# Bridging Ranking Theory and the Stability Theory of Belief

## Abstract

In this paper we compare Leitgeb’s stability theory of belief (Annals of Pure and Applied Logic, 164:1338-1389, 2013; The Philosophical Review, 123:131-171, [2014]) and Spohn’s ranking-theoretic account of belief (Spohn, 1988, 2012). We discuss the two theories as solutions to the lottery paradox. To compare the two theories, we introduce a novel translation between ranking (mass) functions and probability (mass) functions. We draw some crucial consequences from this translation, in particular a new probabilistic belief notion. Based on this, we explore the logical relation between the two belief theories, showing that models of Leitgeb’s theory correspond to certain models of Spohn’s theory. The reverse is not true (or holds only under special constraints on the parameter of the translation). Finally, we discuss how these results raise new questions in belief theory. In particular, we raise the question whether stability (a key ingredient of Leitgeb’s theory) is rightly thought of as a property pertaining to belief (rather than to knowledge).

## Keywords

Belief Probability Lottery paradox Stability theory Ranking theory Knowledge Lockean thesis Odds-threshold## Notes

### Acknowledgments

We would like to thank Hannes Leitgeb for his helpful comments on an earlier manuscript, Hans Rott for his suggestions and critical remarks and an anonymous referee for pressing us to focus more on the novelty of the translation and its consequences. They all helped to improve the quality of the paper.

## References

- 1.Dubois, D., & Prade, H. (1988).
*Possibility Theory: An Approach to Computerized Processing of Uncertainty*. New York: Plenum Press.CrossRefGoogle Scholar - 2.Greene, B. (2012).
*The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos*. London: Penguin Books.Google Scholar - 3.Leitgeb, H. (2013). Reducing Belief Simpliciter to Degrees of Belief.
*Annals of Pure and Applied Logic*,*164*(12), 1338–1389. doi: http://dx.doi.org/10.1016/j.apal.2013.06.015.CrossRefGoogle Scholar - 4.Leitgeb, H. (2014). The Stability Theory of Belief.
*The Philosophical Review*,*123*(2), 131–171. doi: 10.1215/00318108-2400575.CrossRefGoogle Scholar - 5.Leitgeb, H. (2015). The Humean Thesis on Belief.
*Aristotelian Society Supplementary Volume*,*89*(1), 143–185. doi: 10.1111/j.1467-8349.2015.00248.x.CrossRefGoogle Scholar - 6.Levi, I. (1996).
*For the Sake of the Argument: Ramsey Test Conditionals, Inductive Inference and Non-monotonic Reasoning*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 7.Lin, H., & Kelly, K.T. (2012). Propositional Reasoning that Tracks Probabilistic Reasoning.
*Journal of Philosophical Logic*,*41*, 957–981. doi: 10.1007/s10992-012-9237-3.CrossRefGoogle Scholar - 8.Raidl, E. (2014). Probabilité, Invariance et Objectivité, PhD thesis at the University of Paris 1 Panthéon-Sorbonne, IHPST. http://www.theses.fr/s123429.
- 9.Rott, H. (2004). Stability, strength and sensitivity: Converting belief into knowledge.
*Erkenntnis*,*61*(2-3), 469–93. doi: 10.1007/s10670-004-9287-1.CrossRefGoogle Scholar - 10.Rott, H. (2009). Degrees all the way down: Beliefs, non-beliefs and disbeliefs. In Huber, F., & Schmidt-Petri, C. (Eds.)
*Degrees of Belief (pp. 301–339). Dordrecht: Springer*.Google Scholar - 11.Rott, H. (2015a). Stability and scepticism in the generation of plain beliefs from probabilities. Manuscript version of May 26, 2015.Google Scholar
- 12.Rott, H. (2015b). Unstable knowledge, unstable belief. Manuscript version of July 28, 2015.Google Scholar
- 13.Skovgaard-Olsen, N. (2015). The problem of logical Omniscience, the preface paradox, and doxastic commitments.
*Synthese*, 1–26. doi: 10.1007/s11229-015-0979-7. - 14.Spohn, W. (1988). Ordinal Conditional Functions. A Dynamic Theory of Epistemic States. In Harper, W.L., & Skyrms, B. (Eds.)
*Causation in Decision, Belief Change, and Statistics, Vol. 2 (pp. 105–134). Dordrecht: Kluwer*.Google Scholar - 15.Spohn, W. (2012).
*The Laws of Belief: Ranking Theory and its Philosophical Applications*. Oxford: Oxford University Press.CrossRefGoogle Scholar - 16.Spohn, W. (presentation). The Value of Knowledge. https://www.tilburguniversity.edu/upload/87af9554-bf5f-4514-be46-021183a63bf0_Presentation%20Spohn.pdf. Accessed 3 March 2015.
- 17.Yalcin, S. (2011). Nonfactualism about Epistemic Modality. In Egan, A., Weatherson, B., & Yalcin, S (Eds.) (pp. 295–332): Oxford University Press.Google Scholar
- 18.Yalcin, S. (forthcoming). Belief as Question-Sensitive. Philosophy and Phenomenological Research. https://www.academia.edu/26580337/Belief_as_Question-Sensitive. Accessed 19 August 2016.