This was originally part of a post I wrote on logical uncertainty, but it turned out to be post-sized itself, so I'm splitting it off.

Daniel Garber's article Old Evidence and Logical Omniscience in Bayesian Confirmation Theory. Wonderful framing of the problem--explains the relevance of logical uncertainty to the Bayesian theory of confirmation of hypotheses by evidence.

Articles on using logical uncertainty for Friendly AI theory: qmaurmann's Meditations on Löb’s theorem and probabilistic logic. Squark's Overcoming the Loebian obstacle using evidence logic. And Paul Christiano, Eliezer Yudkowsky, Paul Herreshoff, and Mihaly Barasz's Definibility of Truth in Probabilistic Logic. So8res's walkthrough of that paper, and qmaurmann's notes. eli_sennesh like just made a post on this: Logics for Mind-Building Should Have Computational Meaning.

Benja's post on using logical uncertainty for updateless decision theory.

cousin_it's Notes on logical priors from the MIRI workshop. Addresses a logical-uncertainty version of Counterfactual Mugging, but in the course of that has, well, notes on logical priors that are more general.

Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements, by Haim Gaifman. Shows that you can give up on giving logically equivalent statements equal probabilities without much sacrifice of the elegance of your theory. Also, gives a beautifully written framing of the problem.

manfred's early post, and later sequence. Amazingly readable. The proposal gives up Gaifman's elegance, but actually goes as far as assigning probabilities to mathematical statements and using them, whereas Gaifman never follows through to solve an example afaik. The post or the sequence may be the quickest path to getting your hands dirty and trying this stuff out, though I don't think the proposal will end up being the right answer.

There's some literature on modeling a function as a stochastic process, which gives you probability distributions over its values. The information in these distributions comes from calculations of a few values of the function. One application is in optimizing a difficult-to-evaluate objective function: see Efficient Global Optimization of Expensive Black-Box Functions, by Donald R. Jones, Matthias Schonlau, and William J. Welch. Another is when you're doing simulations that have free parameters, and you want to make sure you try all the relevant combinations of parameter values: see Design and Analysis of Computer Experiments by Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn.

Maximize Worst Case Bayes Score, by Coscott, addresses the question: "Given a consistent but incomplete theory, how should one choose a random model of that theory?"

Bayesian Networks for Logical Reasoning by Jon Williamson. Looks interesting, but I can't summarize it because I don't understand it.

And, a big one that I'm still working through: Non-Omniscience, Probabilistic Inference, and Metamathematics, by Paul Christiano. Very thorough, goes all the way from trying to define coherent belief to trying to build usable algorithms for assigning probabilities.

Dealing With Logical Omniscience: Expressiveness and Pragmatics, by Joseph Y. Halpern and Riccardo Pucella.

Reasoning About Rational, But Not Logically Omniscient Agents, by Ho Ngoc Duc. Sorry about the paywall.

And then the references from Christiano's report:

Abram Demski. Logical prior probability. In Joscha Bach, Ben Goertzel, and Matthew Ikle, editors, AGI, volume 7716 of Lecture Notes in Computer Science, pages 50-59. Springer, 2012.

Marcus Hutter, John W. Lloyd, Kee Siong Ng, and William T. B. Uther. Probabilities on sentences in an expressive logic. CoRR, abs/1209.2620, 2012.

Bas R. Steunebrink and Jurgen Schmidhuber. A family of Godel machine implementations. In Jurgen Schmidhuber, Kristinn R. Thorisson, and Moshe Looks, editors, AGI, volume 6830 of Lecture Notes in Computer Science, pages 275{280. Springer, 2011.

If you have any more links, post them!

Or if you can contribute summaries.