Tomás Orozco comments on Definitions of Causal Abstraction: Reviewing Beckers & Halpern

Tomás Orozco 29 Apr 2020 19:04 UTC
1 point
Dear John Wentworth:
I have a doubt regarding the implications of Beckers’ paper on abstractions. I am a lawyer by profession so I’m venturing pretty far afield here, and I hope my question will not be too trivial.
Given that every constructive abstraction is also a $τ$ - abstraction, there must be some surjective function $τ_{U}$ that is compatible with $τ$ . Hence, for constructive abstractions, must there also be mappings $τ_{U}^{1}, \dots, τ_{U}^{n}$ such that, $τ_{U} ({\to u}_{L}) = τ_{U}^{1} ({\to z}_{U}^{1}) \cdot \dots \cdot τ_{U}^{n} ({\to z}_{U}^{n})$ , where ${\to z}_{U}^{i}$ is the projection of ${\to u}_{L}$ onto the variables in ${\to Z}_{U}^{i}$ ? In other words, must there also be a partition of the low-level exogenous variables where each partition is mapped to a distinct high-level variable? I missed $τ_{U}$ in the definition of constructive abstraction..
Thank you!