# Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

## Abstract

Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes versus anti-Bayes debate which is in many ways analogous to what the *classical versus non-classical* debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for *extending* the expressive power of standard Bayesian norms of rationality to second-order uncertainty in a way which is both formally and foundationally conservative.

### Keywords

Second-order uncertainty Bayesian epistemology Admissibility Imprecise probabilities## Notes

### Acknowledgments

I have presented the main ideas of this paper at Kent’s Centre for Reasoning and at the LSE Choice Group in London. I would like to thank Jon Williamson and Richard Bradley for inviting me to speak at those seminars and both audiences for their very valuable feedback. I am very grateful to Gregory Wheeler for his comments on an earlier draft and for many stimulating discussions on the topics covered in this paper. Thanks also to two referees, whose thorough reviews helped me to improve the chapter in many ways. Finally, readers familiar with David Makinson’s work will certainly have spotted a number of terms and expressions which are easily associated with, and sometimes directly coming from, David’s papers and books. I realised this only when the first draft of this chapter was completed. As I started fetching all the originals to give credit where credit was due, it occurred to me that leaving those paraphrases uncredited would probably be the most direct way to express how influential David’s way of doing logic is to my way of thinking about logic. So I stopped.

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