Marcello Truzzi states in an article On the Extraordinary: An Attempt At Clarification, “An extraordinary claim requires extraordinary proof.” This was later restated by Carl Sagan as, “Extraordinary claims require extraordinary evidence.” This is frequently used to argue against things such as “gods, ghosts, the paranormal, and UFOs.”

However, this kind of argument, at least as it is usually made, neglects to take into account the fact that claims themselves are already evidence.

Here is one example: while writing this article, I used an online random number generator to pick a random integer between one and a billion inclusive. The number was 422,819,208.

Suppose we evaluate my claim with the standard that extraordinary claims require extraordinary evidence, and neglect to consider the evidence contained within the claim itself. In this case, given that I did in fact pick a number in the manner stated, the probability that the number would be 422,819,208 is one in a billion. So readers should respond, “Either he didn’t pick the number in the manner stated, or the number was not 422,819,208. The probability that both of those were true is one in a billion. I simply don’t believe him.”

There is obviously a problem here, since in fact I did pick the number in the way stated, and that was actually the number. And the problem is precisely leaving out of consideration the evidence contained within the claim itself. Given that I make a claim that I picked a random number between one and a billion, the probability that I would claim 422,819,208 in particular is approximately one in a billion. So when you see me claim that I picked that number, you are seeing evidence (namely the fact that I am making the claim) which is very unlikely in itself. The fact that I made that claim is much more likely, however, if I actually picked that number, rather than some other number. Thus the very fact that I made the claim is strong evidence that I did pick the number 422,819,208 rather than some other number.

In this sense, extraordinary claims are already extraordinary evidence, and thus do not require some special justification.

However, we can consider another case, a hypothetical one. Suppose that in the above paragraphs, instead of the number 422,819,208, I had used the number 500,000,000, claiming that this was in fact the number that I got from the random number generator.

In that case you might have found the argument much less credible. Why?

Assuming that I did in fact pick the number randomly, the probability of picking 422,819,208 is one in a billion. And again, assuming that I did in fact pick the number randomly, the probability of picking 500,000,000 is one in a billion. So no difference here.

But both of those assume that I did pick the number randomly. And if I did not, the probabilities would not be the same. Instead, the fact that simpler things are more probable would come into play. At least with the language and notation that we are actually using, the number 500,000,000 is much simpler than the number 422,819,208. Consequently, assuming that I picked a number non-randomly and then told you about it,  is significantly more probable than one in a billion that I would pick the number 500,000,000, and thus less probable than one in a billion that I would pick 422,819,208 (this is why I said above that the probability of the claim was only approximately one in a billion; because in fact it is even less than that.)

For that reason, if I had actually claimed to have picked 500,000,000, you might well have concluded that the most reasonable explanation of the facts was that I did not actually use the random number generator, or that it had malfunctioned, rather than that the number was actually picked randomly.

This is relevant to the kinds of things where the postulate that “extraordinary claims require extraordinary evidence” is normally used. Consider the claim, “I was down in the graveyard at midnight last night and saw a ghost there.”

How often have you personally seen a ghost? Probably never, and even if you have, surely not many times. And if so, seeing a ghost is not exactly an everyday occurrence. Considered in itself, therefore, this is an improbable occurrence, and if we evaluated the claim without considering the evidence included within the claim itself, we would simply assume the account is mistaken.

However, part of the reason that we know that seeing ghosts is not a common event is that people do not often make such claims. Apparently 18% of Americans say that they have seen a ghost at one time or another. But this still means that 82% of Americans have never seen one, and even most of the 18% presumably do not mean to say that it has happened often. So this would still leave seeing ghosts as a pretty rare event. Consider how it would be if 99.5% of people said they had seen ghosts, but you personally had never seen one. Instead of thinking that seeing ghosts is rare, you would likely think that you were just unlucky (or lucky, as the case may be.)

Instead of this situation, however, seeing ghosts is rare, and claiming to see ghosts is also rare. This implies that the claim to have seen a ghost is already extraordinary evidence that a person in fact saw a ghost, just as my claiming to have picked 422,819,208 was extraordinary evidence that I actually picked that number.

Nonetheless, there is a difference between the case of the ghost and the case of the number between one and a billion. We already know that there are exactly one billion numbers between one and a billion inclusive. So given that I pick a number within this range, the probability of each number must be on average one in a billion. If it is more probable that I would pick certain numbers, such as 500,000,000, it must be less probable that I would pick others, such as 422,819,208. We don’t have an equivalent situation with the case of the ghost, because we don’t know in advance how often people actually see ghosts. Even if we can find an exact measure of how often people claim to see ghosts, that will not tell us how often people lie or are mistaken about it. Thus although we can say that claiming to see a ghost is good evidence of someone actually having seen a ghost, we don’t know in advance whether or not the evidence is good enough. It is “extraordinary evidence,” but is it extraordinary enough? Or in other words, is claiming to have seen a ghost more like claiming to have picked 422,819,208, or is it more like claiming to have picked 500,000,000?

That remains undetermined, at least by the considerations which we have given here. But unless you have good reasons to suspect that seeing ghosts is significantly more rare than claiming to see a ghost, it is misguided to dismiss such claims as requiring some special evidence apart from the claim itself.