Deeply complicating matters, 0 and 1 can be excluded from theories of probability, and there is a good informal reason to do it, known as Cromwell's rule. The issue is that there no longer are units in such a system; 0 is the disjunctive unit and 1 is the conjunctive unit. This is part of a bigger theme in maths where units are increasingly seen as optional, but Yud is completely unaware of this bigger theme.

[–] blakestacey@awful.systems 7 points 1 year ago* (last edited 1 year ago)

Personally, I've always found Cromwell's rule a deeply boring proposal for screwing with the axioms. Try doing the Dutch-book argument with surreal numbers, then I'll pay attention. :-P

(I would expect that many subjectivist Bayesians would take Cromwell's rule as an addition to the basic rules that are themselves justified by Dutch book or some such means. Not assigning sharp-edged probabilities out of general prudence is a thing an individual gambler can choose to do, if that's the way their tendencies lie, while not being part of the mathematical definition of the subject itself. But, well, 46,656 varieties and all that. Moreover, it is hard to do physics having chopped off the endpoints of the interval without chopping other structures as well. For example, if you don't even allow 0 and 1 to be available as idealizations, you might end up peeling the skin off quantum state space. Some could cope with this, but not Yud, since he demands that all of reality be a single pure quantum state. Insofar as any sense can be made out of Yud's rambles, he is wanting something stronger than Cromwell's rule, anyway, since he wants to forbid probability 1 even for logical implications, which Lindley allowed.)