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...

210y

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Well, care to explain what I did wrong?

310y

One nonconstructive (and wildly uncomputable) approach to the problem is this one: http://www.hutter1.net/publ/problogics.pdf

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 ...

510y

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.

10y15

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.

Nico Roos. A logic for reasoning with inconsistent knowledge. Artificial Intelligence Volume 57, Issue 1, September 1992, Pages 69–103.