That's my point. If you treat all shades of gray the same, the result is insanity. If you treat all shades of gray in any manner that doesn't follow the laws of probability, the result is insanity.

And, answering to the grandparent, if I say “I believe that she does not have cancer” I just mean that the posterior probability of her not having cancer given everything I know is greater than 50%.

You can use "believe" that way, but you can't act like everything is true iff it has a higher than 50% chance. You wouldn't want to leave someone untreated on the basis that they only have a 49% chance of having cancer.