.
Kim Jong II’s neighbor is not Kim Jong II.
I’m positive.
When I appealed a charge for insufficient funds to cover a check once, the bank did not find my assertion that you cannot prove a negative either amusing or convincing. They were certain the money wasn’t there. And they won.
Well, OBVIOUSLY, there are some negatives that can be proven easily.
A simple tape measure will prove that I’m NOT 7 feet tall.
A simple blood test can prove that John is NOT the father of Mary’s baby.
But there are certainly many situations in which proving a negative is quite difficult. In a murder trial, a defendant would be in big trouble indeed if he had to PROVE beyond a reasonable doubt that he did not commit a crime he’s accused of (hence, the burden of proof is on the prosecution).
And, given a problem where there are multiple factors at work, even the most rigorous scientist may have a hard time determining conclusively that one SEEMINGLY unimportant factor is not crucial.
I no longer have any logical view of where this thread is going, or has already gone 
Well, here’s the short of it:
Yes, you can prove a negative so long as its scope is not beyond your universal set, but that holds equally for proving a positive, and is not some mysterious attribute of negation. The notion that you cannot prove a negative might be of modern origin from strong atheists in usenet threads as shown in the link I first gave you.
You will find no such weird constriction as the inability to prove a negative per se in reputable formal logic texts.
Okay?
Many responders have considered this as a question about
formal logic. Considering this a question about scientific
reasoning might be better.
In science one usually formulates a null hypothesis,
such as “This drug has no effect on cholesterol level”, and
then gathers evidence to try to disprove it.
If one has enough evidence against the null hypothesis, then
the custom is to reject the null hypothesis in favor of
an alternative hypothesis. In this example this would
mean saying “This drug does have an effect on cholesterol
level.” It is not the custom to conclude the null
hypothesis is true if there is not enough evidence to reject
it. You may simply not have collected enough evidence. In
this example, you might need to test the drug on more
patients in order for the statistical tests you do on the
data to have enough power to reject a false null hypothesis.
So I think the atheist is saying “The null hypothesis should
be that there is no god. It is the responsibility of the
theist to gather enough evidence to show it should be
rejected. No amount of evidence can prove the null
hypothesis.” And so we arrive at the statement that “No
amount of evidence can prove (the negative): There is no
god.”
Why the null hypothesis should be this and not “There is a
god” is a matter of debate, however. (A debate I am not
prepared to engage in myself.) Reading one of the links
above, it seems the atheist is saying that if you
consider the null hypothesis to be that there is a god,
then you also must be willing to consider the null
hypothesis that there is a unicorn, the null hypothesis that
there is a leperchaun, etc. And I think the argument is
that this is not a sensible way for philosophers to proceed.
With the drug analogy above, it is clear that if the FDA
were to assume that a drug should be used by the public
until enough evidence is gathered to show that it
shoudln’t, medicine would be in a terrible state.
I was once arguing with someone over the validity of the Bible and he made the common fallacious claim that the Bible is the word of God “'cause it sez so right here!” I got a piece of paper and wrote on it, “This is the word of God.” I then said, “Prove this is not true.”
“I know that’s not the word of God because I saw you write it.”
“So you believe the Bible is the word of God because you weren’t there when it was written?”
He couldn’t answer that and stalked off in defeat.
Contrary to assertions made in this thread, the idea that one cannot prove a negative is not new. It is apparently at least four centuries old. This site http://www.panspermia.org/cantest.htm seems to credit (or blame) Pasteur.
(I have yet to figure out how to do block quotes.)
“In the nineteenth century, Louis Pasteur observed that one cannot prove a negative. His principle that life comes only from life is a prime example of a negative principle. Pasteur could defend it only by demonstrating that each counterexample — each alleged example of life from nonlife — was actually the result of incorrect, unsterile procedure. Of course, one experiment could still falsify Pasteur’s principle.” (footnote indices removed.)
This is a example of what Libertarian was saying before people got side tracked into a discussion of formal logic. Pasteur was concerned with the scope of the task of proving his assertion, not its nature.
It’s interesting to note that Pasteur’s principle that complex living things do not spontaneouly erupt from non-living things is generally accepted today, not because Pasteur or anyone else proved it by exhaustive search, but because no one has ever come up with a counter example despite the numerous opportunities available to do so by scientists. In other words, no one has been able to prove the negation of Pasteur’s assertion, so it is accepted as true. That acceptance is assisted, of course, by research that shows that collections of living things have common attributes. Once Pasteur had shown that rats do not spontaneously appear in dirty rags, few reasonable people would have doubted Pasteur’s principle as it applies to mammals. So a completely exhaustive search was not necessary.