## Abstract

Several authors have argued that non-extreme probabilities used in special sciences such as chemistry and biology can be objective chances, even if the true microphysical description of the world is deterministic. This article examines an influential version of this argument and shows that it depends on a particular methodology for defining the relationship between coarse-grained and fine-grained events. An alternative methodology for coarse-graining is proposed. This alternative methodology blocks this argument for the existence of emergent chances, and makes better sense of two well-known subjects of philosophical discussion: the Miners Puzzle and Simpson’s Paradox.

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I am grateful to Jonathan Birch, Luc Bovens, Chloé de Canson, Paul Daniell, Phil Dowe, Christopher Hitchcock, Tyler Millhouse, Jeremy Strasser, Katie Steele, David Watson, James Wills, two anonymous reviewers, and audiences at the 2017 meeting of the Australasian Association of Philosophy in Adelaide, the HPS Seminar at the University of Melbourne, and the LSE Choice Group for their feedback on various versions of this paper.

## Appendix

### Appendix

### Proof of Proposition 1

###
*Proof*

Let *ω* ∈Ω be any possible world. Recall that *ω* is a mapping \(T\rightarrow S\). Let a “conjoined pair” of states (*s*_{i},*s*_{j}) ∈ *S* be any pair of states such that, according to the coarsening function . Let \({\Omega }^{\prime }_{k}\subseteq {\Omega }\) be a set of possible worlds such that, for any pair of worlds \((\omega _{l},\omega _{m})\in {\Omega }^{\prime }_{k}\) and all *t*, the states *ω*_{l}(*t*) and *ω*_{m}(*t*) are conjoined. The set of such sets of worlds \({\Omega }^{\dagger }=\{{\Omega }^{\prime }_{1},{\Omega }^{\prime }_{2},...,{\Omega }^{\prime }_{n}\}\) fully partitions the set of all possible worlds Ω. If the coarsening function *σ*(⋅) is applied to the set of states *S*, then each set of fine-grained worlds \({\Omega }^{\prime }_{k}\) is replaced by a single coarse-grained world . Thus, we can define a bijection from Ω^{‡} into the set of coarse-grained possible worlds , and therefore a bijection from \(\mathcal {A}_{{\Omega }^{\dagger }}\) into .

Finally, define a coarsening function \(\phi (\cdot ):\mathcal {A}_{\Omega }\rightarrow \mathcal {A}_{\Omega }\) such that \(\mathcal {A}_{{\Omega }^{\dagger }}\) is the range of *ϕ*(⋅), i.e. \({\Gamma }_{\phi (\cdot )}=\mathcal {A}_{{\Omega }^{\dagger }}\). It follows that there is a bijection between Γ_{ϕ(⋅)} and . For any \(E\in \mathcal {A}_{\Omega }\) and any \(\phi (E)\in \mathcal {A}_{{\Omega }^{\dagger }}\), if *ω* ∈ *E*, then *ω* ∈ *ϕ*(*E*), since any *ω* is conjoined with itself. Thus, in keeping with my proposed restriction on any coarsening function *ϕ*(⋅), for any \(E\in \mathcal {A}_{\Omega }\), \(E\subseteq \phi (E)\). □

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Kinney, D. Blocking an Argument for Emergent Chance.
*J Philos Logic* (2021). https://doi.org/10.1007/s10992-020-09590-5

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### Keywords

- Chance
- Coarse-graining
- Probability