Skeptic December 2007

You Can Prove a Negative

by Steven D. Hales

A PRINCIPLE OF FOLK LOGIC is that you can’t prove a negative. Skeptics and scientists routinely concede the point in debates about the possible existence of everything from Big Foot and Loch Ness to aliens and even God. In a recent television interview on Comedy Central’s The Colbert Report, for example, Skeptic publisher Michael Shermer admitted as much when Stephen Colbert pressed him on the point when discussing Weapons of Mass Destruction, the comedian adding that once it is admitted that scientists cannot prove the nonexistence of a thing, then belief in anything is possible. Even Richard Dawkins writes in The God Delusion that “you cannot prove God’s non-existence is accepted and trivial, if only in the sense that we can never absolutely prove the non-existence of anything.”

There is one big problem with this. Among professional logicians, guess how many think that you can’t prove a negative? That’s right, zero. 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.

Not only that, but any claim can be expressed as a negative, thanks to the rule of double negation. This rule states that any proposition P is logically equivalent to not-not-P. So pick anything you think you can prove. Think you can prove your own existence? At least to your own satisfaction? Then, using the exact same reasoning, plus the little step of double negation, you can prove that you are not nonexistent. Congratulations, you’ve just proven a negative. The beautiful part is that you can do this trick with absolutely any proposition whatsoever. Prove P is true and you can prove that P is not false.

You can easily construct a valid deductive argument with all true premises that yields the conclusion that there are no unicorns. Here’s one, using the valid inference procedure of modus tollens (Latin for “mode that affirms by denying”):

  1. If unicorns had existed, then there is evidence in the fossil record.
  2. There is no evidence of unicorns in the fossil record.
  3. Therefore, unicorns never existed.

Someone might object that that was a bit too fast — after all, I didn’t prove that the two premises were true. I just asserted that they were true. Well, that’s right. However, it would be a grievous mistake to insist that someone prove all the premises of any argument they might give. Here’s why. The only way to prove, say, that there is no evidence of unicorns in the fossil record, is by giving an argument to that conclusion. Of course one would then have to prove the premises of that argument by giving further arguments, and then prove the premises of those further arguments, ad infinitum. Which premises we should take on credit and which need payment up front is a matter of long and involved debate among epistemologists. But one thing is certain: if proving things requires that an infinite number of premises get proved first, we’re not going to prove much of anything at all, positive or negative.

Maybe people mean that no inductive argument will conclusively, indubitably prove a negative proposition beyond all shadow of a doubt. For example, suppose someone argues that we’ve scoured the world for Bigfoot, found no credible evidence of Bigfoot’s existence, and therefore there is no Bigfoot. This is a classic inductive argument. A Sasquatch defender can always rejoin that Bigfoot is reclusive, and might just be hiding in that next stand of trees. You can’t prove he’s not! (until the search of that tree stand comes up empty too). The problem here isn’t that inductive arguments won’t give us certainty about negative claims (like the nonexistence of Bigfoot), but that inductive arguments won’t give us certainty about anything at all, positive or negative. All observed swans are white, therefore all swans are white looked like a pretty good inductive argument until black swans were discovered in Australia.

The very nature of an inductive argument is to make a conclusion probable, but not certain, given the truth of the premises. That is just what an inductive argument is. We’d better not dismiss induction because we’re not getting certainty out of it, though. Why do you think that the sun will rise tomorrow? Not because of observation (you can’t observe the future!), but because that’s what it has always done in the past. Why do you think that if you turn on the kitchen tap that water will come out instead of chocolate? Why do you think you’ll find your house where you last left it? Again, because that’s the way things have always been in the past. In other words, we use inferences — induction — from past experiences in every aspect of our lives. As Bertrand Russell once pointed out, the chicken who expects to be fed when he sees the farmer approaching, since that is what had always happened in the past, is in for a big surprise when instead of receiving dinner, he becomes dinner. But if the chicken had rejected inductive reasoning altogether, then every appearance of the farmer would be a surprise.

So why is it that people insist that you can’t prove a negative? I think it is the result of two things: (1) Disappointment that induction is not bulletproof, airtight, and infallible, and (2) A desperate desire to keep believing whatever one believes, even if all the evidence is against it. That’s why people keep believing in alien abductions, even when flying saucers always turn out to be weather balloons, stealth jets, comets, or too much alcohol. You can’t prove a negative! You can’t prove that there are no alien abductions! Meaning: your argument against aliens is inductive, therefore not incontrovertible. Since I want to believe in aliens, I’m going to dismiss the argument no matter how overwhelming the evidence against aliens, and no matter how vanishingly small the chance of extraterrestrial abduction.

If we’re going to dismiss inductive arguments because they produce conclusions that are probable but not definite, then we are in deep manure. Despite its fallibility, induction is vital in every aspect of our lives, from the mundane to the most sophisticated science. Without induction we know basically nothing about the world apart from our own immediate perceptions. So we’d better keep induction, warts and all, and use it to form negative beliefs as well as positive ones.

You can prove a negative — at least as much as you can prove anything at all.

Views: 590

Replies to This Discussion

Quibbling over bad examples:

> Why do you think that if you turn on the kitchen tap that water will come out instead of chocolate?

Because my pipes are connected to a pump in the front yard, which I have pulled up myself, and which is sunk into rock that contains water.  How do I know that the aquifer will not suddenly become a moo-quifer?  Because it obeys physical laws.  Same goes for the world turning (aka sunrise) and the persistence of my house's location.

I think the author is playing with us.  When he writes:

> Maybe people mean that no inductive argument will conclusively, indubitably prove a negative proposition beyond all shadow of a doubt.

he is leading us down a garden path.

I'm not a logician, so maybe I am missing something good here.  But it looks to me like the choices are between measureable and not measureable.  If you can measure something, you can prove assertions about it, positive or negative. If you can't measure something, then, like pinball machines, arguments about that subject are for entertainment purposes only.

I'm not sure quite what you mean here by measurable and not measurable. Take your water tap in your kitchen. How do you know that, after having installed it yourself, someone else hasn't come along and tampered with it? Or how do you know that the next time you leave your house, someone won't be standing ready to put it on a truck the moment you are gone? And of course an asteroid could always come and obliterate the Earth by next morning, causing the Earth to stop turning. There is always a logically-possible counter-factual which can disturb what you think is a perfect set up for having knowledge. We can measure the odds of these counter-factuals happening. But even if the odds turn out to be infinitesimal, the logical possibility still exists ("So you're saying there's a chance?"). Whether or not it could actually exist is a different question, one which is easier to handle outside of a philosophy discussion.

I soldered the pipes myself.  It's wet.

I'll leave this one to you smart guys.

Hah, glad to see Andrew in here. As a student of philosophy (BA), I found this article to be very interesting, so thanks Dr. Meaden! I have given epistemology a lot of attention, and I agree with the conclusion here, which is that you can prove a negative so far as you can prove anything. But of course a lot hinges on this. As it turns out, we really can't prove anything at all, not with absolute certainty. It is no small surprise then that the scientific method relies upon skepticism - we believe the evidence gathered under a theory until the theory is proven wrong, which can happen at any place and any time. The distinction which is of vital importance is, in my mind, that between logical possibility and actual possibility. Yes, it is logically possible that the laws of nature will, at some point in the future, cease to be what they are, that the future will cease to resemble the past, and that Jesus will come down from on high and deliver us to salvation. But none of these things is actually possible. The problem is, you can't prove it. Any time you attempt to prove something, you have to do so within the confines of the system which you are operating in, but to prove anything absolutely, you would need to go beyond the system and prove that system to be valid in the lights of a meta-system, which in turn would need to be proven in lights of a meta-meta-system, etc. ad infinitum. A parallel point is that that if you manage to discover a meta-physical world, then that in addition to the physical world becomes the physical world as it would then be known. But in order to prove anything about that new physical world, you would be stuck again in the position of having to find meta-physical support for it. The religious often get themselves tangled up in this infinite regress because they think that their metaphysical construct supports their beliefs about this world, e.g. that the existence of a god supplies the absolute framework for a system of absolute morality. But they conveniently dodge the necessity to explain where that construct came from, etc. The same goes for their arguments concerning the origins of the universe. If something can't come from nothing, then where did god come from? So only he can come from nothing, but nothing else can?


We live in a strange world, and yet we think we can come to some pretty decisive conclusions about it. And I think we can. An example my professor of metalogic gave once - imagine you are tossing a coin, and it comes up heads a few times in a row. You might think well, the next time it is bound to come up tails. But after a dozen more throws, it still only comes up heads. At this point you should be thinking well, this is either a double-headed coin, or some other trick is at work. Now imagine that the coin is thrown trillions upon trillions of times, and always with the same result. You are left to conclude either that this coin will never turn up anything other than heads, or you can stubbornly stick to the hope that next time, next time, its going to turn up tails. Well, we live in a universe which has turned up heads at least that many times, and continues to do so. We can't prove that it will always do so, but we can be reasonably sure beyond anything but the faintest flicker of doubt. We don't need to lose that flicker - it is useful and good, if anything for educational purposes, to keep that specter of skepticism lurking in the corner. But we don't need to entertain the idea that we don't know what we know for sure. We just can't prove it.

About chocolate coming out of pipes, let me paraphrase Mark Twain (I think): if someone offers to bet you that chocolate will come out of your pipes, don't take the bet, because sure as shooting chocolate will come out of them. Conjuring tricks like that are not hard to pull off.

But seriously, folks: according to Karl Popper's dictum that a hypothesis is scientific if it is falsifiable, then progress in science consists more or less entirely of proving negatives, i.e., shooting down hypotheses that fail and leaving us with hypotheses that haven't failed yet. We accept that evolution happened because no evidence has ever turned up to the contrary, and we have no reason to believe that any ever will. But theoretically, it is possible that tomorrow God will show up, part the curtains, and show us that the entire universe is a gigantic hoax. It's just extremely improbable.

I think that the right way to respond to the challenge to prove the nonexistence of God is to accept the existence of God as a working hypothesis, and then ask the theist: now what? Let us suppose that the universe was indeed created by some supernatural entity; how do you propose to prove that that entity is the one that your scriptures describe? The giant leap the creationists make is not in postulating the existence of a creator, but in claiming that the nature of the creator and the eternal rules of conduct that he enjoins us to follow spring from some books written 3000 years ago by ignorant tribesmen.  How do you get from an abstract and incomprehensible creator who caused the Big Bang to some books written around 13 billion years later, and to those particular books and not the many others that make different claims?

Another approach, maybe snappier, is to agree: yes, the universe was created by a divine being named ZARQ. Prove it wasn't.

A principle to keep in mind when debating with creationists, having nothing to do with logic, is that you will never convince them; you are really speaking to the people in the audience who have accepted all this bs uncritically, in the hope of opening their eyes. In the long run, the successful strategy is to give the idiots enough rope.

I am not even a student of logic, but, I think every person posseses some sense of logic. Forty years before when I became an atheist, I had no support from science. Thinking freely and rationally, I came to a conclusion that the rituals we perform are meaningless and the images we worship are non existant and must have come from man's imagination. This was enough for me to become a strong atheist. Logic therefore also helps. Today, however, I can get strong help from science to support my atheism.

We have to be careful to realize that true logic requires the specificity of definitions and axioms. For example, in the disproof of the existence of unicorns presented in the abstract by Dr. Meaden the true logic that was implemented transpired as follows:

1.) An animal has existed only if there is a fossil record of it – Definition.
2.) There is no fossil record of unicorns – Axiom.
3.) Therefore, unicorns have never existed - Logical Conclusion.

However, suppose a definition or axiom were arbitrary. In this case one could seem to prove that unicorns have existed or that horses have not as in the examples below:

Case I.

1.) An animal has existed if there is no fossil record of it - Definition (but an arbitrary one).
2.) There is no fossil record of unicorns - Axiom.
3.) Therefore, unicorns have existed - Logical Conclusion.

Case II.

1.) An animal has existed only if there is a fossil record of it – Definition.
2.) There is no fossil record of horses - Axiom (but an arbitrary one).
3.) Therefore, horses have never existed - Logical Conclusion.

The point is that a logical argument is only as valid as its definitions and axioms. In the layman’s vernacular this precept is expressed as ‘garbage in produces garbage out‘. Accordingly, in order for logic to truly be meaningful the associated definitions and axioms must be self-evident.

With this in mind it is in fact true as Dr. Meaden stated in the abstract that no
professional logician thinks a negative cannot be proven. However, per the above arguments, this is only true if they agree on the beginning definitions and axioms. Dr. Meaden also roughly described the approach to proving a negative when there is agreement on the beginning definitions and axioms. As he knows, in mathematics the approach is called proof by contradiction and in philosophy it is called reductio ad absurdum.

Assuming we want to prove the negative of the existence of a Biblical type god a simple example of it follows:

1.) Assume the opposite of what is to be proven. That is, assume a Biblical type god exists.

2.) It must be perfect in goodness - By definition of a Biblical type god.
3.) It demands to be worshiped - By definition of a Biblical type god.

4.) Good beings don’t demand to be worshiped - By definition of good.

5.) Accordingly, the Biblical type god does not demand to be worshiped - By definition of a Biblical type god (as all good) and 4.).

6.) Therefore, the Biblical type god both demands and does not demand to be worshiped - From 3.) and 5.).

7.) Consequently, the assumption that a Biblical type god exists leads to a self-contradictory proposition and, in that our definitions are self-evident (cannot be compromised), the assumption that a Biblical type god exists cannot be true.

8.) Resultantly, it must be false and, in this, we have proven the negative.

Pertaining to 4.) above - Although it is clear to the objective, theists will say that the demand to be worshiped stems from the desire to be a good father and, as such, is not inconsistent with goodness. Therefore, the proof can be strengthen by using the nature of the being form to demonstrate that the demand to be worshiped is inconsistent with goodness. That is, per Descartes’ time tested definition of a being, it is impossible for one to know absolutely whether one’s perceptions validly reflect an external universe and, as such, demanding that one does in the form of worship is inconsistent with goodness.

It is not inconsistent with science to cite the limitations of the being form in a rational argument. Scientifically speaking nature made us beings and there is no sense in honoring one part of nature and not another. In fact, I submitted the above argument (citing the limitations of the being form to strengthen it) to the Science and Rationalists’ Association of India and they not only published it in their online magazine, The Freethinker, but linked it to their article on the release of Stephen Hawking’s book, The Grand Design.

Even in Mr. Hales argument I see a persistence in the belief that 'wiggle room' exists for superstitions and other instances of "prove my negative." In his case, this belief is couched in platitudes about the mystifying depths of philosophy and epistemology, while the demonstration of negative proofs by purely deductive methods simply acts as a smokescreen because it does not address the root of this popular myth: can you verify the absence of something by observation?


And the answer is, quite simply, yes you can: simply exclude it. As Mr. Hales noted, the key is to use the principle of non-contradiction, which serves so well for deductive negatives. But the methods must be different to account for the basic limit of inductive methods, that they can only accept positives as evidence. In order to prove a negative, then, it is necessary that the negative cannot coexist with the positives found already-- that the existence of A and B together would be contradictory, but either A or B could be found alone. If B is then observed, it serves to exclude the possibility of A completely.


Consider Wanderer's example of a flipped coin which landed heads one trillion times. If you blithely ignore the coin and only tally the result, it is true that you will know nothing useful about it. But an intrepid watcher might turn the coin after it has landed and notice, on the other side, a second heads. As the coin only possesses two sides, and both are marked as heads, there is no room for a tails on that coin. On any future tosses one can know with certainty that no tails can occur, as that would require an (impossible) self-contradiction between the content (2 heads) and metacontent (2 sides) of the coin.


The same inductive methods can be applied to any entity with at least 2 properties, or any system with at least 2 entities. If all the 'space' for a possibility is occupied, then that possibility is necessarily false.


And it doesn't require a complete and exhaustive search; how do you know a cruise ship can't be snuck into your house? Because you can look at the garage door (the largest opening) and know that it's too small. You don't have to check every door, every window, every vent and crack looking for other ways it might fit in-- you've already excluded it by contradiction with "largest opening". Searches need only be as long as it takes to find an exclusionary conflict.



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