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.
Hi Bejnamin! Have some links you might already have read.
For an introduction to the topic, you can't go wrong with the scholarly literature: Gaifman's "Reasoning with limited resources and assigning probabilities to arithmetical statements."
I will also happily recommend my own posts on the subject, beginning at the very beginning, here.
For breadth, you might also check out Abram Demski's description, which is good to think about though unsatisfying to me. There's also some discussion on lesswrong and a MIRI document uses a slightly modified version, somewhere.
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 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.
Hi Ben,
If you have not already seen them, you should probably read this and this, which are two ways to define a coherent probability measure on sentences. The first link my construction, and the second link is Abram Demski's. Abram and I are working on this problem in the MIRIxLosAngeles group.
You seem to be doing something different from what we are doing, in that our probability functions assign probability 1 to all provable statements, and try to answer the question of what to do with statements which are neither provable nor disprovable. (If you want your probability function to represent a probability measure on models of your logical system, then you cant just give them all probability 1/2)
For both of our constructions, the probability assignments are approximable in the sense that we have a computation which updates probabilities and converges to the probability in our construction (but never equals it exactly) and both of our computations work by updating as you observe proofs of sentences.
In particular I have a procedure which (I only conjecture right now) converges to my probability assignment by noticing over and over again "I can prove that exactly one of A, B, and C is true, but their probabilities do not sum to one" and then shifting the probabilities of these sentences accordingly. (This is not written up in the post I linked to, but I can explain it more if you like.)
If you do look at them, feel free to ask me questions about either construction, or if you would like I can talk to you about what we've done in a skype call or something. I have a few conjectures that I would like to prove related to my construction.
Also, neither one of us pays any attention to proof length, so it would seem reasonable for you to ignore what we did and go your own direction. However, if you find yourself thinking mostly about how to assign probabilities to unprovable things, then it is worth looking at.
I do have one suggestion on notation. I suggest you reserve the term "probability measure" for something more strict than just assignments of probabilities to sentences. Say "probability function" or "probability assignment." I would say a "probability measure on sentences" should represent a way to choose a single sentence at random, and a "probability measure on models" should be a way to choose a single model of your axioms at random.
I am excited to see what you guys do!
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.
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Nico Roos. A logic for reasoning with inconsistent knowledge. Artificial Intelligence Volume 57, Issue 1, September 1992, Pages 69–103.