## Proving Negatives and Dealing With Absence of Evidence and Evidence of Absence

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]}

### An Example of Proving a Negative in the Sense that People Mean it When they Say the Phrase

Putting aside negatives we can prove with certainty for a second, consider the following:

To prove a negative, we simply have to provide sufficient evidence that a proposition (statement or claim) is true. In other words, we have to show that it is very likely the case, we don’t have to show it is true with absolute certainty.

This can be accomplished by providing evidence of absence (not argument from ignorance). For example, a strong argument that proves that it is *very likely* Unicorns don’t exist involves showing that there is no evidence of Unicorns existing (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.

We could then put forth a scientific theory, based on empirical data, that says “Unicorns don’t exist.”

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

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 Logical Truths and the Evidence of Absence

To clarify the above, in terms of proving negatives with certainty and uncertainty, **we can prove some negatives with certainty** (like necessary logical truths such as “A is not B” and double negatives like “I don’t not not exist”), **but generally “proving negatives” means providing compelling evidence that shows it is very likely that something isn’t the case** (it means “proving” probable truths, not certain ones).

This type of “inductive” reasoning can produce scientific truths using evidence of absence (it can show that it is likely that something isn’t the case via scientific testing that shows a lack of evidence), but the reality is there isn’t much we can actually prove true or false for certain (negative or positive) using evidence or a lack-there-of.

For example, all empirical data points to me sitting at my desk writing this, but maybe “I’m in a virtual simulation and my senses are tricking me?”

Yet, despite any confusion we may have in terms of certainty, “everything is ultimately either true or it isn’t despite our inability to know for sure.”

Below we explain all the above and more in detail, including the ways in which proving negatives is and isn’t possible. First, here are some examples related to proving negatives:

**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). 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.” This is also a logical rule that relates to deductive reasoning.^{[4]}**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 a negative with probability using induction**: We can show something is highly certain using evidence of absence, but we can’t know for sure (that said, the same is true for proving positives with evidence; this is the nature of induction). Ex. Santa cannot be real and not real at the same time. There is no evidence to suggest Santa is real. Therefore it is highly likely Santa is not real. Simply put, induction doesn’t prove negatives with certainty; but it can produce highly certain scientific and logical conclusions.

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

### 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: we 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.

Likewise, 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.

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

With that in mind, and as noted above, we can’t actually prove positives very well either.

Most proofs (positive or negative) rely on inductive evidence, and induction necessarily always produces probable conclusions.

In other words, 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, we might be in the Matrix, etc).

**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.”

### The Bottom Line on Proving Negatives

The bottom line here is:

- We can actually prove some negatives with certainty (using deduction), and generally speaking this type of proof is a foundational aspect of logic.
- “The Absence of Evidence is not the Evidence of Absence”… although it is often a really, really, strong hint (the same works for correlation and causation). In other words, a lack of evidence is a sort of evidence, but it doesn’t prove anything with certainty.
- As far as induction goes, “really strong hints” with “lots of logical truths backing them up”… are types of “proofs” (putting aside the definition of mathematic proof and treating proof as ” sufficient evidence or a sufficient argument for the truth of a proposition.”)
^{[5]}

For one last example before moving on:

If we know Ted must be in Room A or Room B, and we have always seen Ted in Room A, and Ted never has gone into Room B and doesn’t have a key to Room B, and no person in the history of humankind has ever provided evidence of Ted being in Room B, we can be very confident in a logical inductive argument that concludes with a very high degree of certainty that Ted is in Room A.

In these ways, “we can prove a negative… just not with certainty in some cases” (although, again, we can’t prove much with certainty outside of “tautological and analytic statements a priori” anyway).

**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.

## What Does Proving a Negative Mean?

With the above introduction covered, let’s start at the beginning again and cover some details.

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] he doesn’t, you can’t find evidence to prove it. There is an “absence of evidence.”

## How to Prove a Negative

Now that we know the task and have the basics down, let’s discuss how to prove a negative and the ways in which we can and can’t prove negatives.

To do that, let’s answer an important related question, “what is the law of contradiction / non-contradiction?”

The law of contradiction / non-contradiction states that a proposition (statement) cannot be both true and not true (that nothing can be both true and false).

…in other words, one of the fundamental laws of logic (the laws of thought featured below) is a provable negative proposition (and is thus an example of proving a negative).

The rule of contradiction is saying that a statement CANNOT be both true and not true (unlike the positive rule of identity that says “whatever is, is.”)

To help that sink in, here are the laws of thought.

**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. So the following logic works:

- If Santa was real there would likely be some evidence of Santa (not certain).
- There is no evidence of Santa (we should assume this as certain for the example).
- Therefore we can reasonably infer that Santa is not real (a likely truth inferred using induction).

Now, to be fair, that is an inductive argument (a very strong and cogent one, but an inductive argument none-the-less). What that means is that we didn’t produce a certainty, we produced a probability.

In other words, we can “prove a negative” (we do it all the time), but we can’t prove something with certainty without direct proof…

… with that in mind, almost all arguments are inductive and therefore probable.

If we find a person hovering over a dead body with the murder weapon and they say “I did it” we still don’t know for sure that they did it.

If we find zero proof of Santa ever, we still don’t know for sure he doesn’t exist.

However, in both cases we can be very certain of our conclusions, and we can prove our conclusions using logic.

If to prove something is to prove absolute certainty, then only tautological forms of deduction are valid and induction (and all other reasoning methods are useless). <— This creates a very strange and existential world where we can’t trust our senses and theories are meaningless.

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. <— This is how science works.

With all that said, details aside, the blanket statement “we can’t prove a negative” is wrong either way.

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

Of course, even in the way it is meant, that lack of evidence doesn’t imply lack of existence, even that isn’t exactly right. No one ever in the history of mankind having evidence of Santa is itself… pretty strong evidence.

“You can’t prove a negative” #logic.

## Summary of the Different Ways to Prove a Negative

To summarize the above, when we have zero evidence of something, the absence of evidence itself is not the evidence of absence. However, proper scientific evidence of absence can be a fairly strong hint in terms of inductive reasoning (all that non-evidence is a big hint).

If this was all there was to it, we might reasonably conclude that we “can’t prove a negative with absolute certainty (and we would also note that we can’t prove a positive with much certainty either).”

However, we have to also consider that:

**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 as Well**: 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**: We can prove A is not B by noting that every A is A, nothing can be A and not A, and everything is either A or not A. Therefore B is not A. We proved that B is not A, we proved a negative.