Thanks for those links! They both look very intresting, and I’ll read them in depth.
As you mention, you are doing something slightly diffrent. You are assigning probability 1 to all the provable sentances, and then trying to investagate the unprovable ones. I, on the other hand, am taking the unprovable ones as just that, unprovable, and focusing on assigning probability mass to the provable ones.
I think the question of how to assign probability mass to provable, yet not yet proven, statments is the really important part of logical uncertanty. That’s the part that is handwaved away in discussions of, say, UDT, and so is the part that I want to focus on.
About your suggestion on notation: Yes, I was being slightly casul with notation there. By construction, it is a measure, I think, as always gives probabilities in the range [0,1], and it obeys the law of the excluded middle. I didn’t actually prove that the measure of multiple independant sentances is equal to the sum of the measures, but I think it follows… More work is needed on this. At the moment, this only gives probabilities to individual sentances, and not to gtoups of sentances, so technically that wouldn’t work at all. The obvois next step is tpo try to extend it in order to be able to do this. But until that is done, you are correct that it is abuse of notation to call it a measure.
I do not think it is a measure. If A B and C are all unprovable, undisprovable, but provably disjoint sentences, then your system cannot assign probability of A or B or C equal to P(A)+P(B)+P(C) because that must be 3⁄2.
I think that the thing that makes logical uncertainty hard is the fact that you cant just talk about probability measures (on models) because by definition a probability measure on models must assign probability 1 to all provable sentences.
That’s a good point, and I concede that you are right. At the moment, it’s more of a “probability assignment”, as you said, rather than a probability measure. More work needs to be done on the subject, and hopefully we will progress along these lines at the MIRIx workshop.
Thanks for those links! They both look very intresting, and I’ll read them in depth.
As you mention, you are doing something slightly diffrent. You are assigning probability 1 to all the provable sentances, and then trying to investagate the unprovable ones. I, on the other hand, am taking the unprovable ones as just that, unprovable, and focusing on assigning probability mass to the provable ones.
I think the question of how to assign probability mass to provable, yet not yet proven, statments is the really important part of logical uncertanty. That’s the part that is handwaved away in discussions of, say, UDT, and so is the part that I want to focus on.
About your suggestion on notation: Yes, I was being slightly casul with notation there. By construction, it is a measure, I think, as always gives probabilities in the range [0,1], and it obeys the law of the excluded middle. I didn’t actually prove that the measure of multiple independant sentances is equal to the sum of the measures, but I think it follows… More work is needed on this. At the moment, this only gives probabilities to individual sentances, and not to gtoups of sentances, so technically that wouldn’t work at all. The obvois next step is tpo try to extend it in order to be able to do this. But until that is done, you are correct that it is abuse of notation to call it a measure.
I do not think it is a measure. If A B and C are all unprovable, undisprovable, but provably disjoint sentences, then your system cannot assign probability of A or B or C equal to P(A)+P(B)+P(C) because that must be 3⁄2.
I think that the thing that makes logical uncertainty hard is the fact that you cant just talk about probability measures (on models) because by definition a probability measure on models must assign probability 1 to all provable sentences.
That’s a good point, and I concede that you are right. At the moment, it’s more of a “probability assignment”, as you said, rather than a probability measure. More work needs to be done on the subject, and hopefully we will progress along these lines at the MIRIx workshop.