Researched by Thomas DeMichelePublished - July 26, 2017

Last Updated - March 1, 2021

## Proving Negatives

Thesaying “you can’t prove a negative” isn’t accurate. Proving negatives is a foundational aspect oflogic (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 alluded 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” 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 an 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 negative**: “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 positive (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 from 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 that 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 negative**:“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 – Can’t 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.

While it is true that the absence of evidence isn’t the evidence of absence, the blanket statement “we can’t prove a negative” is arguably is misleading if not fully incorrect (especially when we are talking about deductive logic like the law of contradiction).