Thanks for linking to your own proposal! I’m posting this before actually digesting it, so I may reply again later.
I agree that I’m tunneling on the mutually exclusive and exhaustive case, and the characterization “simpler sentences have probabilities closer to 1/2” is an accurate characterization of Abram’s scheme. I’m not so sure about Paul’s—I’m pretty sure it’s not minimizing a weighted entropy, but a cross-entropy, which tries to make the probability proportional to the pre-prior.
As for other examples, there are a variety of more dilletantish cases (though I’m not really one to talk, of course) of people just trying to port Solomonoff induction on sentences or on models, and similar proposals of things like decreasing probability as a function of logical depth (example).
I’d defend my focus on mutually exclusive and exhaustive sets by saying that they show up everywhere and are always a basis we can use to represent knowledge. For example, if I had a list of 7 sentences and knew that exactly 5 were true, I can completely characterize my knowledge by probabilities of the 7 choose 5 mutually exclusive and exhaustive possibilities.
That said, it’s certainly possible there are things I’m missing because of this focus.
Thanks for linking to your own proposal! I’m posting this before actually digesting it, so I may reply again later.
I agree that I’m tunneling on the mutually exclusive and exhaustive case, and the characterization “simpler sentences have probabilities closer to 1/2” is an accurate characterization of Abram’s scheme. I’m not so sure about Paul’s—I’m pretty sure it’s not minimizing a weighted entropy, but a cross-entropy, which tries to make the probability proportional to the pre-prior.
As for other examples, there are a variety of more dilletantish cases (though I’m not really one to talk, of course) of people just trying to port Solomonoff induction on sentences or on models, and similar proposals of things like decreasing probability as a function of logical depth (example).
I’d defend my focus on mutually exclusive and exhaustive sets by saying that they show up everywhere and are always a basis we can use to represent knowledge. For example, if I had a list of 7 sentences and knew that exactly 5 were true, I can completely characterize my knowledge by probabilities of the 7 choose 5 mutually exclusive and exhaustive possibilities.
That said, it’s certainly possible there are things I’m missing because of this focus.