Blocking an Argument for Emergent Chance

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.

Appendix

1.1 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 $E\in \mathcal {A}_{\Omega }$, $E\subseteq \phi (E)$. □