Colyvan

). If I were omniscient I would even be able to tell you that it is exactly k/N (interpreting probability as the percentage, i.e. the chance to randomly select one of the k gold medalists > 180cm out of all N gold medalists.) but only in a perfect world.I guess what I am trying to say is this: If you want a numerical value for quantifying uncertainty from me, at least give me a precise hypothesis I can work with, then I'll see what I can do. If you want to specify a fuzzy membership function for what you mean by 'rainy days', I can try to work with that, too, but it has to be you giving me the membership function. I will not start guessing what you mean.

I really think that we should base probability (and possibility) distributions on our data. I do not know what a fair coin is supposed to be but I know how to count outcomes. And probability itself does not exist. Is it anything more than an idealized model (satisfying the Kolmogorov axioms) to which I can try and fit my data? And since (by its own definition) I will not be able to exactly deduce it from finite data, I have to think about how to quantify (more or less) plausible families of probability distributions, a.k.a. imprecise probabilities. This is what we all do all the time. Even the early scientists with their deficient notion of a set were working with a precise description that worked for their purposes until they found out it that with respect to some other problems, the definition needed refinement.

This was quite some rambling and I think somebody must have phrased this more elegantly (and more coherently) before me. What do you think?

Scott wrote:

But couldn't Holmes' fictionality likewise support a contrary probability of 1?

Maybe you are avoiding a larger point (not sure that Mark actually makes it) that probabilists hold that their calculus is appropriate for all pronouncements, including the ones that have no evidence either way. They have no problem at all with setting a probability for pronouncements about one-off events such as whether OJ actually killed those people, or hypothetical events involving as fictional characters, or even ethereal concerns such as whether God exists. (Talk about being non-scientific!)

In his paper about his squizzles, Roger Cooke, like you, protests that we have to know what all the words mean, so pronouncements about whether the slithy toves did or did not gyre and gimble in the wabe might fail in the clairvoyant test (clarity test). But words can have meanings despite there being dispute about them. And just because something is vague in the philosophical sense does not mean it is not clear. The number of rainy days this last October in Liverpool is perfectly clear, but it depends on arbitrarily specifying the number of millimeters of precipitation in the definition of 'rainy'. We talk about uncertainty of the third kind only to admit that it is absurd to say, as for instance the India Meteorological Department does, that a day with 2.51 mm of rain is a rainy day but one with 2.49 mm of rain is not. Using fuzzy structures, one could handle a range of definitions all at once and project the implications of that arbitrariness through subsequent calculations. It's not a matter of having to guess the definition, it's a matter of being clear about the vagueness (borderline-ness).

Lotfi may not have been a very nice man, but his idea to bring those non-scientific statements into the realm of science, by accounting for the arbitrariness, is a good one. He was wrong about a lot of things, but this idea seems reasonable to me, and it might be important. For example, the definition of an endangered species is set internationally by the IUCN criteria, but they are necessarily arbitrary. Species that are near--but technically not--endangered are not eligible for certain legal protections and conservation funding. It can therefore make sense for a motivated conservationist to go out and kill a few animals if doing so could tip the species over the line to reap those advantages. Such tyrannies of arbitrary reactions to vagueness are serious problems in many areas. I confess I've not seen a lot of applications of fuzzy methods successfully used to practically resolve or handle the arbitrariness, but I don't read in this area anymore so I might be missing them.

Now you're starting to talk crazy. What do you mean you don't know what a fair coin is supposed to be? I think you know. Kolmogorov never said probability doesn't exist, quite the opposite. Di Finetti said that craziness, and his narrowness on the subject was to say that he only wants to talk about subjective probability. As if flippable coins don't exist!

I do agree with you about this: "And since (by its own definition) I will not be able to exactly deduce it from finite data, I have to think about how to quantify (more or less) plausible families of probability distributions, a.k.a. imprecise probabilities."

Bring me Schrodinger's head!