All of sophist's Comments + Replies

Perhaps it might be wiser to use measures (distributions), or measures on spaces of measures, or iterate that construction indefinitely. (The concept of hyperpriors seems to go in this direction, for example.) Consider the following propositions. P1: The recently minted U.S. quarter I just vigorously flipped into the air landed heads on the floor. P2: A ball pulled from an unspecified urn containing an unspecified number of balls is white. P3(x): The probability of P2 is x Part of the problem is the laxness in specifying the language, as I mentioned. For example, if the language we use is rich enough to support self-referring interpretations, then it may not even be possible to coherently assign a truth value--or any other probability, or to know whether that is possible. But even ruling out Goedelian potholes in the landscape and uncountably infinite families of propositions, the contrast between P1 and P2 is problematic. P1 is backed up by a vast trove of background knowledge and evidence, and our confidence in asserting Prob(P1) = 1/2 is very strong. On the other hand, background knowledge and evidence about P2 is virtually nil. It is reasonable as a matter of customary usage to assume the number of balls in the urn is finite, and thus the probability of P1 is a rational number, but until you start adding in more assumptions and evidence, one's confidence in Prob(P2) < x for any particular real number x seems typically to be very much lower than for P1. Summarizing one's state of knowledge about these two propositions onto the same scale of reals between 0 and 1 seems to ignore an awful lot that we know about the relative state of knowledge vs. ignorance with respect to P1 and P2. An awful lot of knowledge is being jettisoned because it won't fit into this scheme of definite real numbers. To make the claim Prob(P2) = 1/2 (or any other definite real number you want to name) just does not seem like the same kind of thing as the claim Prob(P1) = 1/2. It feels

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