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Nico Roos. A logic for reasoning with inconsistent knowledge. Artificial Intelligence Volume 57, Issue 1, September 1992, Pages 69–103.
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.
I appreciate the links. I haven't read Gaifman's paper before, so I'll go ahead and read that.
Anyhow, I won't cotton to any method of assigning a logical probability that takes longer than just brute-forcing the right answer. For this particular problem I think a bottom-up approach is what you want to use.
I see the sentiment there, and that too is a valid approach. That said, after trying to use the bottom-up approach many times and failing, and after seeing others fail using bottom-up approaches, I think that if we can at least build a nonconstructive top-down theory, that would be a starting point. After all, Solomonoff Induction is completely top down, yet it's a very powerful theoretical tool.
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 am a long time LessWronger (under an anonymous pseudonym), but recently I've decided that it is finally time to bite the bullet, abandon my few thousand karma, and just move over to my real name already.
Back in the day, when I joined LessWrong for the first time, I followed my general policy of anonymity on the Internet. Now, I'm involved with the Less Wrong community enough that I find this anonymity holding me back. Thus the new account.
Edit: For my first post on this new account, I posted a few of my thoughts on logical uncertainty.