Extraordinary evidence although it might be intrinsically ordinary, must be extraordinarily in its implications. If it is offered as proof that a natural law has been violated, then the violation of a natural law — and nothing else — must be the only way to account for it.
In other words, "Extraordinary claims require extraordinary evidence" is a kind of restatement of Occam's razor. If our conventional beliefs can account for the evidence, then we don't need any new beliefs.
Conventional beliefs as I have here defined them would include any consensus of the scientific community. Although science affirms things that seem to go beyond common experience, the scientific method leading to such affirmations is ultimately based on common experience. This is what the principle of replicability is all about. Any person can, in principle, do any of the experiments on which any scientific theory, no matter how counterintuitive, is based.
If this seems to stack the deck in favor of conventional thinking, then so be it. Our survival as a species historically has depended, and still does depend, on our being reluctant to change our beliefs when our beliefs have served us well. Of necessity, it ought to be difficult to convince us that lessons we have learned and confirmed by long experience should be unlearned.
But the value of being resistant to changing one's mind does not imply a greater value to being immune to it. The challenge is to figure out when we cross the line between prudent resistance and imprudent obstinance. It is a challenge not easily met, and there is no good formula universally agreed upon that anyone can use to define the line.
What we could all agree to, though, is to give each other credit for some good faith.
Every one of us, skeptic and believer, holds to certain ideas that it would be extremely difficult to convince us are wrong. For good or ill, we are not going to believe that something happened contrary to those ideas just because a few people say it happened. Rationalists are by no means the only people who insist on extraordinary evidence for extraordinary claims.
Where we are perhaps trying to be different is in our attempt to objectivize the distinction between the credible and the incredible. When we call a claim extraordinary, we are merely noting that you are asking us to abandon a fundamental belief about how the universe works. We are no more ready than you are to do that just on somebody's say-so, no matter how reputable that somebody is.
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