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Negative propositions / Proving a negative

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Guest CrookedTimber

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Guest CrookedTimber

Thank you in advance for taking the time to answer my question.

I am new to Objectivism and have some visceral problems with it. Obviously, that may just be due to the limitations in my exposure to its principles or my misunderstanding of them altogether.

This is my question: How does Objectivism make sense of negative assertions about physical reality (eg., "atoms are not the most basic constituents of physical reality)? Obviously, such statements have cognitive significance, and we can understand what is meant by them.

My superficial understanding of Objectivism is that it denies that we can have real knowledge about things that do no exist in physical reality. An assertion about something that can't theoretically be pointed to directly would be "arbitrary" under the terms of its system.

By definition, a negative concept does not point to anything directly.

So, what exactly are negative concepts "about"? What is the "substance" or "nature" of a denial?

(You could say that a negative assertion is just a re-translation of a positive assertion [eg. sub-atomic particles are the basic constituents of physical reality], but doesn't that beg the question? The two statements are two different kinds of intellectual acts. In short, what __is__ "negativity"?)

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Read Introduction to Objectivist Epistemology, and the section in the apendix where Ayn Rand discusses the concept of "nothing".

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Negation is the denial of an affirmative statement. To assert that "A is not B", is to assert that "it is false that A is B". A statement is false if it contradicts the facts of reality. Thus, negation essentially reduces to contradiction.

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  • 3 years later...

*** Merged topic ***

I always hear, especially in discussions about God, that it is impossible to prove a negative. I always just took that statement for granted, but now that I've thought about it, is it really true?

Could somebody explain it? When we say, "prove a negative" does it also include proving a negative WRONG? Or does it just mean proving a negative true?

For example, if somebody said, "The computer you're using doesn't exist," I could prove that wrong by saying, "well what am I typing on?"

I think I'm just confused about the terminology...

Edited by softwareNerd
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You can logically prove a negative only if you can show that the positive claim is self-contradictory. Then again, you can only logically prove a positive claim by showing that its denial is self-contradictory, which is a pretty high bar for any theist to clear.

However, when you mean by "proof" a system by which to decide which beliefs to hold, you can prove a negative by showing that you have no reason to believe the positive. Because there is no reason to believe in god, in a very straight-forward and simple way you prove that one ought not to believe in god.

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I always hear, especially in discussions about God, that it is impossible to prove a negative. I always just took that statement for granted, but now that I've thought about it, is it really true?
There is a small germ of truth to the claim (and remember that germs are small anyhow). First you have to sort out what "a negative" means. People may erroneously use that expression to refer to a sentence with a negative word in it like "no", "not", "never" and so on. Here, then, is "a negative":

This letter >A< is not "B".

Clearly, I can in fact prove this negative (see below on "prove"). The only sensible meaning behind the slogan that I can think of is that you cannot prove a negative existential over an unbounded domain (FYI, that's untrue). For example, you can't prove that unicorns don't exist although you can prove that unicorns don't exist right now in the Oval Office (but you can't prove that there isn't a horse's ass there right now). The essence of the "can't prove a negative" canard is that falure to find a thing in a particular place or time does not mean that the thing does not exist at some other place or time, and you cannot inspect all places and all times.

It matters what you think a "proof" is. I assert that there does not exist any number in the set of rational numbers which is equal to the square root of -2. For example, -1.4 is not that number. I could continue inspecting numbers that aren't it until the cows come home, and I'll never produce it -- clearly, I must be incompetent at guessing. A fancier way to do this is to show that such a thing would be contradictory. You could then prove that there are no cold-blooded mammals that reproduce asexually, and you can similarly also show that god cannot exist.

The urge to prove "a negative" is misguided and arises from the irrational admission of the arbitrary into the sphere of logic, by certain types of people. Before considering how to prove that there are no unicorns, you should inspect the evidence that there are unicorns, and unsurprisingly you'll find that there is none. Rational people don't give any consideration to arbitrary claims, they only consider claims for which there is evidence.

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I always hear, especially in discussions about God, that it is impossible to prove a negative. I always just took that statement for granted, but now that I've thought about it, is it really true?

Could somebody explain it? When we say, "prove a negative" does it also include proving a negative WRONG? Or does it just mean proving a negative true?

For example, if somebody said, "The computer you're using doesn't exist," I could prove that wrong by saying, "well what am I typing on?"

I think I'm just confused about the terminology...

Proving a negative is incorporated within the larger principle of logic: the burden of proof. I wrote this in a paper: "The “burden of proof” principle states that the person who asserts the existence of something must present evidence which supports his claim, and further that a person must never attempt to prove a negative."

The burden of proof principle, to my understanding, is itself based on the logical rule that contradictions don't exist (Aristotle's Law of Non-contradiction, which could be considered a corollary to Objectivism's Identity axiom).

It is a contradiction to present evidence which follows from the non-existence of something. If something doesn't exist, it has no effects upon reality; no effects "follow" from its non-existence, because in order to have effects on reality, it must have already existed in reality. You should not "prove a negative," the burden of proof principle says, precisely because it cannot be done--you literally could not present evidence which follows from something's non-existence.

In your example, your opponent's assertion is contradictory; he's conceding that you're using it, but argues that it doesn't exist. I would consider it an arbitrary assertion, not even a false one, and so not worth my time. He's also stealing the concept of "using."

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From the article Upton linked:

Yes, Virginia, you can prove a negative, and it’s easy, too. For one thing, a real, actual law of logic is a negative, namely the law of non-contradiction. This law states that that a proposition cannot be both true and not true. Nothing is both true and false. Furthermore, you can prove this law. It can be formally derived from the empty set using provably valid rules of inference. (I’ll spare you the boring details). One of the laws of logic is a provable negative. Wait … this means we’ve just proven that it is not the case that one of the laws of logic is that you can’t prove a negative. So we’ve proven yet another negative! In fact, “you can’t prove a negative” is a negative — so if you could prove it true, it wouldn’t be true! Uh-oh.
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I think there may be some misunderstanding as to what it means to say that "one can't prove a negative." Proof is based on evidence that is logically organized into a non-contradictory chain of thought. If you have no evidence, then you can't prove anything, which is why you cannot prove that something does not exist. If it does not exist, then you have no evidence to point to and reason from in order to show that it doesn't exist.

There cannot be any evidence that a non-existing pixie sitting on my shoulder is telling me how to write this essay. You can look at me or take a picture and see that there is no pixie sitting there, but that is not proof that it does not exist -- because you don't have any evidence to come to the conclusion that it doesn't exist.

That's why the onus of proof is on he who asserts the positive. If someone wants to claim that there is a pixie sitting on my shoulder telling me how to write this essays, then it is up to him to show me (or anyone else) the evidence. That lack of evidence doesn't prove anything.

Similarly, one cannot say that it has been proven that the Loch Ness monster does not exist or that aliens from outer space do not exist. What evidence do you have that they don't exist, and what would you count as evidence in those cases anyhow? There are no footprints, DNA samples, photographs, or other such evidence of the non-existence of any of this. I mean, if I take a picture of my living room and one does not see the Loch Ness monster in that photograph, does that prove that it doesn't exist? What if I take a picture of the lake itself and there is no Loch Ness monster in it, does that prove that it doesn't exist? What if I don't see an alien when I go outside at night in a vacant wooded area, does that prove that aliens don't exist? How many such empty photographs must I take to show that neither the Loch Ness monster nor aliens exist?

Once one has evidence, one can prove that such and such is the case, but the lack of evidence does not prove anything -- it just means that you don't have evidence, and if you don't have evidence then it can be treated as the arbitrary.

So, the existence of God, the Loch Ness monster, or aliens is arbitrary until there is evidence. The lack of evidence doesn't prove anything.

Edited by Thomas M. Miovas Jr.
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I think there may be some misunderstanding as to what it means to say that "one can't prove a negative." Proof is based on evidence that is logically organized into a non-contradictory chain of thought. If you have no evidence, then you can't prove anything, which is why you cannot prove that something does not exist. If it does not exist, then you have no evidence to point to and reason from in order to show that it doesn't exist.

...

That's why the onus of proof is on he who asserts the positive. If someone wants to claim that there is a pixie sitting on my shoulder telling me how to write this essays, then it is up to him to show me (or anyone else) the evidence. That lack of evidence doesn't prove anything.

I agree with this. I think this is very similar to my point, if not the same thing.

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This has probably been answered before, but what about proofs which do not require direct empirical observation to be true? Instead, for example, mathematical proofs? I may be misunderstanding but it seems to me that if you can deduce certain laws from the empty set (which you can) that not all proofs have an empirical root. I guess we could call them features of logic or rules about the structure of logic rather than facts about reality. But I'm still confused. Thoughts?

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This has probably been answered before, but what about proofs which do not require direct empirical observation to be true?
Such proofs fall into two categories. The first is "proof of the true"; they do require observation. (There's no real meaningful between direct and indirect observation, and there are no "non-empirical observations"). Typically, though, those foundational observations are not mentioned and as a kind of shorthand they are just assumed. To take a numeric truth, "3+2=5". To prove this, you have to among other things establich concepts such as "successor", "combining", "equivalence", "two", "three", "five" etc. This has to be done by observation, for instance generalizing from observing two kitties and two puppies and two rats and so on.

There could be a symbolic-replacement sequence using meaningless symbols like ₥ or ↂ, but even if you can construct a well-formed derivation using certain rules and these symbols, the propositions have no meaning (no reference) so they aren't true, which stretches the concept of "proof".

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This has probably been answered before, but what about proofs which do not require direct empirical observation to be true? Instead, for example, mathematical proofs? I may be misunderstanding but it seems to me that if you can deduce certain laws from the empty set (which you can) that not all proofs have an empirical root. I guess we could call them features of logic or rules about the structure of logic rather than facts about reality. But I'm still confused. Thoughts?

By the way, in my earlier reply to the thread, I was not trying to imply that no one had given the right answer, it's just that sometimes things have to be re-iterated to make them clear.

I haven't studied set theory, so I can't comment on it specifically, but basically there can be no operators or operations or processes without something there to operate on or to process. In other words, there is no logic without content -- logic, in Objectivism, means non-contradictory identification, and if there isn't anything there to identify, then logic isn't involved.

To take a simple mathematical example, say addition, while 0+0=0 is true, that is not based upon the operator addition operating on nothing. It comes about due to observation -- i.e. if one has an empty box and doesn't put anything in it, then it is still empty.

Sometimes I think the way operators are handled in mathematics is very Kantian. That is, they are handled as if our mind is able to process without there being anything to process, or rather that the operator is a function of the mind in itself without making any reference to existence.

The process of addition came about because it was observed that if you put one apple on the table, then put on another one, then one has two apples on the table. Doing abstractions from abstractions, we were able to come up with simple arithmetic, and then algebra, and then more complicated mathematics. However, it is not as if someone came up with a process that didn't have anything whatsoever to do with reality in any way whatsoever, and somehow, voila! someone found a way to use it. If he came up with a process not based on reality, then what was he processing in order to come up with the process? This is like Kant saying that our minds come up with things that have nothing to do with the real neumenal world, and that we can only be aware of things due to our processing but this has nothing to do with reality.

All of the higher level mathematics, including vector calculus and partial differential equations (that I have dealt with) and set theory (which I haven't had much to do with) ultimately come down to observations or abstractions derived from observations. There is no mathematical operation of that which does not exist or that is not derived from evidence and observation, both of reality and what man's mind can do.

Now, you can probably prove that a set is empty, just as one can prove that 5+(-5)=0 but that is not the same thing as operating on nothing whatsoever. One cannot be conscious if there is not anything there for one to be conscious of.

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I always hear, especially in discussions about God, that it is impossible to prove a negative. I always just took that statement for granted, but now that I've thought about it, is it really true?

Could somebody explain it? When we say, "prove a negative" does it also include proving a negative WRONG? Or does it just mean proving a negative true?

For example, if somebody said, "The computer you're using doesn't exist," I could prove that wrong by saying, "well what am I typing on?"

I think I'm just confused about the terminology...

Your formulation of the problem is not correct. The logical issue is not one of proving a negative statement false, but of proving false a positive statement for which no evidence is given. The archtypical example is "God exists" say the religionists. "If you don't believe God exists, prove he doesn't" they say. "Well," you say, "where is the evidence he exists?" There is none. The issue is that one cannot prove the nonexistence of a positive assertion for which no evidence exists.

As you demonstrate, it is quite easy to prove "the computer you're using doesn't exist" is false by pointing to evidence. But that is not the issue of the impossibility of proving a negative. How would you prove the negative if there was no evidence for the positive statement that "you are using a computer"?

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I always hear, especially in discussions about God, that it is impossible to prove a negative. I always just took that statement for granted, but now that I've thought about it, is it really true?

Could somebody explain it? When we say, "prove a negative" does it also include proving a negative WRONG? Or does it just mean proving a negative true?

For example, if somebody said, "The computer you're using doesn't exist," I could prove that wrong by saying, "well what am I typing on?"

I think I'm just confused about the terminology...

There is a negative and I will prove it. There do not exist positive integers m, n with no common factor such that (m/n)^2 = 2.

Proof. Suppose such integers existed. The m^2 = 2*n^2. Hence m is even. Let m = 2*k. Substituting, we get

4*k^2 = 2*n^2. Dividing by 2, we get 2*k^2 = n^2. But this implies n is also even which contradicts the assumption that m, n have no common factor (other than 1, of course). QED.

A negative has been proven.

Bob Kolker

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