Self-Referential Probability

Abstract

This thesis focuses on expressively rich languages that can formalise talk about probability and which have sentences that say something about the probability of themselves. For example:

(π): “The probability of the sentence labelled π is not greater than 1/2.”

Such sentences lead to philosophical and technical challenges. For example seemingly harmless principles, such as an introspection principle, lead to inconsistencies with the axioms of probability in this framework. Throughout the thesis, our motivating concept of probability is as measuring an agent’s degrees of belief.

This thesis investigates the following two questions relevant to such frameworks:

• How can one develop a formal semantics for this framework? Working with possible world structures we develop a Kripke-style fixed-point semantics with strong-Kleene evaluation scheme, and a supervaluational evaluation scheme (which results in imprecise probabilities). We also consider how to develop a revision theory semantics.
• What rational constraints are there on an agent once such expressive frameworks are considered? We investigate how to apply the Dutch book and accuracy arguments to cases involving such self-referential sentences. In particular we show a number of bullets one has to bite when adopting a consequentialist perspective on the accuracy argument.