When we say extraordinary claims require extraordinary evidence, the amount of evidence neccessary is probably determined by the method of evaluation. If your neighbor told you that he watered his plants yesterday, it would be a reasonable claim to believe. But if he told you he fed his pet three headed 50 foot alien, it would probably not be reasonable to believe at this particular point. Your neighbor would have to elaborate and give you evidence that this 50 foot three headed alien exists and is his pet. So in this sense, the evidence required to believe his statement required something beyond what was otherwise neccessary for a natural claim, hence "extraordinary." The claim didn't fit our prior well established body of knowledge and therefore required some extra proof of its soundness.
It's quite amusing to hear self-proclaimed "rationalists" attempt to make use of this quote, since the claim that "extraordinary evidence" is required is fundamentally illogical. Because that which is supernatural must interact with the natural in order to be perceived, most supernatural activity will leave natural footprints which are capable of being evaluated by fully natural means. A poltergeist is supernatural, while a vase smashed by a poltergeist, a video of a vase being smashed by an invisible force, and an audio recording of an observing scientist watching a vase being smashed by an invisible force are all natural things that could be provided as evidence for the supernatural.
A proper scientific study of the supernatural, as proposed by the likes of Daniel Dennett, will look no different and provide evidence that is no more extraordinary than the evidence that is provided for any natural claim. Whether one is studying the utility of prayer, Vitamin C, or surgery in curing cancer, the means and the evidence produced will be the same.
As for the example of the watered plants and the alien pet, Ninja Rabbit has created a very poor analogy. It's not a proper comparison. He's assuming knowledge on the one part and denying it on the other; how do we know that the plants are actually his neighbor's? We don't; perhaps he doesn't own them or even the house. How do we know if he watered them or not? We don't. Perhaps they're plastic. Ninja Rabbit is making a series of baseless assumptions based on the fact that because he knows other people own plants that they water, and because he thinks that his neighbor has no motivation to lie to him, his neighbor must be telling him the truth.
The reason the analogy is poor is obvious if we consider a more equitable version of it. Ninja Rabbit lives next door so he can see the plants. He knows they exist and are in decent health, so he concludes someone must be watering them. His neighbor claims to be doing so, so he accepts the claim. Now, if he also saw the three-headed 50-foot alien next door on as regular a basis and it appeared to be in good health, his neighbor's claim to own it and feed it would be no more remarkable than his claim to have watered his plants.
It should be clear to everyone, now, that "extraordinary claims" require no extra proof of their soundness, the only difference between an "extraordinary claim" and an ordinary one is that there is usually less tangential knowledge surrounding what is described as an extraordinary one. But that tangential knowledge should not be confused with evidence in itself, especially when it does not even support the specific claim being made.
How do you know if the evidence you are presented with matches the claim? How do you know the smashed vase wasn't a trick to make it seem supernatural? What is more likely, natural forces leaving natural footprints? Or supernatural forces leaving natural footprints? How would you be able to tell? There seems to be a problem with Vox's argument but I need help to articulate it. Can anyone here help me?