This post explores a draft paper by Paul Christiano et al (06/10/2013 draft), produced at a workshop hosted by the Machine Intelligence Research Institute. My intent is to go thoroughly through the main argument, to ensure that I fully understand it. Also, on my first or second reading, I had some worries that the argument seemed to produce something for nothing, possibly violating a “conservation of depth” in mathematics. I am no longer worried about this.
Towards the end of this post, I discuss the non-constructive nature of the proof given, but show that it can at least be modified so as not to rely on the Axiom of Choice.
A one-sentence summary of the paper is: Although a logical theory can’t contain its own Truth predicate, it can contain its own “subjective probability function” which assigns reasonable probabilities to sentences of the theory, including of course statements about the probability function itself.
Or to quote the conclusion of the paper:
Tarski’s result on the undefinability of truth is in some sense an artifact of the infinite precision demanded by reasoning about complete certainty.