Is the statement "Claims demand evidence" always true?

[Mudhammam]

 But the question is how can one know which?  I do not believe that you can go from experience to an understanding of the reality that underlies it.  You cannot know that the Matrix is not consistently managing the falling of apples or the resolution of quantum particles.  You cannot know that the Mind of God isn't constantly feeding to us images of things, their properties, and their dynamic relationships.  Nor can I know that there is more than a physical Universe, which perhaps is philosophically complex to be exclusively self-supporting, meaning that there are no other universes and no Creator at all.  This is because, hypothetically, all possible frameworks which are capable of providing experiences might possibly contain the organizational principles (all things attract by gravity, certain forces cause particles to interact in certain ways, etc.) with which we are familiar.

As a purely negative or skeptical conclusion, I agree with you.  We cannot know such truths with complete certainty.  But is there a spectrum in which some methods are objectively more likely to be closer to fact in their description of "the reality that underlies" experience?  Absolutely.  Is this a move that can only be made for pragmatic reasons?  I don't think so.  Logic and empirical verification go a long way, or so I think we have most reason to believe.  And either we have reason to be confident in the truths that we establish -- whether these should be conceived as absolute or provisional -- or, au contaire, to be skeptical of them, as you suggest.