After thinking about your comment, I think this observation comes close to the core of our disagreement:

By saying that a probability based only on common-sense reasoning is meaningless, I think what you're really trying to do is set a minimum quality level.

Basically, yes. More specifically, the quality level I wish to set is that the numbers must give more useful information than mere verbal expressions of confidence. Otherwise, their use at best simply adds nothing useful, and at worst leads to fallacious reasoning encouraged by a false feeling of accuracy.

Now, there are several possible ways to object my position:

• The first is to note that even if not meaningful mathematically, numbers can serve as communication-facilitating figures of speech. I have conceded this point.

• The second way is to insist on an absolute principle that one should always attach numerical probabilities to one's beliefs. I haven't seen anything in this thread (or elsewhere) yet that would shake my belief in the fallaciousness of this position, or even provide any plausible-seeming argument in favor of it.

• The third way is to agree that sometimes attaching numerical probabilities to common-sense judgments makes no sense, but on the other hand, in some cases common-sense reasoning can produce numerical probabilities that will give more useful information than just fuzzy words. After the discussion with mattnewport and others, I agree that there are such cases, but I still maintain that these are rare exceptions. (In my original statement, I took an overly restrictive notion of "common sense"; I admit that in some cases, thinking that could be reasonably called like that is indeed precise enough to produce meaningful numerical probabilities.)

So, to clarify, which exact position do you take in this regard? Or would your position require a fourth item to summarize fairly?

I think what's confusing you is an intuition that if someone gives a probability, you should be able to take it as-is and start calculating with it. [...] So in a sense, those sorts of probabilities are less meaningful - you discard them when they compete with better probabilities, or at least weight them less - but there's still a nonzero amount of meaning there.

I agree that there is a non-zero amount of meaning, but the question is whether it exceeds what a simple verbal statement of confidence would convey. If I can't take a number and start calculating with it, what good is it? (Except for the caveat about possible metaphorical meanings of numbers.)