Proving a negative?

[DinosaurKnight]

A quick question I'd like to run through the Randi forums. I know there are a lot of smart people here; maybe someone can help me out.

It's often said that you can't prove a negative. This doesn't seem right to me. It seems to me that it's very easy to prove a negative in some cases. But I've had no formal training in logic, so it's possible (maybe even likely) that my line of thinking is flawed. I've had people try to tell me that it is, but they haven't been able to tell me why. At least, not very convincingly.

It goes like this: I propose a geometric figure called a "squircle." It's a two-dimentional figure which contains all of the properties of both a square and a circle.

However, the properties of a square and circle are mutually exclusive. They cannot simultaleously exist in the same figure. A square, by definition, has four corners. A circle, by definition, has zero corners. Therefore, a squircle must, by definition, have four corners and zero corners. This is clearly impossible. It's impossible on Earth, it's impossible on Mars, and it's impossible in the Andromeda galaxy.

Therefore, the squircle does not exist. I can say this with certainty, because four does not equal zero, and four cannot equal zero under any circumstances.

I believe that I just successfully proved a negative. Am I mistaken? If so, why?

Thanks in advance for any help.

[rjh01]

The problem with proving a negative is that there may be a possibility you have overlooked. For example if I tell you 1+1 =3 in certain conditions you cannot prove me wrong. But I can prove this true.


Solution
Increase the value of 1 to 1.4, but show only whole numbers.

[TobiasTheViking]

But God could make a squircle couldn't he? i mean in the bible PI is exactly three. So.... Why not?

Can you proove that God can't make a squircle?

[Curnir]

Prove that I didn't shoot JFK.

-You weren't born yet.

-Well I could have used my timemachine.

-No such thing.

-Prove it.

etc.


mmm Red Dwarf.. now that was a nice series.

[TobiasTheViking]

hehe, yeah, great episode.

[Beady]

I believe that I just successfully proved a negative. Am I mistaken? If so, why?

Thanks in advance for any help.

I'm not sure whether to call your proposition a false dichotomy, or a strawman. Probably the former. Thing is, you have invented a situation which exists solely for the purpose of proving the situation. As such, it is completely circular and therefore illogical.

It is also unscientific. One of the characteristics of science is that it is concerned solely with the natural world. As you have demonstrated, the "sqircle" (and any of the constructs by M C Escher) cannot exist in the natural world. Therefore, the squircle is not a scientific proposition.

Therefore, your proposition proves nothing, either logically or scientifically.

[Operaider]

Basicly you can show that something is pretty much impossible as far as we know things to be, but you can never prove that it couldn't happen due to some some event we could never foresee.

[Anacoluthon64]

The essential difficulty with proving a negative is that you would have to examine every single instance that ever existed, exists, or ever will exist of the phenomenon in question in order that the proof is valid. For example, consider trying to prove the statement about yourself, "I am not a thief." In order to prove it, you would have to account for every single microsecond of your life - past, present and future - and corroborate your assertions in this regard. On the other hand, to prove the opposite statement, i.e. "I am a thief," you need only prove a single instance where you stole something. Do you see the asymmetry?

'Luthon64

[arthwollipot]

Oh and just as an aside to the main topic, the definition of a circle is not that it has no corners - it could equally have an infinite number of corners. The definition of a circle is the collection of points that are equidistant from a common point.

Back to the topic...

[LW]

i mean in the bible PI is exactly three.

You may find this page (http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm) illuminating.

[JohnF_73]

I propose a geometric figure called a "squircle." It's a two-dimentional figure which contains all of the properties of both a square and a circle.

However, the properties of a square and circle are mutually exclusive. They cannot simultaleously exist in the same figure. A square, by definition, has four corners. A circle, by definition, has zero corners. Therefore, a squircle must, by definition, have four corners and zero corners. This is clearly impossible. It's impossible on Earth, it's impossible on Mars, and it's impossible in the Andromeda galaxy.

Therefore, the squircle does not exist. I can say this with certainty, because four does not equal zero, and four cannot equal zero under any circumstances.

I believe that I just successfully proved a negative. Am I mistaken? If so, why?

I don't think so. The saying 'you cannot prove a negative' is often misapplied, it seems to me.
You can prove some negatives, indirectly, by trying to prove the counter-statements.
Like, a negative "x != 6" could be proved by, proving two related things. 1) x has a unique value. 2) x=5 (or some other value which wasn't 6).

I'm sure the statement that you cannot prove a negative has some specific meaning in boolean logic, but that like the theory/hypothesis thing, most laymen don't understand the distinction.

[chocolatepossum]

I thnk the OP is correct. Any apriori truth that you can prove can be extended to be proof of a negative. If i can prove a triangle's internal angles add up to 180 degrees, then surely I can prove that they don't add up to 190.

This doesn't apply to empirical proof, but that is a different concept from a prioiri truth entirely, at least as I understand it.

[TobiasTheViking]

You may find this page (http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm) illuminating.
Ehm, had nothing to do with the significant digit, since it states both radius(or diameter, i can't remember) and circumference.

[TobiasTheViking]

Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

Diameter = 10 cubits
Circumference = 30 cubits.

[LW]

Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

Diameter = 10 cubits
Circumference = 30 cubits.

Note this:

Zeroes at the end of a number are significant only if they are behind a decimal point as in (c). Otherwise, it is impossible to tell if they are significant. For example, in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four.

You are assuming in your computation of "Bible Pi" that the figures in 1 Kings 7 have two significant digits. If they don't, then it is an error to claim that the Bible asserts that pi is exactly 3.

[TobiasTheViking]

Note this:



You are assuming in your computation of "Bible Pi" that the figures in 1 Kings 7 have two significant digits. If they don't, then it is an error to claim that the Bible asserts that pi is exactly 3.
prove to me that they didn't mean pi is exactly three in the bible.

[Anacoluthon64]

There are two rather important points that seem to be missed in this discussion. The first is that the philosophical dictum actually talks of the impossibility of proving a universal negative, which compounds the issue. Clearly, it is impossible to exhaust all possible circumstances in which a phenomenon can arise and therefore one cannot show that in each case the occurrence is not of a particular type or class, unless the prescribed phenomenon cannot, by definition, arise. In this case you have effectively asked for a square circle, which cannot occur logically owing to the definitions of "square" and "circle." Proving a universal negative requires omniscience.

The second point is that axiomatic formal systems - e.g. geometry, arithmetic, logic - admit of proof or disproof of a given proposition (provided that it is decidable in that formal system) because the rules of inference and axioms on which the system is based determine what is permitted and what is not. However, real life doesn't seem to behave like an axiomatic formal system, so that one can in practice only really assign probabilities (from extremely unlikely to virtual certainty) when attempting to evaluate the truth or otherwise of a statement. The laxity of common language use invariably blurs these difficulties.

'Luthon64

[Paul C. Anagnostopoulos]

"You can't prove a negative" is too generic a claim. A better claim is "You can't prove many kinds of universal negatives" or something like that.

I may not be able to prove the universal "there are no elephants anywhere," but surely I can prove that there are no elephants in my living room.

~~ Paul

[TobiasTheViking]

"You can't prove a negative" is too generic a claim. A better claim is "You can't prove many kinds of universal negatives" or something like that.

I may not be able to prove the universal "there are no elephants anywhere," but surely I can prove that there are no elephants in my living room.

~~ Paul
You can prove there are no elephants in my living room?

Ok, 3... 2... 1... GO.

[Dr B]

In addition to all the excellent points above, the op actually does nothing to prove a negative. What it actually does is recruit two positives, that is the known geometrical properties of circles and sqaures - so these are based on positive evidence from shapes.

I have heard a similar argument based around 'cats'. Someone once said to me, if my cat is in the kitchen I can prove it is not in my bedroom, thus 'proving a negative'. But it does not for all the reasons given above by everyone else and the fact that the evidence for your cat being in the kitchen is positive evidence. So you do not disprove it is in the bedroom, you prove it is in the kitchen (and thus by default is highly unlikely to be in the bedroom at the same time). Again it is the presence of positive evidence that guides the thinking.

[Curnir]

You can prove there are no elephants in my living room?

Ok, 3... 2... 1... GO.

what type of elephants?

[TobiasTheViking]

what type of elephants?
prove that i have no elephants of any type in my room :D

[Correa Neto]

You can prove there are no elephants in my living room?

Ok, 3... 2... 1... GO.
First things first. Can you prove you have a room?

[TobiasTheViking]

First things first. Can you prove you have a room?
you prove i don't have a room.

I mean, i might have one on mars, can you prove i don't.

[Correa Neto]

You are the one making the claim :D!

Prove that I have to prove you have a room!

[TobiasTheViking]

You are the one making the claim :D!

Prove that I have to prove you have a room!
No, this is about proving a negative.
You prove that i don't have to prove that i have a room.

[Correa Neto]

Denialist!

Now, on a more serious side, back to the OP...

If an experiment provides no data that fits with a certain hypothesis, and assuming the experiment was well designed and propeprly carried out, it can be used to prove a negative.

The "can't prove a negative" line has been bit too overhyped in some cases. And in other cases it becomes ridiculous... A last ditch.

[petre]

Hrmn, perhaps the next time someone claims "You can't prove a negative" you should insist they back up their claim with evidence and demonstrate that no provable negative proposition exists :)

[LW]

prove to me that they didn't mean pi is exactly three in the bible.

Immediately after you prove to me that the Hebrews were aware of the concept of pi.

[TobiasTheViking]

Immediately after you prove to me that the Hebrews were aware of the concept of pi.
Oh, you are no fun anymore.. Anyways, i think we showed the point.

[Skiltch]

You can probably prove negatives that are contrapositives of positives. (e.g., if I'm in Chicago I am in Illinois. Therefore, given that I'm not in Illinois and the earlier statement, I can prove I am not in Chicago).

[blutoski]

A quick question I'd like to run through the Randi forums. I know there are a lot of smart people here; maybe someone can help me out.

It's often said that you can't prove a negative. This doesn't seem right to me. It seems to me that it's very easy to prove a negative in some cases. But I've had no formal training in logic, so it's possible (maybe even likely) that my line of thinking is flawed. I've had people try to tell me that it is, but they haven't been able to tell me why. At least, not very convincingly.

It goes like this: I propose a geometric figure called a "squircle." It's a two-dimentional figure which contains all of the properties of both a square and a circle.

However, the properties of a square and circle are mutually exclusive. They cannot simultaleously exist in the same figure. A square, by definition, has four corners. A circle, by definition, has zero corners. Therefore, a squircle must, by definition, have four corners and zero corners. This is clearly impossible. It's impossible on Earth, it's impossible on Mars, and it's impossible in the Andromeda galaxy.

Therefore, the squircle does not exist. I can say this with certainty, because four does not equal zero, and four cannot equal zero under any circumstances.

I believe that I just successfully proved a negative. Am I mistaken? If so, why?

Thanks in advance for any help.


This question does come up, and it's quite reasonable. The problem partly revolves around the vagueness of the terms "prove" and "a negative."

First, let's look at "prove": we're also told that science doesn't really prove things, but assigns likelihoods to them. So, extremely unlikely explanations are de facto disproven, but not completely. In this sense, the statement holds, but is not very informative.


Next: what's "a negative?" Ultimately, the best way to make this statement hold is to say that it's "the nonexistence of something" (as opposed to, say, negative numbers, which we know exist)

There's still a problem with this. For example, I can prove that there are no whole numbers between 0 and 1, because that follows from the definition of whole numbers.

The cat story above is an application of the logical 'truth through contradiction', but it does depend on proving that a cat cannot be in two places at once. ie: the critique requires proving a negative, which can only be done in this case by applying common-sense.


The above scenarios are too hypothetical, though.


The most common real-world applications of disproof by skeptics are of the debunking type, showing that :


a supernatural event did not happen
a secret conspiracy is not responsible for something
a creature does not exist
a health modality does not work


I agree that these cannot be "disproven scientifically", but they can be assigned a degree of chance so small as to be laughable, or at least, skeptics assign more plausibility to another explanation and use it as reality. That is: you don't have to disprove something to argue for another explanation, if it's more likely, or more useful. Personally, I argue for building a case for a better explanation.

There's a second level to this statement, too: Sagan didn't mean it this way, but after years of experience, I'd like to add a corollary: "You can't prove a negative to a true believer, so don't bother trying."

[UrsulaV]

Math is not my strong suit, and logic not particularly either, but in practical terms, I think people have been known to hide behind the whole "you can't prove a negative" phrase.

If I say mammoths can fly, then yes, it may be logically impossible to prove they can't, since you can't prove a negative. But saying "Well, you can't prove a negative, so you can't say they didn't fly!" is pretty silly, when any idiot could assemble a pretty good case that mammoths were not winging majestically 'cross the steppes on the morning breeze.

It's generally not possible to prove a negative perfectly, beyond a shadow of a doubt. You can't even prove perfectly that there's not an elephant in your living room (after all, it could be an invisible, intangible elephant, or a very very small elephant, or an elephant contortionist capable of hiding behind the sofa, or...) but in practical terms a quick glance around the room will usually satisfy most viewers to whether or not there's an elephant present.

So I'd say perfectly, no, but practically, often yes.

[Hellbound]

Bah, easy to prove a negative.

Just hold the positive lead firmly in one hand, and with the other hand grasp the suspected negative.

Assuming a sufficient voltage, the negative will be proven.

Seriously, a specific negative can be disproven (i.e. "There is a blue car in the parking lot.") It's the general negatives ("There are no blue cars") that cause problems, because they are open ended. The argument can always be made that you haven't looked in every possible location.

[TobiasTheViking]

"Well, you can't prove a negative, so you can't say they didn't fly!"
yet people so often DO make claims like "well, can you prove elephants can't fly? oh, you can't HAH, i win". Hence why skeptics often say "can't prove a negative".

It comes up way too often to be funny :(

[gnome]

Any negative is just the negation of a counter-proposition... "the sun will come up tomorrow" vs. "the sun will not come up tomorrow".. they can often be rephrased so that it's unclear which "negative" is off the hook for burden of proof.

The key, I think, is in the proposition itself--you compare it to its negation--which one has a limited enough scope to be demonstrated.

For example, the proposition "All horses are colors besides red."... it is easy to prove the negative, "Not all horses are colors besides red." by finding a red horse... a lot easier than checking all horses and demonstrating no redness.

[John Jackson]

It really boils down to whether a claim is (potentially) falsifiable.

'There's not a dog in my garage' is proveable because we know dogs exist and if there was a dog in the garage it could be detected.

'There's not an invisible dragon in my garage' is not proveable because there's no way it can be tested - it's essentially a meaningless claim.

'Aliens do not exist' - this is a potentially proveable claim (if entirely impractical) because if aliens did exist they could be detected.

So, it is possible to prove a negative when the claim is falsifiable, but it's not possible to prove a negative if the claim is not falsifiable. Non-falsifiable claims are meaningless; so claims to prove/disprove them are also meaningless.

It's related to the 'absence of evidence is not evidence of absence' saying.

Absence of evidence certainly is evidence (although not necessarily proof) of absence for a falsifiable claim. For a non-falsifiable claim absence of evidence is meaningless.

[UrsulaV]

yet people so often DO make claims like "well, can you prove elephants can't fly? oh, you can't HAH, i win". Hence why skeptics often say "can't prove a negative".

It comes up way too often to be funny :(

Yeah, that's the problem, really...

[articulett]

Bah, easy to prove a negative.

Just hold the positive lead firmly in one hand, and with the other hand grasp the suspected negative.



I can always trust the JREF forum to teach me new things.

[morningstar2651]

P1. All dogs are mammals.
P2. No fish are mammals.
C. No fish are dogs.

[SirPhilip]

Can you proove that God can't make a squircle? Can you prove a flying spaghetti monster can't. Tit for tat, and all that. :rolleyes:

[John Jackson]

P1. All dogs are mammals.
P2. No fish are mammals.
C. No fish are dogs.

What about dogfish? :p

[Alkatran]

When people say "You can't prove a negative" what they actually mean, I hope, is "You can't prove something doesn't exist in the physical world."

The 'physical' qualification is important because you can easily prove things don't exist within the confines of mathematics. For example, I can prove there is no even prime number greater than 2, because such a number would be divisible by 2 and therefore not prime.

The 'doesn't exist' is important because you can't possibly search the entire universe. You can prove there's no pink, non-invisible in your backyard in plain sight right now just by looking.

[arthwollipot]

I almost wept when I first encountered Euclid's proof of the infinitude of primes. It was so simple.

[gnome]

I almost wept when I first encountered Euclid's proof of the infinitude of primes. It was so simple.

Another Euclid fan! Yes, this proof was "truly marvelous". And small enough to fit in a margin to boot!

In your face Euler!

[Alkatran]

Another Euclid fan! Yes, this proof was "truly marvelous". And small enough to fit in a margin to boot!

In your face Euler!

I think you mean "In your face Fermat". Unless you're referring to e^(ix) = -1

[gnome]

I think you mean "In your face Fermat". Unless you're referring to e^(ix) = -1

Nah, I meant Euler because his famous proof of the infinitude of primes wasn't nearly as marvelous :)

[blutoski]

prove that i have no elephants of any type in my room :D

Now you've opened it to a problem. If I claim I have planted an embryonic elephant in your room, how would you prove I didn't? Methodical microscopic analysis of every cubic millimeter of your room. Could I have planted it 10cm deep into your mattress? On the back of your drywall in the ceiling? How long would it take? Would it be practical?

Or is the burden of proof on me to prove it's there?

[blutoski]

It really boils down to whether a claim is (potentially) falsifiable.

I think it's more complicated than that: ease of falsifiability is an issue. As in my elephant embryo example, some debunkings are very difficult, bordering on impossible. Nessie's endurance is certainly a consequence of the Loch's size and water opacity - it's not easy to 'see' the lake is empty of monsters, even though in principle, it could be done. I think even draining it would not satisfy TBers, since there's a theory that it's connected to other lochs, caverns, and even the ocean, via tunnels.


Also: we may disagree with woos about what constitutes disproof. The existence of a ghost is disprovable to the woos by using a medium (eg: "You say there's a ghost here, but I have disproven it by asking my spirit guide, and she says there isn't"), but skeptics don't consider this a valid falsification technique.

[John Jackson]

I think it's more complicated than that: ease of falsifiability is an issue. As in my elephant embryo example, some debunkings are very difficult, bordering on impossible.

I tried to anticipate that with my example of proving that aliens don't exist.

Even though such claims are entirely impractical, they are fundamentally different in essence to unfalsifiable claims.

So I'm not disagreeing with you; but I do think there's a fundamental difference between claims that are impossible to test and those which are impractical to test.