Self-Referential Probability and Rationality

This thesis argues for attitudes an agent ought to adopt towards two problematic examples of probabilistic self-reference. In particular, I look at a case of self-referential probability I refer to as the Probabilistic Liar, due to its similarities to the Liar paradox. The Probabilistic Liar emerges when an agent's credence can act as evidence for the truth of the proposition. Examples of self-reference turn out to be problematic for traditional Bayesian accounts of rationality.

I develop an account of how rational agents ought to respond to the Probabilistic Liar by suspending judgment. Suspended judgment is an attitude more naturally talked about in traditional all-or-nothing belief models. I argue for suspended judgment in a credal framework and in particular that suspended judgment is a determinate attitude that should be represented by imprecise credences. This gives a principled way of weakening the requirement that a rational agent's degrees of beliefs ought to be probabilistically coherent.

Once a solution to the Probabilistic Liar has been given a new question emerges. Can we give another example of problematic probabilistic self-reference in terms of the suspended judgement attitude? That is, can we give a Revenge problem. I explore how a Revenge problem can be generated for my account and how a rational agent can respond by having indeterminate attitudes.
Finally, I argue that both the Probabilistic Liar and Revenge problems are cases of indeterminacy. I then look at the normative question of what attitude an agent ought to adopt towards cases of indeterminacy. Drawing on the attitudes I have argued for in the thesis, I argue for a pluralist answer to the normative question.