# Epistemic-State Parallelism: Translating Between Probabilities and Ranks

## Abstract

This paper contributes to the investigation of the nature of the relation between probability theory and ranking theory. The paper aims at explaining the structural harmony between the laws of probability theory and those of ranking theory in a way that respects the foundational dualistic attitude developed by Spohn in *The Laws of Belief*. The paper argues that the so called *atomic translation family* satisfies the desiderata and does so in the ‘best’ possible way. On the one hand, the atomic translation can be seen as ensuring maximal order agreement between rankings and probability—more would lead to trivialising dualism. On the other hand, the atomic translation also is *assumption minimal*, i.e., a minimal set of justifiable correspondences or translation principles suffice to derive the translation family, and this result is relatively robust.

## Notes

### Funding

Deutsche Forschungsgemeinschaft, FOR 1614.

## References

- Aczél, J. (1966).
*Lectures on functional equations and their applications*. New York: Academic Press.Google Scholar - Capotorti, A., Coletti, G., & Vantaggi, B. (1998). Non additive ordinal relations representable by lower or upper probabilities.
*Kybernetika*,*34*(1), 79–90.Google Scholar - Capotorti, A., & Vantaggi, B. (2000). Axiomatic characterization of partial ordinal relations.
*International Journal of Approximate Reasoning*,*24*(2–3), 207–219.CrossRefGoogle Scholar - Dubois, D. (1984). Steps to a theory of qualitative possibility. In
*Proceedings of the 6th international congress on cybernetics and systems*(pp. 147–152). AFCET Publication, Paris.Google Scholar - Dubois, D. (1986). Belief structures, possibility measures and decomposable set-functions.
*Computers and Artificial Intelligence*,*5*, 403–416.Google Scholar - Dubois, D. (1988). Possibility theory: Towards normative foundations. In B. Munier (Ed.),
*Risk, decision and rationality*(pp. 601–614). Hingham, MA: Reidel.CrossRefGoogle Scholar - Fitelson, B. (2017). Two approaches to doxastic representation. In
*Presentation at the APA*.Google Scholar - Goldszmidt, M., & Pearl, J. (1992). Reasoning with qualitative probabilities can be tractable. In D. Dubois, M.P. Wellman, B. D’Ambrosio, P. Smets (Eds.),
*Proceedings of the 8th conference on uncertainty in artificial intelligence*, (pp. 112–120). San Mateo, CA: Morgan Kaufmann.Google Scholar - Goldszmidt, M., & Pearl, J. (1996). Qualitative probabilities for default reasoning, belief revision, and causal modeling.
*Artificial Intelligence*,*84*, 57–112.CrossRefGoogle Scholar - Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971).
*Foundations of measurement*. New York: Academic Press.Google Scholar - Leitgeb, H. (2013). Reducing belief simpliciter to degrees of belief.
*Annals of Pure and Applied Logic*,*164*(12), 1338–1389.CrossRefGoogle Scholar - Leitgeb, H. (2014). The stability theory of belief.
*The Philosophical Review*,*123*(2), 131–171.CrossRefGoogle Scholar - Leitgeb, H. (2015). The humean thesis on belief.
*Aristotelian Society Supplementary*,*89*(1), 143–185.CrossRefGoogle Scholar - Raidl, E. (2014).
*Probabilité, Invariance et Objectivité*, Ph.D thesis, University Sorbonne-Panthéon (IHPST), Paris.Google Scholar - Raidl, E., & Skovgaard-Olsen, N. (2017). Bridging ranking theory and the stability theory of belief.
*Journal of Philosophical Logic*,*46*(6), 577–609.CrossRefGoogle Scholar - Spohn, W. (1988). Ordinal conditional functions. A dynamic theory of epistemic states. In W. L. Harper and B. Skyrms (Eds.),
*Causation in decision, belief change, and statistics*(Vol. 2, pp. 105–134). Dordrecht: Kluwer.Google Scholar - Spohn, W. (2012).
*The laws of belief: Ranking theory and its philosophical applications*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Scott, D. (1964). Measurement structures and linear inequalities.
*Journal of Mathematical Psychology*,*1*, 233–247.CrossRefGoogle Scholar - Wong, S. K. M., Yao, Y. Y., Bollmann, P., & Bürger, H. C. (1991). Axiomatization of qualitative belief structure.
*IEEE Transactions on System, Man, and Cybernetics*,*21*, 726–734.CrossRefGoogle Scholar