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Wonderful Phenomena Demand Wonderful Evidence

by Daniel Loxton, Dec 11 2012

Growing up in a family of silvicultural contractors, much of my parents’ dinner table conversation revolved around the low bid system, which made the treeplanting industry a deeply exasperating, boom-and-bust roller-coaster. The challenge was how to work out bids which would cover the costs of the work and allow us to keep a roof over our heads through the winter, while also somehow outbidding people who didn’t know how to do that. Maddeningly, startup treeplanting companies were constantly jumping in with an incomplete appreciation of the complexity and risks (and therefore true costs) of running large contracts. The inexperienced routinely underbid the experienced—leaving us hungry—and then went promptly out of business. So I grew up with the lesson that things are always more complicated than they look at first glance—a lesson my work in skepticism has never failed to confirm.

At art school, I learned another lesson no less applicable to understanding paranormal and fringe claims: originality isn’t really a thing. Too many people have been sharing ideas for too long to support the conceit that our own ideas offer much in the way of substantial novelty.

I’m reminded of both lessons right now, as my research takes me a little deeper into skeptical history. Case in point: the well known and often repeated skeptical slogan, “Extraordinary claims demand extraordinary evidence.”

“Jos. F. Rinn” (from the Bain Collection at the Library of Congress Prints and Photographs Division)

Wonderful Phenomena

Among the early skeptical thinkers, authors, and activists I’m exploring at the moment, Joseph F. Rinn must be among the most fascinating. A lifelong friend of Harry Houdini, Rinn had the financial resources, conjuring skills, and vast personal experience in psychical investigation (as it was then called) to make him perhaps the most powerful voice for skepticism active during the first two decades of the twentieth century. “In the field of psychical research, my friend Harry Houdini and I were destined to play similar and, in some respects, parallel roles,” Rinn recalled late in his life. “Although both of us were sons of poor parents, he rose to the highest position in the amusement world and I, in my line, in the commercial field. All through our lives we were the outstanding foes of psychic impostors.”1

In a typical 1911 Washington Post article, Rinn went after those scientists who endorsed the allegedly supernatural feats of spirit mediums. He argued, “Wonderful phenomena demand wonderful evidence in their support, and no such evidence exists, as at no time have mediums produced phenomena under conditions where fraud has been impossible.”2

Rinn’s “wonderful phenomena” line caught my attention, of course. It turns out this was a line he used more than once, and probably often. In 1906, for example, Rinn told the New York Times, “Wonderful phenomena need wonderful evidence in their support to be of any value so far as truth is concerned, otherwise they are valueless.”3

Although rarely discussed today, Rinn’s work certainly influenced the late-twentieth century skeptical movement. Could the modern “extraordinary claims” slogan have been based on Rinn’s line?

Extraordinary Claims

Well, let’s have a look—keeping in mind that this a tentative, preliminary exploration. (If you have citations or insight to add, please do note them in the comments.)

Credit for popularizing the phrase “extraordinary claims demand extraordinary evidence” is usually given to astronomer Carl Sagan,4 with many sources clarifying that Sagan paraphrased this slogan from an earlier saying attributed to sociologist (and founding Skeptical Inquirer editor) Marcello Truzzi.5 A few sources, looking deeper, note that similar sentiments were earlier expressed by French mathematician Pierre-Simon Laplace6 and Scottish philosopher David Hume.7

All that is roughly true, but the devil (and the fun, for nerds like me) is in the details. To begin with, it was not clear to me when Sagan first used the exact phrase “extraordinary claims demand extraordinary evidence,” as he preferred throughout his career the formulation “extraordinary claims require extraordinary evidence.” Like “billions and billions,” he did eventually use the “demand” phrase that is now commonly attributed to him, but not often, and not (it seems) early in the game. The earliest source I have in which Sagan is directly quoted to that effect is a 1994 New York Times Magazine feature about alien abduction researcher John Mack:

“I tried to argue that on issues of this importance, extraordinary claims demand extraordinary evidence,” Sagan says. “And John would have none of that. He was quite content with anecdotal cases and his judgement that these people must be telling the truth because they are so extremely distraught.8

Sagan was, on the other hand, regularly using that “require” line by 1974, when it appeared in his book Broca’s Brain.9 This use predates by a year the earliest articulation of this sentiment by Truzzi that I am aware of (though it may well be the case that either or both men may have used some version or versions of the line at still earlier dates).

In 1975, Truzzi made an argument in a letter to Parapsychology Review that closely paralleled Rinn’s argument in 1911:

It is the basic character of science to be both conservative and demand parsimony in its explanations. Extraordinary claims require extraordinary proof. If a witness tells us he has seen two men shaking hands last Saturday, this is simply not requiring of the same level of proof as someone who states he saw two men floating in the air in defiance of the laws of gravity. Both may be correct, but scientific proof in each case is not the same. The first witness might be lying and the second telling the truth, but science is more willing to accept the testimony of the former with less supporting evidence than the latter since the latter would demand a radical reconstruction of our view of nature.10

Truzzi employed a number of minor variations on this “proof” formulation, including, “An extraordinary claim requires extraordinary proof,”11 and “when such claims are extraordinary, that is, revolutionary in their implications for established scientific generalizations already accumulated and verified, we must demand extraordinary proof.”12 (He explicitly linked this with Hume’s comments on miracles.) Since the 1970s, versions of the Truzzi-associated “proof” line have lived alongside Sagan’s “evidence” line, with some authors preferring one over the other, and at least one 1995 Skeptical Inquirer article employing both.13 Among these, the earliest use of the canonical “extraordinary claims demand extraordinary evidence” version I’m aware of within the modern (post-1975) skeptical literature dates to 1980 (in which no attribution is given).14

Of course, Rinn’s “wonderful phenomena” saying tells us that variations on this theme were part of the main thrust of skeptical thought long before Sagan or Truzzi were born. Moreover, it seems that early twentieth century versions of this sentiment were neither limited to Joseph Rinn, nor limited to skeptics. For instance, consider this passage from a pro-paranormal book from 1911:

Extraordinary as are the claims made, that psychic images—invisible to the human eye—can be photographed, they pale into insignificance before that which claims not only that these images but written messages come on photographic plates which have never been exposed. Extraordinary claims demand extraordinary evidence, and this is given as freely as possible within my limited space. I could readily devote the entire book to psychography, but the following must suffice.15

Likewise, similar constructions had currency amongst nineteenth century religious apologists, either in defense of their own faiths or in criticism of others. Here is an example from 1895:

Extraordinary claims always call for extraordinary proof.
Christ, the King, makes the most wonderful claims. He deliberately claims divinity. He declared (1) that he came from God; (2) that he existed with the Father before the world was; (3) that he was one with the Father; (4) that he fully revealed the Father in his own personality; (5) that he was a divine King, and as such he claimed the homage an4 allegiance of men; (6) that he was going to the Father when he should leave this world; (7) that he would come a second time to call forth the dead from their graves and sit in judgment upon an assembled universe. … When the King arrived, he proceeded to substantiate his most extraordinary claims by the most wonderful miraculous works and his still more wonderful teachings. In him a long line of prophecy was seen to have a perfect fulfillment.16

And another from 1878:

Upon the supposition that Revealed Truth is possible, it is to a certain extent probable. So far, then, miracles in attestation of it ought to be performed, because they afford extraordinary proof commensurate with extraordinary claims; because, as we have shown, they do not violate the laws of nature, and because human confidence could not in any other way be so well secured; therefore, if revelation be possible, the position of the infidel is impossible in the nature of things, and this shows the absurdity of its assumption.17

This last author, a clergyman, used variants on the “extraordinary claims” line in other writing, including this 1868 theological critique of Spiritualism (which he considered Satantic, or more colorfully, “sensualizing Pantheism taught through necromancy”18):

When Christianity began its career, it had to produce extraordinary evidence for extraordinary claims. So modern Spiritualism, being an advance upon Christianity, has to present supernatural props in the method chosen by spirits of telegraphing from the Spirit-world through the nervous system of living agents suited to their purpose. And what matters it what may be the means employed, so long as the end is secured?19

In 1856, that same clergyman wrote an entire book attacking the new Spiritualist religion on theological grounds, which included this line:

We contend that extraordinary claims must be sustained by corresponding proof, and that under circumstances which must be satisfactory to reasonable minds.20

Other Christians employed the same construction in their arguments against what they perceived to be theological wrong-headededness, as in this 1877 passage from a critique of “Irvingism”:

No one can doubt the sincerity of the writer, and as he is well aware that extraordinary claims need to be supported by extraordinary proofs, he narrates not only some events which seem to be of a purely subjective character, but some also which, if duly attested, look as if they must be supernatural. But may not the writer have been led away by feeling into being a bad judge of evidence?21

Here is similar line from from another author in 1824, at which point it was turned against the Quakers:

If any enter claims to the succession of such high offices, let us see also the succession of their gifts. Do any demand our faith and obedience as prophetesses now? Let us see the commanding proofs of their miraculous or prophetical powers? This is what the church demanded in every age; and this was unhesitatingly complied with by every true prophet. It is as reasonable now as in ancient times. Extraordinary claims can rest only on extraordinary proofs. Ordinary claims are supported by common proofs. The Friends have none of the apostolic powers to display. Their female prophets can not work miracles, no, nor predict.22

Laplace and Hume

Clearly the “extraordinary claims ” concept was out in the wild long before Sagan or Truzzi. Sources that link the concept to Pierre-Simon Laplace and David Hume are no doubt correct to do so. Both influential thinkers considered this idea in some detail.

Writing in 1814, Laplace argued,

From what precedes, we ought generally to conclude that the more extraordinary the event, the greater the need of its being supported by strong proofs. For those who attest it, being able to deceive or to have been deceived, these two causes are as much more probable as the reality of the event is less. We shall see this particularly when we come to speak of the probability of testimony.23

He added,

We see thus how the probability of the falsehood increases in the measure that the deed becomes more extraordinary.24

Laplace continued,

Applying this conclusion to all extraordinary deeds it results from it that the probability of the error or of the falsehood of the witness becomes as much greater as the fact attested is more extraordinary. Some authors have advanced the contrary on this basis that the view of an extraordinary fact being perfectly similar to that of an ordinary fact the same motives ought to lead us to give the witness the same credence when he affirms the one or the other of these facts. Simple common sense rejects such a strange assertion ; but the calculus of probabilities, while confirming the findings of common sense, appreciates the greatest improbability of testimonies in regard to extraordinary facts.25

And, as Hume had done earlier, Laplace argued, “There are things so extraordinary that nothing can balance their improbability.”26

Writing in the mid-eighteenth century (c.1739) Hume developed these ideas in depth—not only of the effective impossibility of confirming the occurrence of a miracle through testimony, but of the general problems of testimony, evidence, and claims of the merely very weird. Much of his thinking may be summed up in a line that is often cited as an equivalent to (or inspiration for) the modern “extraordinary claims” slogan:

A wise man, therefore, proportions his belief to the evidence.27

Altogether less pithy, but no less relevant to modern skeptical concerns about witnesses to allegedly paranormal events, was this warning:

We entertain a suspicion concerning any matter of fact, when the witnesses contradict each other; when they are but few, or of a doubtful character; when they have an interest in what they affirm; when they deliver their testimony with hesitation, or on the contrary, with too violent asseverations. There are many other particulars of the same kind, which may diminish or destroy the force of any argument, derived from human testimony.

Suppose, for instance, that the fact, which the testimony endeavours to establish, partakes of the extraordinary and the marvellous; in that case, the evidence, resulting from the testimony, admits of a diminution, greater or less, in proportion as the fact is more or less unusual.28

Hume distinguished between events that are simply bizarre or unexpected, and those that apparently overturn well-tested and previously reliable natural law:

No Indian, it is evident, could have experience that water did not freeze in cold climates. This is placing nature in a situation quite unknown to him; and it is impossible for him to tell à priori what will result from it. It is making a new experiment, the consequence of which is always uncertain. One may sometimes conjecture from analogy what will follow; but still this is but conjecture. And it must be confessed, that, in the present case of freezing, the event follows contrary to the rules of analogy, and is such as a rational Indian would not look for. The operations of cold upon water are not gradual, according to the degrees of cold; but whenever it comes to the freezing point, the water passes in a moment, from the utmost liquidity to perfect hardness. Such an event, therefore, may be denominated extraordinary, and requires a pretty strong testimony, to render it credible to people in a warm climate: But still it is not miraculous, nor contrary to uniform experience of the course of nature in cases where all the circumstances are the same. The inhabitants of Sumatra have always seen water fluid in their own climate, and the freezing of their rivers ought to be deemed a prodigy: But they never saw water in Muscovy during the winter; and therefore they cannot reasonably be positive what would there be the consequence.29

So where does that leave us? Well, I have no idea who should get the credit for this most popular of skeptical slogans—no one? everyone?—but I’m not at all surprised to learn that it’s complicated. This little blog post can hardly be an exhaustive exploration of this concept; it’s probably much more complicated than I imagine. But we could have predicted that in the first place, based on the depth and fractal complexity of history.

Indeed, as a wise man once remarked, “I have yet to see any problem, however complicated, which when looked at in the right way did not become still more complicated.”[CITATION NEEDED]


  1. Rinn, Joseph. Sixty Years of Psychical Research. (The Truth Seeker Company, 1950.) p. 1. Unfortunately, Rinn was on the wrong side of the science in his vocal opposition to smallpox vaccination. “Call Vaccination a Medical Sham. Speakers Tell Brooklyn Philosophical Association It Doesn’t Bring Immunity from Smallpox.” The New York Times. February 6, 1911. p. 4
  2. “Ghosts and Sprits Made to Order. Mr. Joseph Rinn, Friend and Critic of Professor Hyslop, Scoffs at Psychic Phenomena and Describes How He Himself Has Reproduced Mediums’ Feats by Trickery.” The Washington Post. December 10, 1911. p. M8
  3. “Topics of the Times.” The New York Times. January 17, 1906. p. 10
  4. Brezsny, Rob. “Free Will Astrology.” The Village Voice (New York). December 7–13, 2005. p. 120
  5. Shermer, Michael. “Skeptical Sayings: Wit and Wisdom from Skeptics Past and Present.” Skeptic. Vol. 9. No. 2, 2002. p. 25/li>
  6. “Frequently Asked Questions.” (Accessed December 10, 2012)
  7. Shermer (2002); Pigliucci, Massimo. “Do Extraordinary Claims Really Require Extraordinary Evidence?” Skeptical Inquirer. Vol. 29, No. 2. March/April 2005. p. 14
  8. Rae, Stephen. “John Mack.” The New York Times Magazine. March 20, 1994. pp. 30–33
  9. Sagan, Carl. Broca’s Brain. (New York: Ballantine Books, 1993.) p. 73
  10. Truzzi, Marcello. Letter to the editor. Parapsychology Review. November- December 1975, pp. 24-25. (My thanks to Timothy Binga and the Center for Inquiry Libraries for providing me with a copy of this difficult-to-find source.)
  11. Truzzi, Marcello. “On the Extraordinary: An Attempt at Clarification.” Zetetic Scholar. Vol. 1, No. 1. 1978. p. 11
  12. Truzzi, Marcello. “Editorials.” The Zetetic. (aka Skeptical Inquirer.) Vol. 1, No. 1. Fall/Winter, 1976. pp. 3–6
  13. McLaughlin, J. P. “On Leaping and Looking and Critical Thinking: Boulder Critical Thinking Workshop.” Skeptical Inquirer. Vol. 19, No. 3. May/June 1995. p. 7
  14. Mazurek, Kas and Brian Titley. “Teaching Critical Thinking: How to foster a critical scientific attitude toward anomalous claims.” Skeptical Inquirer. Vol. 5, No. 2. Winter 1980-81. p. 55
  15. Coates, James. Photographing the Invisible: Practical Studies in Spirit Photography, Spirit Portraiture, and other Rare but Allied Phenomena. (London. L. N. Fowler & Co., 1911.) p. 339
  16. Zollars, Ely Vaughan. The Great Salvation. (The Standard Publishing Company, 1895) pp. 78–79
  17. Gordon, William. The Science of Revealed Truth Impregnable, as Shown by the Argumentative Failures of Infidelity and Theoretical Geology. (Board of Publication of the Reformed Church in America, 1878.) pp. 120–121
  18. Gordon, William R. “Spiritualism. The More Specious of All Present Forms of Infidelity.” Prophetic Times and Watch Tower. Volume 6, Issue 2. February, 1868. p. 26
  19. Ibid. p. 27
  20. Gordon, William R. A Three-Fold Test of Modern Spiritualism. (New York: Charles Scribner, 1856.) pp. 260–261
  21. “On the Movement Commonly Called Irvingism.” The Guardian, February 14, 1877. p. 228
  22. Brownlee, Craig. A Careful and Free Inquiry into the True Nature and Tendency of the Religious Principles of the Society of Friends, Commonly Called Quakers. (Philadelphia: J. Mortimer, 1824) p. 201
  23. Laplace, Pierre Simon. Frederick Truscott and Frederick Emory, trans. “A Philosophical Essay on Probabilities.” (New York: J. Wiley, 1902.) p. 17
  24. Ibid. p. 113
  25. Ibid. p. 114
  26. Ibid. p. 119
  27. Hume, David. Antony Flew, ed. An Enquiry Concerning Human Understanding. (La Salle, Illinois: Open Court, 1993.) p. 144
  28. Ibid. p. 146
  29. Ibid. pp. 147–148

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18 Responses to “Wonderful Phenomena Demand Wonderful Evidence”

  1. Joseph says:

    “A thousand phenomena present themselves daily which we cannot explain, but where facts are suggested, bearing no analogy with the laws of nature as yet known to us, their verity needs proofs proportioned to their difficulty. A cautious mind will weigh well the opposition of the phenomenon to everything hitherto observed, the strength of the testimony by which it is supported, and the errors and misconceptions to which even our senses are liable.”

    – Thomas Jefferson, letter to Dr. Daniel Salmon, February 15, 1808

    • “A thousand phenomena present themselves daily…”

      Oh, hey, neat. That’s a new one by me!

    • “No testimony is sufficient to establish a miracle…”

      This one I had. It’s often cited in relation to the extraordinary claims line, but like Laplace’s “There are things so extraordinary that nothing can balance their improbability,” it seems to make a slightly different or further point that even extraordinary evidence isn’t always enough. That’s a relevant argument to be sure, but I think not quite the same argument.

  2. Other Paul says:

    I suppose one might settle it by convention, in that a decision might be made, and consensus reached, as to what constitutes sufficient pith-depth. Since clearly the central idea has been around for rather longer than its conciseness of expression. Much like Sod’s Law.

  3. Max says:

    “At art school, I learned another lesson no less applicable to understanding paranormal and fringe claims: originality isn’t really a thing.”

    Hey, you stole that from the Bible. “There is nothing new under the sun.”

    • Other Paul says:

      Hey, you stole that from the Bible. “There is nothing new under the sun.”

      I’d be impressed if Old King Sol was the first to have expressed that thought (however pithily).

      Then again, the first one who did, pretty much instantly contradicted herself.

  4. Trimegistus says:

    I had not known about Joseph Rinn. Thanks for bringing an unsung hero of skepticism to a wider audience.

    As to the “wonderful phenomena” quote, I’m betting on LaPlace as the originator of the parallel formation which has come down to us today. Hume definitely had the idea, but your cite from LaPlace looks like the soource of the deft construction we all love and repeat.

  5. DoctorAtlantis says:

    I went through a similar research of the oft quoted “People tend to remember the hits and forget the misses.” Turns out that’s an oldie but a goodie too. And WAY older than I expected.

    • Yes, it sure is! One of my favorite iterations goes back to c.1580:

      Besides, nobody keeps a record of their erroneous prophesies since they are infinite and everyday; right predictions are prized precisely because they are rare, unbelievable and marvellous.
      —Michael de Montaigne. M. A. Screech, trans. “On Prognostications.” Michael de Montaigne: The Complete Essays. (London: Penguin Books, 2003.) p. 44

      • John Wilkins says:

        It’s Francis Bacon, I think in the New Organon. From memory, “Men mark it when they hit and do not mark it when they miss.”

  6. tmac57 says:

    “Talk of big kill from hunt need big skin”
    -Zog 14,500 BP

  7. Janet Camp says:

    We have a concept in Anthropology called “independent invention”. Multiple people (cultures) can get similar ideas around the same time which results in similar “advances” seen throughout the world.

    Since good minds were exploring the world around them, it is not too hard to think that a number of people came to the conclusion that miraculous-seeming events had no logical explanation and they would begin to challenge that idea.

    • There’s every likelihood that variations of this saying have been cooked up by multiple authors independently, but Hume casts such a long shadow that diffusion seems even more likely in many cases.

      The clergyman William Gordon, for example, seems to have known Hume (and disliked him quite a bit). He wrote,

      Open infidelity now no longer rejoices in the intellectual power of giants. They are all dead, though their cause yet lives ; and is furious in its weakness. The war-worn weapons of Porphory and Celsus, of Gibbon, Hume, Voltaire, and Paine, and others sharing their inglorious fame, are wielded now by feebler hands unguided by a competency of wit and skill; and furthermore, by frequent smiting on the bosses of her buckler, their edges have been badly battered. Hence they are unfit for service, unless relaid, and ground by “spirit- hands,” and used under the direction of “spirit-brains,” because the human have been worn out in the service.
      —Gordon, William R. A Three-Fold Test of Modern Spiritualism. (New York: Charles Scribner, 1856.) pp. 346

      For his part, Marcello Truzzi cited Hume in this context as early as 1976, writing,

      And when such claims are extraordinary, that is, revolutionary in their implications for established scientific generalizations already accumulated and verified, we must demand extraordinary proof. This is especially true for claims bordering on the miraculous, for Hume long ago showed us that false witness is a far more likely explanation for a miracle than is the contradiction of an established, lawlike generalization.
      —Truzzi, Marcello. “Editorials.” The Zetetic. (aka Skeptical Inquirer.) Vol. 1, No. 1. Fall/Winter, 1976. pp. 3–6

  8. Chris says:

    Excellent post. Fwiw, a couple of copy editing notes: you have Hume writing in the ‘mid-sixteenth century (c. 1739)’ – obviously that should be eighteenth century. In the same sentence you have ‘effective impossibility of establishing verifying a miracle’ – either there is a missing word (probably ‘or’) or one of either ‘establishing’ or ‘verifying’ should be deleted. (Feel free to delete this comment also.)

  9. d brown says:

    “independent invention” Somebody said when its time to steam there are steam boats, somewhere. When the teck is there it happens. Its not the idea but the how. Arthur C Clark said something like Extraordinary Evidence in the early 60’s about UFOS. If enough people make enough clams one of them will hit.

  10. Austerity says:

    Though not strictly relevant to tracking the origin of the “slogan”, there is an interesting side story involving Thomas Bayes and the probability theorem named for him. It is interesting because the idea expressed in the slogan is also embodied in a statistical theorem that is in wide use today in the sciences.

    Sometime in the 1740’s (probably) Bayes wrote “An Essay towards Solving a Problem in the Doctrine of Chances”, where he set forth the earliest version of the idea that would eventually be developed, by others, into Bayes’ Theorem. It is a theorem that attempts to infer causes from observed evidence (often referred to later as inverse probability). In 1763 Richard Price had the paper, including Price’s own commentary, published with the Royal Society. This was two years after Bayes’ death.

    At roughly the same time, 1767, Price sent Hume a copy of his own “Four Dissertations” which used Bayes’ method to argue against Hume’s statement that “No testimony is sufficient to establish a miracle.”

    In 1774 Laplace published his own early version of inverse probability, but only in words. An equation would have to wait until 1812 when Laplace put the theorem in its modern form.

    So why is this only a side story and not part of the main story? First, Bayes’ paper was only discovered by Price in 1761 when Bayes died, so Hume would not have been familiar with it when he was writing. Though Hume did respond positively to Price’s “Four Dissertations” it isn’t clear if he was specifically referencing the inverse probability argument, or if it effected subsequent editions of his writing. Second, though Laplace wrote after the publication of the Bayes paper it is unlikely he knew about it since continentals were not in the habit of reading lowly Brits at the time. He probably had heard of Bayes by 1812, and perhaps before, but most researchers seem to think Laplace’s work was mostly independent. Finally, Bayes examined cases where prior probability was not important (i.e. there was nothing concerning cases where any claims whee “extraordinary”). Laplace did more or less the same but he clearly understood the need to consider prior probability (he just never worked out any examples). Still, it is interesting that questions Hume was struggling with ended up not just in some slogans but in a mathematical theorem that, after many years of being sort of the bastard child of probability theory, is widely used in the sciences.

    Bayes’ theorem is attempting to answer the question how likely is my hypothesis given the evidence I have. The key pieces of information needed to get an answer are (1) how likely is the hypothesis to start with given what I already know (prior probability), and (2) what is the relative likelihood of my cause producing this evidence compared to all other hypotheses. So a hypothesis with a low prior probability can only be considered likely if the evidence is such that only that hypothesis, and not more mundane ones, are likely to produce the evidence we have. I think you’ll find this to be a fine formalization of all the quotes above.

    Hume’s formulation in “On Miracles” may be the best though: “No testimony is sufficient to establish a miracle [a low prior probability event based on our past experience and knowledge of the world], unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish [unless the evidence is much less likely on any mundane hypothesis]”

    And finally, “Extraordinary claims [claims with a low prior probability based on our past experience and knowledge of the world] demand extraordinary evidence [evidence that is much less likely on any mundane hypothesis].”