Cyan: I certainly admit that the ease of the math may be part of my reaction. Maybe if I was far more familiar with the theory of functional equations I’d find Cox’s theorem more elegant than I do.
(I’ve read that if one makes a minor tweak to Cox’s theorem, just letting go of the real number criteria and letting confidences be complex numbers, the same line of derivation more or less hands you quantum amplitudes. I haven’t seen that derivation though, but if that’s correct, it makes Cox’s theorem even more appealing. QM for “free”! :))
The vulnerabiliy ones though actively motivate the criteria rather than a list of reasonable sounding properties an extention to boolean logic ought to have. The basic criteria, I guess, could be describes as “will you end up in a situation in which you’d knowingly willingly waste resources without in any way benefiting your goals?”
So each step is basically a “Mathematical Karma is going to get you and take away your pennies if you don’t follow this rule.” :)
But yeah, the bit about only reasonable extention of vanilla logic does make the Cox thing a bit more appealing. On the other hand, that may actively dissuade some from the Bayesian perspective. Specifically, constructivists, intuitionists in particular, for instance, may be hesitant of anything too dependant on law of excluded middle in the abstract. (This isn’t an abstract hypothetical. I’ve basically ended up in a friendly argument a while back with someone that more or less had them rejecting the notion of using probability as a measure of belief/subjective uncertainty because it was an extention of boolean logic, and the guy didn’t like law of excluded middle and basically was, near as I can make out, an intuitionist. (Some of the mathematical concepts he was bringing up seem to imply that))
Personally, I just really like the whole “math karma” flavor of each step of a vulnerability argument. Just a different flavor than most mathematical derivations for, well, anything, that I’ve seen. Not to mention, in some formulations, getting decision theory all at once with it.
Cyan: I certainly admit that the ease of the math may be part of my reaction. Maybe if I was far more familiar with the theory of functional equations I’d find Cox’s theorem more elegant than I do.
(I’ve read that if one makes a minor tweak to Cox’s theorem, just letting go of the real number criteria and letting confidences be complex numbers, the same line of derivation more or less hands you quantum amplitudes. I haven’t seen that derivation though, but if that’s correct, it makes Cox’s theorem even more appealing. QM for “free”! :))
The vulnerabiliy ones though actively motivate the criteria rather than a list of reasonable sounding properties an extention to boolean logic ought to have. The basic criteria, I guess, could be describes as “will you end up in a situation in which you’d knowingly willingly waste resources without in any way benefiting your goals?”
So each step is basically a “Mathematical Karma is going to get you and take away your pennies if you don’t follow this rule.” :)
But yeah, the bit about only reasonable extention of vanilla logic does make the Cox thing a bit more appealing. On the other hand, that may actively dissuade some from the Bayesian perspective. Specifically, constructivists, intuitionists in particular, for instance, may be hesitant of anything too dependant on law of excluded middle in the abstract. (This isn’t an abstract hypothetical. I’ve basically ended up in a friendly argument a while back with someone that more or less had them rejecting the notion of using probability as a measure of belief/subjective uncertainty because it was an extention of boolean logic, and the guy didn’t like law of excluded middle and basically was, near as I can make out, an intuitionist. (Some of the mathematical concepts he was bringing up seem to imply that))
Personally, I just really like the whole “math karma” flavor of each step of a vulnerability argument. Just a different flavor than most mathematical derivations for, well, anything, that I’ve seen. Not to mention, in some formulations, getting decision theory all at once with it.