Update: See Reflection in Probabilistic Logic for more details on how this result relates to MIRI’s research mission.
In a recent blog post we described one of the results of our 1st MIRI Workshop on Logic, Probability, and Reflection:
The participants worked on the foundations of probabilistic reflective reasoning. In particular, they showed that a careful formalization of probabilistic logic can circumvent many classical paradoxes of self-reference. Applied to metamathematics, this framework provides (what seems to be) the first definition of truth which is expressive enough for use in reflective reasoning.
In short, the result described is a “loophole” in Tarski’s undefinability theorem (1936).
An early draft of the paper describing this result is now available: download it here. Its authors are Paul Christiano (UC Berkeley), Eliezer Yudkowsky (MIRI), Marcello Herreshoff (Google), and Mihaly Barasz (Google). An excerpt from the paper is included below:
Unfortunately, it is impossible for any expressive language to contain its own truth predicate True…
There are a few standard responses to this challenge.
The first and most popular is to work with meta-languages…
A second approach is to accept that some sentences, such as the liar sentence G, are neither true nor false…
Although this construction successfully dodges the “undefinability of truth” it is somewhat unsatisfying. There is no predicate in these languages to test if a sentence… is undefined, and there is no bound on the number of sentences which remain undefined. In fact, if we are specifically concerned with self-reference, then a great number of properties of interest (and not just pathological counterexamples) become undefined.
In this paper we show that it is possible to perform a similar construction over probabilistic logic. Though a language cannot contain its own truth predicate True, it can nevertheless contain its own “subjective probability” function P. The assigned probabilities can be reflectively consistent in the sense of an appropriate analog of the reflection property 1. In practice, most meaningful assertions must already be treated probabilistically, and very little is lost by allowing some sentences to have probabilities intermediate between 0 and 1.
Another paper showing an application of this result to set theory is forthcoming.