Say I asked you to give evidence for the existence of atoms, and so you stood up, went to the sink, filled a clear glass with water, threw in a dash of soot from the fireplace and said “Voilà!” I look at you in puzzled confusion, and so you put a wooden ball in my left hand and a lead ball in my right and then say “there, d’ya get it now?”
So far as evidence is what you point to in order to prove your case, you’ve done just what I’ve asked. The jitters of the soot in water were one of the clinching pieces of evidence for atomism, and the differing weights of identical volumes was one of the most ancient pieces of evidence for it. While there’s no reason to take these examples as showing us something common to everything we call evidence, they do show that it is possible that evidence be a lot more than just what you point at to prove your case.
The word “evidence” therefore sometimes (often?) means not “something you show me” but “something you show me while telling me a story about it” or “something you show me while giving me an argument about it”, and since there’s no intrinsic limit on how complicated a story might be, how abstruse an argument might get, how much IQ, study, humility, or moral rectitude might be required to accept a story or argument, the demand for “plain evidence” for everything seems to involve a good deal of ignorance about what exactly evidence is.
The humdrum thing you point at (soot jiggling in a jar) might have no proportion to the magnitude of the thing it is evidence for (the composition of all bodies in the universe). So what are we to make of the claim that extraordinary claims require extraordinary evidence? This obviously isn’t true if evidence is just “what one points to to prove the case”: jiggling soot is hardly an extraordinary thing to point to, though a claim about the composition of the entire universe clearly is. So it must mean that an extraordinary claim requires an extraordinary story or argument in defense of it.*
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*Though one supposes that whatever proved an extraordinary claim would be ipso facto extraordinary, making the claim tautological. The “extraordinary claims” postulate is thus not a standard for judgment, but merely tells us that if an extraordinary conclusion is given, the proof itself is extraordinary. The claim is this no more interesting than saying “if an action ends illegally, it is illegal”.
Just Thomism 
Matthew McCormack said,
August 23, 2015 at 12:10 am
So, ‘evidence’ is evidence against positivism. There needs to be comprehension to have evidence.
So to add my two cents. As I have understood it. Evidence is not proof, and I always understood the 2nd ‘extraordinary’ in ‘extraordinary claims require extraordinary evidence’ as meaning a lot more evidence than usual, not that the evidence itself needs to be unusual. The evidence needs to be more than usual because the claim does not fit in with the currently accepted meta view, and the evidence needs to solidly connect it from many angles to what is accepted. This is because theories are underdetermined by evidence.
A non-extraordinary claim would not require ‘extraordinary’ evidence but would be accepted more readily because it was consonant with the accepted view. An extraordinary claim challenges the accepted view and because theories are underdetermined by evidence it is not as readily accepted.
I think the idea of extraordinary evidence being necessary for extraordinary claims could be true of the acceptance of heliocentrism in the 16th century.
Just my two cents.