Abstract
Halpern and Hitchcock appealed to the normality of witness worlds to solve the problem of isomorphism in the Halpern-Pearl definition of actual causality. This paper first proposes a new isomorphism counterexample, called “bogus permission,” to show that their approach is unsuccessful. Then, to solve the problem of isomorphism, I propose a new improvement over the Halpern-Pearl definition by introducing default worlds. Finally, I demonstrate that my new definition can resolve more potential counterexamples than similar approaches in the current literature, including the Lewisian causal dependence, Menziesian causal dependence, and modified version of the Halpern-Pearl definition. Some other advantages of my definition are also discussed.
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Notes
For simplicity, I followed the literature that omits the explicit reference to the exogenous variables in the following discussion.
Based on the so-called default role of normal causality in that paper.
SE refers to structural equations.
I changed the original formation slightly.
Halpern’s modified definition states that \({B}_{1}=1\wedge {B}_{2}=1\wedge C=1\) is the actual cause, and \(C=1\) is part of a cause. Halpern seems to think this understanding is enough. However, I believe that philosophers aim to find the complete definition of actual causality. Either the part of a cause should be counted as the cause, or it should not. By applying ACmod, neither of \({B}_{1}=1,{B}_{2}=1,C=1\) is the cause.
References
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Acknowledgements
I would like to thank Feng Ye, Mu Liu, Jiangjie Qiu, Hao Li, Sisi Yang, Ge Song, and members of the Reading Group: Causal Reasoning and Causal Modeling for their helpful comments and discussions.
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FZ contributed to the study conception and design. The draft of the manuscript was written by FZ.
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Appendix 1
Appendix 1
Proofs of theorems require a property of interventionist counterfactuals that have been proved by Galles and Pearl (1997).
Composition. If \(\left( {M,\vec{u}} \right) \vDash \left[ {\vec{X} = \vec{x}} \right]\vec{W} = \vec{w}\), then: \(\left( {M,\vec{u}} \right) \vDash \left[ {\vec{X} = \vec{x},\vec{W} = \vec{w}} \right]\vec{Y} = \vec{y}\), if and only if, \(\left( {M,\vec{u}} \right) \vDash \left[ {\vec{X} = \vec{x}} \right]\vec{Y} = \vec{y}\).
Theorem 1
Let \(\overrightarrow{X}=\overrightarrow{x}\) be \(X=x\), \(\varphi\) be \(Y=y\), and they are Boolean variables, the following conditions hold:
-
(1)
If they have Menziesian causal dependence, they satisfy AC2def (a).
-
(2)
If they satisfy AC2def and for any \(x^{\prime}\), \(\left( {M,\vec{u}^{d} } \right) \vDash \left[ {X = x^{\prime}} \right]\vec{W} = \vec{w}\) where \(\overrightarrow{W}=\overrightarrow{w}\) is the contingencies, then they have Menziesian causal dependence.
Proof:
(1) Let \(\overrightarrow{W}\) be \(\varnothing\). AC2def (a) trivially holds.
(2) Assuming that \(X = x\) and \(Y = y\) satisfy AC2def, and \(\left( {M,\vec{u}^{d} } \right) \vDash \left[ {X = x} \right]\vec{W} = \vec{w}\), we at least have \(\left( {M,\vec{u}^{d} } \right) \vDash \left[ {X = x,\vec{W} = \vec{w}} \right]Y = y\) and \(\left( {M,\vec{u}^{d} } \right) \vDash \left[ {X = x} \right]\vec{W} = \vec{w}\). By composition, \(\left( {M ,\vec{u}^{d} } \right) \vDash \left[ {\vec{X} = \vec{x}} \right]Y = y\). Since \(X = x\) and \(Y = y\) satisfy AC2def, we have there is a \(X = x^{\prime}\) such that \(\left( {M ,\vec{u}^{d} } \right) \vDash \left[ {X = x^{\prime},\vec{W} = \vec{w}} \right]\neg \left( {Y = y} \right)\). Therefore, \(\left( {M_{X = x^{\prime}} ,\vec{u}^{d} } \right) \vDash \left[ {\vec{W} = \vec{w}} \right]\neg \left( {Y = y} \right)\). Since \(\left( {M_{X = x^{\prime}} ,\vec{u}^{d} } \right) \vDash \vec{W} = \vec{w}\), again by the composition, \(\left( {M ,\vec{u}^{d} } \right) \vDash \left[ {X = x^{\prime}} \right]Y = y^{\prime}\).
Theorem 2
Let \(\overrightarrow{X}=\overrightarrow{x}\) be \(X=x\), and \(\varphi\) be \(Y=y\), and they are Boolean variables, if AC2def holds when taking \(\left( {M,\vec{u}} \right) \vDash \vec{W} = \vec{w}\) as contingencies and \(\left( {M,\vec{u}} \right) \vDash X = x\), then for the same \(\vec{W} = \vec{w}\), they satisfy AC2mod (b), and hence AC2org (b).
Proof:
Assuming that \(X = x\) and \(Y = y\) satisfy AC2def when taking \(\left( {M,\vec{u}} \right) \vDash \vec{W} = \vec{w}\) as the contingency, for all subsets \(\vec{Z}^{\prime}\) of \(\vec{Z}\) and \(\vec{z}^{*}\) such that \(\left( {M,\vec{u}} \right) \vDash \vec{Z}^{\prime} = \vec{z}^{*}\), we have \(\left( {M_{{\vec{W} = \vec{w}}} ,\vec{u}^{d} } \right) \vDash \left[ {X = x,\vec{Z}^{\prime} = \vec{z}^{*} } \right]Y = y\). Let \(\vec{Z}^{\prime}\) be \(\vec{Z}\), since \(\left( {M,\vec{u}} \right) \vDash \vec{W} = \vec{w}\), we have \(\left( {M_{{\vec{Z} = \vec{z}^{*} ,\vec{W} = \vec{w}}} ,\vec{u}^{d} } \right) = \left( {M_{{\vec{W} = \vec{w}}} ,\vec{u}} \right) = \left( {M,\vec{u}} \right)\). Since \(\left( {M_{{\vec{Z} = \vec{z}^{*} ,\vec{W} = \vec{w}}} ,\vec{u}^{d} } \right) \vDash \left[ {X = x} \right]Y = y\), \(\left( {M,\vec{u}} \right) \vDash \left[ {X = x} \right]Y = y\). Because \(\left( {M,\vec{u}} \right) \vDash X = x\), we have \(\left( {M,\vec{u}} \right) \vDash Y = y\). s\(\left( {M,\vec{u}} \right) \vDash \vec{W} = \vec{w} \wedge X = x \wedge \vec{Z}^{\prime} = \vec{z}^{*}\) for all subsets \(\vec{Z}^{\prime}\) of \(\vec{Z}\) and \(\vec{z}^{*}\) such that \(\left( {M,\vec{u}} \right) \vDash \vec{Z}^{\prime} = \vec{z}^{*}\). Therefore, for the same \(\vec{W} = \vec{w}\) and \(\vec{Z}^{\prime} = \vec{z}^{*}\), \(\left( {M_{{\vec{W} = \vec{w}}} ,\vec{u}} \right) \vDash \left[ {X = x,\vec{Z}^{\prime} = \vec{z}^{*} } \right]Y = y\).
proof from AC2mod (b) to AC2org(b) is trivial.
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Zhu, F. A New Halpern-Pearl Definition of Actual Causality by Appealing to the Default World. Axiomathes 32 (Suppl 2), 453–472 (2022). https://doi.org/10.1007/s10516-021-09613-z
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DOI: https://doi.org/10.1007/s10516-021-09613-z