Abstract
The paper focuses on a recent challenge brought forward against the interventionist approach to the meaning of counterfactual conditionals. According to this objection, interventionism cannot in general account for the interpretation of right-nested counterfactuals, the problem being its strict interventionism. We will report on the results of an empirical study supporting the objection, and we will extend the well-known logic of actual causality with a new operator expressing an alternative notion of intervention that does not suffer from the problem (and thus can account for some critical examples). The core idea of the alternative approach is a new notion of intervention, which operates on the evaluation of the variables in a causal model, and not on their functional dependencies. Our result provides new insights into the logical analysis of causal reasoning.
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Notes
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This is true for recursive causal models, the only ones that this paper will discuss. For their definition: from \(\mathcal {S}\), define a relation \(\rightarrowtail \) on the set of variables \(\mathcal {U}\cup \mathcal {V}\) by writing \(X \rightarrowtail Y\) if and only if X is among the parents of Y (the structure \(\langle \mathcal {U}\cup \mathcal {V}, \rightarrowtail \rangle \) is called \(\mathcal {S}\)’s induced causal graph). Let \(\rightarrowtail ^+\) be the transitive closure of \(\rightarrowtail \) (so \(X \rightarrowtail ^+ Y\) indicates that Y is causally dependent on X). A causal model is said to be recursive when \(\rightarrowtail ^+\) is a strict partial order (there are no circular dependencies between the variables).
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In [13]’s general setting, the selection function returns a set of models. However, for the possible antecedents of counterfactuals considered in our fragment of her language, the selected model is uniquely defined.
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Fisher also considers another example, involving the counterfactual “If the match were struck and it lit, then if it hadn’t been struck, it would have lit”. This is not a good example to make his point, as it contains a conjunction of cause (striking the match) and effect (the match lights) in the antecedent. For the counterexample to work, Fisher needs this conjunction to be interpreted as two independent interventions. However, it could be that “and” is interpreted causally in this case: “If the match were struck and because of that it lit, ...”. But then the fact that the match lights would be introduced as a causal consequent of the striking of the match and not as an independent intervention.
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We ignore other possible variables, as they will not affect the relevant predictions made.
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Fisher discusses in [12] an alternative definition of intervention, dubbed “side-constrained intervention”, but admits that this variation is not really targeting the root of the problem.
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“Y is causally affected by Z” intuitively means changing the value of Z may change the value of Y under some setting of variables. Formally, it means there exists some variables \(\overrightarrow{V}\), \(\overrightarrow{v}\in \mathcal {R}(\overrightarrow{v})\), and some distinct value \(y,y'\in \mathcal {R}(Y)\), such that the value of Z forced by setting \(\overrightarrow{V},Y\) to \(\overrightarrow{v},y\) is different from its value forced by setting \(\overrightarrow{V},\overrightarrow{Y}\) to \(\overrightarrow{v},y'\).
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Recall: the model is recursive. Hence, \(\mathcal {S}\)’s induced causal graph induces, in turn, a chain of sets of variables \(S_0\subseteq \cdots \subseteq S_n\) such that \(S_0=\mathcal {U}\cup \overrightarrow{V}\), \(S_n=\mathcal {U}\cup \mathcal {V}\) and, for any \(S_i\) and \(S_{i+1}\), the value of variables in \(S_{i+1}\setminus S_i\) can be calculated from the causal dependencies and the value of variables in \(S_{i}\).
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Note: \(\mathcal {A}^{\overrightarrow{V}=\overrightarrow{v}}\) may not comply with the causal dependencies in \(\mathcal {S}\).
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When the original assignment \(\mathcal {A}\) complies with the causal dependencies in \(\mathcal {S}\), both strategies produce the same result. This is the only case relevant for Briggs’ purposes.
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Proofs were omitted due to space limitations, but are available online https://www.dropbox.com/s/0i0xy416rs5dmor/Lori_Proofs.pdf?dl=0.
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If the model allows interventions on exogenous variables, the example can be modelled with only two variables: the exogenous one S and the endogenous one L. We use the additional U, as in the literature it is common to allow interventions only on endogenous variables.
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In [15] there are no causal violations; thus, \(V=v\) is equivalent to \([\;](V=v)\), and axioms A1 and A2 suffice. This is not the case in our setting, as causal violations might occur; hence the need of A10–A11.
extends the basic causal language (e.g., [References
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Schulz, K., Smets, S., Velázquez-Quesada, F.R., Xie, K. (2019). A Logical and Empirical Study of Right-Nested Counterfactuals. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_19
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