A post on my own blog following a MIRIx workshop from two weekends ago.

http://qmaurmann.wordpress.com/2014/08/10/meditations-on-l-and-probabilistic-logic/

Reproducing the intro:

This post is a second look at The Definability of Truth in Probabilistic Logic, a preprint by Paul Christiano and other Machine Intelligence Research Institute associates, which I first read and took notes on a little over one year ago.

In particular, I explore relationships between Christiano et al’s probabilistic logic and stumbling blocks for self-reference in classical logic, like the liar’s paradox (“This sentence is false”) and in particular Löb’s theorem.

The original motivation for the ideas in this post was an attempt to prove a probabilistic version of Löb’s theorem to analyze the truth-teller sentences (“This sentence is [probably] true”) of probabilistic logic, an idea that came out of some discussions at a MIRIx workshop that I hosted in Seattle.

I've been learning math lately; specifically I've been reading MIRI's recent research preprints and the prerequisite material. In order to actually learn math, I typically have to write it down again, usually with more details and context. I started a blog to make my notes on these papers public, and I think they're of high enough quality that I ought to share them here.

Note: my use of the pronoun "we" is instilled habit; I am not claiming to have helped develop the core ideas herein.

• Löb's Theorem and the Prisoner's Dilemma is an account of the LaVictoire et al paper Robust Cooperation in the Prisoner's Dilemma.
• Details in Provability Logic is a technical followup to the above, which goes into the details of modal logic needed for the LaVictoire et al paper; namely the normal form theorem, the fixed point theorem, and the decidability of GL via Kripke semantics.
• Definability of Truth in Probabilistic Logic goes through the Christiano et al paper of the same name. It's a little rougher around the edges on account of being the first blog post I ever wrote (and being produced more hastily than the other two). I note that the construction doesn't truly require the Axiom of Choice.