## Proving Negatives

The saying “you can’t prove a negative” isn’t accurate. Proving negatives is a foundational aspect of logic (ex. the law of contradiction).^{[1]}^{[2]}^{[3]}^{[4]}

Furthermore, if you define “proof” as something that only requires us to show that something is very likely, then you can prove a negative this way as well.

Below I’ll cover the different ways to prove negatives, including the two just mentioned, and I will cover why the statement “you can’t prove a negative” has weight to it despite this.

## What Does Proving a Negative Mean?

So first, “what does proving a negative mean?”

It means proving something isn’t true. For example, “proving Santa Claus doesn’t exist.”

If Santa did exist, you could find evidence and prove it, but because [spoiler] it is very likely he doesn’t exist, you can’t find evidence to prove he doesn’t with certainty, you can only find evidence that suggests he doesn’t (you can only find “evidence of absence”).

## The Absence of Evidence and the Evidence of Absence – What Do People Mean When they Say “You Can’t Prove a Negative”?

In general, and putting aside those who misunderstand the concept, when people use the phrase “you can’t prove a negative” they mean: you can’t prove negatives with certainty based on the absence of evidence alone (the absence of evidence is not *necessarily* the evidence of absence).

For example, having no proof of Bigfoot doesn’t prove that he isn’t real with certainty, it just means you can’t find evidence that he is real.

Likewise, it is hard to provide proof that a giant flying invisible unicorn doesn’t exist… because there is no evidence of such a thing and thus our best evidence is an absolute lack of evidence.

We can only “prove” that which there is no evidence for with a high degree of probability (by considering the lack of evidence and some rules of logic).

However, while the above is true (one reason the “you can’t prove a negative” saying has weight to it), the reality is we can’t prove positives very well either.

Most proofs (positive or negative) rely on inductive evidence, and induction necessarily always produces probable conclusions and not certain ones.

So for example, if we had Santa on tape admitting he was Santa… it would still only be very strong evidence (it wouldn’t prove he Santa was real with certainty; our senses could be tricking us, the video could be fake, the person may be lying, we might be in the Matrix, etc).

In other words, we could argue that proving both positives and negatives rely on likelihood and not certainty.

**TIP**: Learn about how induction and deduction work. The certain proofs are deductive, the likely proofs are inductive.

**Absence of evidence and the evidence of absence**: Absence of evidence is an ambiguous term. If it is absence from ignorance, in that no one has ever carefully studied the matter, then it means next to nothing. If it is absence despite careful empirical study done in-line with the scientific method, then the absence of evidence itself can be considered a type of scientific evidence. If we inspect the room over and over and there is never any mice in the room, we can conclude with a high degree of certainty from the absence of mice that the room is not infested with mice. Here absence of evidence (or “the evidence of absence despite our looking for it” more specifically) is a type of evidence. If we keep checking and don’t see evidence of Santa, we can be highly confident that there is no Santa. See “Evidence of absence.”

## An Example of Proving a Negative With Likelihoods and the Evidence of Absence

As eluded to above, one way to “prove” a negative with a high degree of certainty is to show enough evidence of absence.

Consider the following argument:

**If to “prove” a something we simply have to provide sufficient evidence that a proposition (statement or claim) is very likely true**.**Then, to prove a negative, we only have to show that it is very likely the case and we don’t have to show it is true with absolute certainty**.

Under those conditions, we DO NOT have to observe empirically that which cannot be observed (for example, we don’t have to see a Unicorn not existing to know it doesn’t exist, we just have to show compelling evidence of its non-existence).

Thus, proving a negative in this sense can be accomplished by providing evidence of absence (not argument from ignorance, but scientific evidence of absence gathered from scientific research that shows absence).

For example, a strong argument that proves that it is *very likely* Unicorns don’t exist on earth involves showing that there is no evidence of Unicorns existing on earth (no fossils, no eye witness accounts, no hoofprints, nothing).

If we did a serious scientific inquiry, searching for Unicorn fossils, and turned up nothing, it would be a type of evidence for the non-existence of Unicorns. If no one could show scientific data pointing toward unicorns to combat this, then at a point it would become a good theory and we could put forth a scientific theory, based on empirical data, that says “Unicorns don’t exist on earth.”

At that point, the burden of proof would be on those who believe in Unicorns to prove that Unicorns do in fact exist (the burden would be on them to prove the theory of non-existent Unicorns false by providing a better theory).

In other words, if we accept that a proof can have a degree of uncertainty, we can argue that it is possible to use the evidence of absence to prove negatives.

However, since we didn’t prove the non-existence of unicorns with certainty, below we deal with the law of contradiction and using double negatives to provide more certain proofs.

**TIP**: Science can’t actually prove anything with 100% certainty. Essentially, “all we know for sure is that we know nothing for sure.” This is because all testing of the outside world involves inductive reasoning (comparing specific observations to other specific observations), and inductive reasoning is by its nature uncertain (for example the statement “I see a horse there, therefore horses exist on earth” could be wrong if your eyes aren’t working AKA if your measuring device is off, if you are in a simulation, if that isn’t actually a horse, or if this isn’t actually earth, etc…. in short it is a more compelling argument than the unicorn argument, but still something we can poke holes in). Meanwhile, logically certain truths are generally pure analytic a priori (they are generally tautologically redundant and necessarily true facts; for example, “since A is A” therefore “A is not B.”) With those logical truths we have the positive side “A is A” and the negative side “A is not B.”

Is the popular “you can’t prove it doesn’t exist” a good argument?

**Does this prove God does or doesn’t exist?** Proving the existence of God (or the non-existence) is loosely related to this line of reasoning, but it is sort of outside of the sphere of what we are talking about here. If one claims, “all that is is, but God exists outside of that” then the argument for God becomes ontological, theological, metaphysic, and faith-based. Faith-based metaphysical arguments don’t require scientific empirical evidence… unless they try to posit something that can be debunked by empirical science (in that case, arguments for faith instead of reason tend to be logically “weak,” in that they lack supporting evidence).

## An Introduction to Proving Negatives With Necessary Logical Truths

Above we “proved” an argument using likelihoods (not certainty). The idea was to show that using evidence to prove a positive and the evidence of absence to prove a negative were both valid.

With that said, **we can actually prove some negatives with certainty** (for example, necessary logical truths such as “nothing can both be and not be” and double negatives like “I do not not exist”).

Here are some examples of proving negatives with logical truths:

**The Law of Contradiction itself is a negative**: “Nothing can be A and not A.” Ex. Ted can’t be in Room A and not in Room A (and therefore, if Ted is in Room A, then Ted is not in Room B). We are using a positive to prove a negative with the law of contradiction, but we are proving a negative. This is a rule used in deductive reasoning and is a necessarily true logical rule.**The Modus Tollens also proves a negatives**: “If P, then Q. Not Q. Therefore, not P.” Ex. “If the cake is made with sugar, then the cake is sweet. The cake is not sweet. Therefore, the cake is not made with sugar.” Not every argument of this structure is true, but we are proving a negative… as we are simply trying to prove “not P.” This is also a logical rule that relates to deductive reasoning.^{[5]}**Proving a negative with certainty using double negatives**: Any true positive statement can be made negative and proved that way. Ex. I do not not exist; or Every A is A, nothing can be A and not A, everything is either A or not A, therefore A is not not A. These prove a negative with certainty, but are somewhat redundant (rephrasing “A is A” as “A is not not A” is tautological).**Proving Impossibility**. In mathematics there are different ways to prove a problem can’t be solved. For example, because π is non-algebraic, and only a subset of the algebraic numbers can be constructed by compass and straightedge, you can’t square a circle with a compass and straightedge. Here you could argue that we are again first proving a positives (that π is non-algebraic, that only a subset of the algebraic numbers can be constructed by compass and straightedge, etc)… but ultimately it is another example of providing negatives despite this.

## How to Using the Above Logical Arguments To Structure an Argument that Attempts to Prove a Negative

As noted above, the law of contradiction states that a proposition (statement) cannot be both true and not true (unlike the positive rule of identity that says “whatever is, is.”)

That “law” is part of three laws that comprise the “laws of thought.”

Those laws are:

**The Law of Identity**: Whatever is, is; or, in a more precise form, Every A is A. Ex. Whatever is true about Santa is true about Santa.**The Law of Contradiction**: Nothing can both be and not be; Nothing can be A and not A. Ex. Santa cannot be real and not real at the same time.**The Law of Excluded Middle**: Everything must either be or not be; Everything is either A or not A. Ex. Santa must be real or not real.

In other words, Santa is either real or not real, there is no in-between.

With that covered, we can now apply the following Modus Tollens style logic on top of the idea that “Santa is either real or not”:

- If Santa was real there would likely be some evidence of Santa (not certain).
- There is no evidence of Santa that we have found (this has more weight if we truly look for the evidence).
- Therefore we can reasonably infer that Santa is very likely not real (a likely truth inferred using inductive reasoning based on the absence of evidence).

Here we could try to prove that it is impossible for Santa to be real, but that is aside the point. The point is, once we have to prove something empirically, once we start dealing with the evidence of absence, we introduce likelihoods.

## Ultimately the Argument For Proving Negatives or Not Depends on How We Define “Prove”

In mathematics and logic, when we replace empirical evidence for numbers and symbols, we can prove negatives all day.

However, when we go to prove negatives in the material world using empirical evidence, we have to deal with evidence of absence, and thus end up dealing with likelihoods and not certainties.

Therefore, in many ways, the argument that “you can’t prove a negative” relies on how we define the term “prove.”

If to prove something is to prove absolute certainty, then only tautological forms of deductive logical truths, like A is A, are valid. Meanwhile, induction is invalid and thus even the empirical evidence we gather to prove positives is called into question (because “what if we can’t trust our sense”).

If we on the other hand can consider overwhelming evidence that draws a highly certain conclusion as proof until better evidence comes along, then we can prove negatives.

However, if we say, yes we can consider overwhelming evidence, but only the evidence of presence and not the evidence of absence. Then, well, we get the old “you can’t prove a negative line.”

With all that said, since there are arguments for providing a negative with the evidence of absence, and since there are aspects of deductive logic which involve providing negatives, I would still put fort the idea that the saying “we can’t prove a negative” is ultimately misleading if not flat out wrong.

“You can’t prove a negative” #logic.

## Summary of the Different Ways to Prove a Negative

**The Law of Contradiction proves a negative with certainty**: Nothing can both be and not be; Nothing can be A and not A.**The Modus Tollens also proves a negatives**: “If P, then Q. Not Q. Therefore, not P.”**We Can Use Inductive Reasoning to Provide a Likely Proof**: We can show evidence of absence as proof of likelihood.**We Can Also use Double Negation**: Simply converting a positive statement into a double negative.**We can generally use a mix of all the above.**

**TIP**: For more reading, see: “You Can Prove a Negative ” Steven D. Hales Think Vol. 10, Summer 2005 pp. 109-112.

James Randi Lecture @ Caltech – Cant Prove a Negative. Skepticism is very useful, here is a good discussion on the ways in which we should understand the truth behind the “you can’t prove a negative” idea.

Doesn't beleive this myth.THE OLD PSEUDOLOGICAL STATEMENT “I CAN’T PROVE A NEGATIVE” has been used mostly by those who have a hard time dealing with truths that they either can’t handle or don’t want to. In this current age and post truth climate of “fake news” and fact manipulation this fallacious statement is used by a lot of media people and reporters when truth thirsty people aren’t letting them do their jobs the way they want. I like this essay so much I posted it on Facebook.

Glad to hear the feedback. Thank you for sharing it!

Supports this as a Fact.1. The article states, ” it is hard to provide proof that a giant flying invisible pink unicorn name Terry-corn isn’t… because it isn’t and thus our best evidence is the absolute lack of evidence.”

This is a poor example. Invisible pink (or any other color) anything’s, are easy to disprove.

Pink is a frequency in the electromagnetic spectrum. If something is invisible, by definition it is not reflecting any frequency at all.

If it’s pink, it’s frequency is being reflected and it cannot be invisible.

For this old argument to work, you have to posit some mechanism which is impeding the reflection of the color from the view of some observer. But that doesn’t make the unicorn invisible in itself. That only adds a factor that hides it from light-dependent vision systems like human eyes.

It may seem like quibbling but it’s important.

2. You also state: “If one claims, “all that is is, but God exists outside of that” then the argument for God becomes ontological, theological, metaphysic, and faith-based”

This is wrong. That claim that God exists outside of nature – all ELSE that exists is a logical conclusion, not a faith-based claim.

By definition, God created ALL ELSE that exists and therefore he NECESSARILY exists “outside”, “beyond”, or transcendent to it.

“Faith-based metaphysical arguments don’t require scientific empirical evidence…”

a) Your entire article is based on metaphysics. Logic and reason are metaphysical!

b) You have an incorrect idea of what proper faith is. Faith is trust. Trust must be based on evidence of trustworthiness.

And that is of course, the very kind of faith promoted in Christianity for example.

Nowhere are people demanded to just blindly believe without reason, in the bible. On the contrary, we are told, “Come let us reason together” – by God.

We are also told that Christ left his followers “many infallible proofs” of his resurrection. He had no such stupid idea of faith that they ought to just believe it without evidence.

Your view in this looks like the ubiquitous error made by atheists everywhere concerning the meaning of faith as some sort of blind, irrational leap in the dark. That is not real faith. That is stupidity.

That blind kind of “faith’ can exist, of course. But that kind is foolishness.

“unless they try to posit something that can be debunked by empirical science (in that case, arguments for faith instead of reason tend to be logically “weak,” in that they lack supporting evidence)”

Anything that can be debunked by empirical or other forms of evidence cannot be logically weak. Lacking supporting evidence does equate to “weak”. An argument may be air-tight logically, yet without any empirical evidence. The proof of God, for example, is logical, not empirical. Unless one denies the validity of logic, that proof, if well reasoned, is as good as any other. Else we ought to abandon mathematics which is based on logic.

Moreover, you seem to be neglecting the fact that you cannot trust your own faculties of reason without having some good evidence that they are presenting the real world, correct logic etc.

You cannot test your brain using your brain.

So how do we know that our own faculties of reason are trustworthy? Under a purely materialist view, we do not and cannot. The reliability of reason itself must be taken as an article of faith. An axiom of reason, if you will. Else nothing is knowable at all. We may well all be nothing but clumps of cells in a Matrix, being fed illusory images and sensations of some algorithmic origin, in such a case. The idea that we are all mere bags of chemicals, sacks of meat, packs of neurons, destroys any possibility of objective reasoning being known to be reliable. It destroys objective rationality.

Thanks for the thoughtful response. I’ll think on it.

If all we know for sure is that we know nothing for sure, how can we be sure we know nothing for sure?

I’m sure I do not know…. or am I?

Supports this as a Fact.Horse hockey.

You cannot prove that something does not exist. “There is no God.” Show me proof.

The phrase refers to the concept of a scientific proof, not winning an argument. A scientific proof is considered absolute. Some proofs, like maths and geometry, are arguments that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to prove. In statistics, you present a hypothesis, you test it, you either say you can’t show it to be true or that you have demonstrated it to be (QED). THAT is a proof, not “dogs are the best pet because more people own dogs than any other animal.”

You cannot, logically, prove a negative. A reductio ad absurdum example is “Prove that Thomas DiMichele is not a murderer.” Unless you can produce incontrovertible evidence that Mr DiMichele has not ever, at any nanosecond of his life committed homicide, you cannot prove that statement. Another: “No one goes there anymore.” First, the lack of time frame is obvious. Second, the only way to prove that is to go there to observe, which then disproves that statement.

Best: “You can’t prove a negative” is not a fact. While i can expound on my examples of a scientific proof into a proof of it’s own, I cannot say there is NOT evidence showing it is NOT a fact. Lack of positive evidence is not the same as disproving it.

A preponderance of evidence is NOT proof. It is simply an argument to support the general validity of a statement.

Supports this as a Fact.Modus Tollens Fallacy:

Proposed: “If P, then Q. Not Q. Therefore, not P.”

Fallacy Proof:

If you behead the King, then he will die.

If the King does not die. Therefore, he will not be beheaded.

Proposed: “If P, then Q. Not P. Therefore, not Q.”

Fallacy Proof:

If you behead the King, then he will die.

You don’t behead the King. Therefore, the King won’t die.

This made me laugh. Lots of big words and references to the mathematics of logic… which I have taken and use professionally on a daily basis. You lose all credibility with; double negatives like “I don’t not not exist”… this is a triple negative, genius. How about; “since A is A” therefore “A is not B.” This is in correct logic and proves nothing. If B=A, which you did not state that is was not, then “A is not B” is a false statement. Or; “if Ted is in Room A, then Ted is not in Room B”, “if Ted is in Room A” is proving a positive. Now if you said “Ted is not in room D”, that is a negative… does that mean he is in room A? Modus Tollens is a fallacy of propositional logic by denying the antecedent. This ‘Denying the Antecedent’ is non-validating, which means that not every argument of that form is valid. This doesn’t mean that every argument that denies the antecedent is invalid; rather, it means that some arguments of that form are invalid. Since proof exists is in the real world, how many rooms are there out there? A million? A billion? Thus proving this logic flawed. For the rest of these examples, it is just as misleading… “If P…“; P a positive, “induction doesn’t prove negatives”; this as an argument to prove negatives? This one made me laugh the hardest. “Proving a negative with certainty using double negatives”… as you, yourself pointed out that double negatives cancel each other out (as they do in mathematics) with this example; if “A is A” as “A is not not A.” Thus “A is A” = “A is not not A” then this statement is actually “Proving a negative with certainty using a positive”. This is too funny! Or better yet (or worse in this case), the comment; “<-This is how science works"… maybe, but not logic… which is the argument of topic. I am not sure you understand the OR and AND of logic correctly. The positive aspect of an argument is an OR. If I can show empirical evidence that something exists, (Unicorn Bones, DNA, baby unicorns) then I have proven my case. It does not matter how many people have not found bones, have not found babies, or have not found DNA, a single proof is all that is needed. A negative is an AND. To prove a negative you have to cross off everything in the list to prove it. Thus if I said “there is no such thing as Unicorns” I would have to have looked everywhere and every when, to prove this is the case.

You cannot prove a negative, period. Take this sentence; "Thus, to prove a negative, we only have to show that it is very likely the case." Not true… implication of proof, is still not proof. Given the example above, you would have to examine all planets, in all solar systems, for all previous existing eras, since the beginning of time, to prove Unicorns do not now exist and have never existed. This limited scope of proof, proves only your limited scope of true understanding. Since this is not possible, proving that they do not exist and that have never existed, is not possible. Again, implication of proof, is still not proof. Just because you have looked in one microscopic section of the universe, within a miniscule sliver of time and found no evidence, this does not prove Unicorns do not, or have not ever existed. “At that point, the burden of proof would be on those who believe in Unicorns to prove that Unicorns do in fact exist.” Really? At what point? When you haven’t yet proved anything? “I looked in my closet and did not see any, so it is up to you to prove they do exist.” So the argument for “proving a negative” is to have the opposing side of the argument prove a positive? Hilarious. Proof is obtained through empirical evidence. This does not mean lack of empirical evidence proves it does not exist. Modus Tollens Fallacy. Talk about “an argument from ignorance”, I laughed all the way through this convolution of dribble.

Interesting counterpoints. I’ll have to read this over carefully.

UPDATE: I read this over. It was a rather insulting (at times going from academic to ad hominem) and insanely long paragraph… but, that aside, it was also full of interesting counter arguments and a few very valid points (like the one about double negatives… my example was incorrect). I think this all boils down to a debate between “how do we define proof” regarding most of what I’m talking about, and to some degree semantics in other cases. For example with the Unicorn, it is “how do we define proof” (basically I say we can provide a very likely reasonable proof, a very compelling inductive argument; and you say “that is not proof,” only certain empirical evidence is proof…) and meanwhile for example with double negation we seem to be in an argument of semantics (you say basically not-not-P doesn’t cut it, I say but it does). Anyway, all that aside, I appreciate the feedback as 1. it helped me to make my argument stronger, and 2. it points out where I should clearly state the opposing view (as I think many people will ultimately agree with your stance, and under certain criteria I could make an argument for it as well).

ALSO:

Instead of:

If you behead the King, then he will die.

If the King does not die. Therefore, he will not be beheaded.

Try:

If you behead the King, then he will die.

If the King is not dead. Therefore, he was not be beheaded.

Same structure, a different wording, point being we are trying to prove the King was not beheaded.