There is further discussion of this in Appendix C; will this be discussed in connection with Chapter 1, or at some later time in the sequence? For example, in Appendix C, it turns out that desideratum 1 subdivides into two other axioms: transitivity, and universal comparability. The first one makes sense, but the second one doesn't seem as compelling to me.
It is indeed an extremely interesting question! Perhaps it would be wiser to use complex numbers for instance.
But intuitively it seems very likely that if you tell me two different propositions, that I can say either that one is more likely than the other, or that they are the same. Are there any special cases where one has to answer "the probabilities are uncomparable" that makes you doubt that it is so?
Great! I'm looking forward to this! :)
I'm a little late, but I'm in!
Very interesting! But I have to read up on the Appendix A4 I think to fully appreciate it...I will come back if I change my mind after it! :-)
My own, current, thoughts are like this: I would bet on the ball being white up to some ratio...if my bet was $1 and I could win $100 I would do it for instance. The probability is simply the border case where ratio between losing and winning is such that I might as well bet or not do it. Betting $50 I would certainly not do. So I would estimate the probability to be somewhere between 1 and 50%...and somewhere there is one and only one border case in between, but my human brain has difficulty thinking in such terms...
The same thing goes for the coin-flip, there is some ratio where it is rational to bet or not to.