Let's see if we can try to hug the query here. What exactly is the mistake I'm making when I say that I believe such-and-such is true with probability 0.001?

Is it that I'm not likely to actually be right 999 times out of 1000 occasions when I say this? If so, then you're (merely) worried about my calibration, not about the fundamental correspondence between beliefs and probabilities.

Or is it, as you seem now to be suggesting, a question of attire: no one has any business speaking "numerically" unless they're (metaphorically speaking) "wearing a lab coat"? That is, using numbers is a privilege reserved for scientists who've done specific kinds of calculations?

It seems to me that the contrast you are positing between "numerical" statements and other indications of degree is illusory. The only difference is that numbers permit an arbitrarily high level of precision; their use doesn't automatically imply a particular level. Even in the context of scientific calculations, the numbers involved are subject to some particular level of uncertainty. When a scientist makes a calculation to 15 decimal places, they shouldn't be interpreted as distinguishing between different 20-decimal-digit numbers.

Likewise, when I make the claim that the probability of Amanda Knox's guilt is 10^(-3), that should not be interpreted as distinguishing (say) between 0.001 and 0.002. It's meant to be distinguished from 10^(-2) and (perhaps) 10^(-4). I was explicit about this when I said it was an order-of-magnitude estimate. You may worry that such disclaimers are easily forgotten -- but this is to disregard the fact that similar disclaimers always apply whenever numbers are used in any context!

In any case, you seem to concede that these numbers ultimately don't convey any more information than various vague verbal expressions of confidence. If you want to make the latter more systematic and clear, I have no problem with that, but I see no way to turn them into actual numbers without introducing spurious precision.

Here's the way I do it: I think approximately in terms of the following "scale" of improbabilities:

(1) 10% to 50% (mundane surprise)

(2) 1% to 10% (rare)

(3) 0.1% (=10^(-3)) to 1% (once-in-a-lifetime level surprise on an important question)

(4) 10^(-6) to 10^(-3) (dying in a plane crash or similar)

(5) 10^(-10) to 10^(-6) (winning the lottery; having an experience unique among humankind)

(6) 10^(-100) to 10^(-10) (religions are true)

(7) below 10^(-100) (theoretical level of improbability reached in thought experiments).