Abstract
While structural equations modeling is increasingly used in philosophical theorizing about causation, it remains unclear what it takes for a particular structural equations model to be correct. To the extent that this issue has been addressed, the consensus appears to be that it takes a certain family of causal counterfactuals being true. I argue that this account faces difficulties in securing the independent manipulability of the structural determination relations represented in a correct structural equations model. I then offer an alternate understanding of structural determination, and I demonstrate that this theory guarantees that structural determination relations are independently manipulable. The account provides a straightforward way of understanding hypothetical interventions, as well as a criterion for distinguishing hypothetical changes in the values of variables which constitute interventions from those which do not. It additionally affords a semantics for causal counterfactual conditionals which is able to yield a clean solution to a problem case for the standard ‘closest possible world’ semantics.
Notes
Not just any counterfactual is a causal counterfactual. While the question of which counterfactuals are causal is a question to be decided by theory rather than stipulation, at the least, causal counterfactuals must be non-backtracking (see Lewis 1979) and they must relate distinct events (see Kim 1973 and Lewis 1986).
A word on notation: throughout, I’ll be using uppercase letters (\(A, B, C, \dots , Z\)) to represent variables, and the corresponding lowercase letters (\(a, b, c, \dots , z\)) to stand for the values of those variables. Functions will be denoted with ‘\(\phi\)’, with subscripts added to indicate which variable the function is associated with. Vectors of variables or variable values will be represented with boldface or calligraphic letters (\({\mathcal {U}}, {\mathcal {V}}, {\mathbf {X}}, {\mathbf {x}}, {\mathbf {PA}}(V)\), etc.). At times I will use the function name alone to denote the entire structural equation—for instance, I will write ‘\(\phi _Y\)’ to denote the structural equation ‘\(Y \,{:}{=}\,\phi _Y(X_1, \dots , X_N)\)’. Propositions will be denoted with upright uppercase letters \((\mathrm {A}, \mathrm {B}, \mathrm {C}, \dots , \mathrm {Z})\). I’ll also be abusing set-theoretic notation, \(\in , \cup , \subseteq , -\), and so on, by applying it to vectors of variables (or variable values).
‘\(\lceil x \rceil\)’ is the function which rounds \(x\) up to the closest integer.
It’s worth noting that the functions \(\phi _V\) must be non-constant. A constant function from one variable to another does not represent any kind of determination of of the latter variable by the former. We should also require that the domain of each structural equation include the entire image of their parent variables’ structural equations, and only that image.
In general, there will be many such vectors. It won’t matter for my purposes which one ‘\({\mathbf {PA}}(V)\)’ refers to. Pick one. Likewise for the other vectors of variables or variable values I discuss here.
See Cartwright (2009).
Here and throughout, I’m using ‘actual’ as an indexical like ‘here’, and not as a rigid designator for the actual world.
From the standpoint of Lewis (1973)’s account of counterfactuals, it will appear that, by adopting this general framework, I am tolerating the so-called limit assumption—the assumption that, for any arbitrary antecedent \(\text {A}\) and world \(w\), there is a set of most similar \(\text {A}\)-worlds (see Lewis 1973; Stalnaker 1980). Appearances are deceiving. The limit assumption is not needed for any of my arguments here. In Lewis’s framework, for any case in which the limit assumption fails and \(\text {A} \;\square \!\!\rightarrow \text {C}\) is true, we can just define \(f({\text {A}}, w)\) to be the largest sphere centered on \(w\) containing at least one \(\text {A}\) world and throughout which the material conditional \(\text {A} \supset \text {C}\) is true. So long as \(\text {A} \;\square \!\!\rightarrow \text {C}\) is true, there will be some such sphere. If it is false, of course, there won’t be such a sphere, so this won’t do as an account of the truth conditions of these counterfactuals. However, I am not interested in providing truth conditions for these counterfactuals. Rather, I am interested in the question of whether the truth of a set of counterfactuals is sufficient to guarantee the correctness of a structural equations model. And this trick will tell us what we can infer from the truth of \(\text {A} \;\square \!\!\rightarrow \text {C}\) at a world \(w\), on Lewis’s account.
\((f1)\) corresponds to Lewis’s 2nd condition on the selection function; and \((f2)\) is a weakening of Lewis’s 4th condition. See Lewis(1973, p. 58). I’ve used the weaker \((f2)\), rather than Lewis’s stronger constraint—\({\text{ if }} {\text {A}} \,\models \, {\text {B}} {\text{ then }} f({\text {B}}, w) \,\cap {\text {A}} = f({\text {A}}, w)\)—not because I’m skeptical of the stronger constraint, but rather because the weaker constraint is all that I need avail myself of in developing the counterfactual account, and I think it best to minimize my assumptions where possible. Indeed, in developing the nomic sufficiency account in Sect. 4, I will only require \((f1)\) and \((f3)\). See fn 15.
Throughout, I use ‘\(@\)’ to denote the actual world.
Above, I didn’t define \(f\) for sets of worlds. Let’s say that \(f({\text {B}}, f({\text {A}}, w))\) is the union of \(f({\text {B}}, w')\) for every \(w' \in f({\text {A}}, w)\).
The condition is minimally sufficient for the effect just in case no subset of the condition is also sufficient for the effect. The minimal sufficient condition is occurrent iff it actually obtained on the occasion in question.
The values of \({\mathbf {PA}}(V)\) are sufficient, and not (or not necessarily) necessary, for the value of \(V\) because two different assignments of values to \({\mathbf {PA}}(V)\) could get mapped by \(\phi _V\) to the very same value of \(V\). \({\mathbf {PA}}(V)\) must be minimally sufficient for \(V\)’s value because we require that \(\phi _V\) be a non-constant function of each of its parents. See fn 5.
In fact, the nomic sufficiency account will ultimately require us to assume strictly less about this selection function than the counterfactual account did. For the theorems proved in the appendix only require that \(f\) satisfy \((f1)\) and \((f3)\); so the condition \((f2)\), along with any other constraints on the selection function, are unnecessary on the nomic sufficiency account.
At least, Lewis thinks that it achieves the same results; but see Elga (2001) for a persuasive argument to the contrary.
The case currently under discussion provides a counterexample to the equivalence.
\(x\, \veebar\, y\) is the exclusive ‘or’, which is \(1\) iff \(x \ne y\).
Thanks to an anonymous reviewer for pressing me to consider this objection.
For one: we could simply emend the definition of eclipsing by removing condition (1).
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Acknowledgments
I am indebted to Gordon Belot, Allan Gibbard, Jim Joyce, Brian Weatherson, and an anonymous reviewer for helpful conversations and feedback on this material.
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Appendix
Appendix
Define the rank of a variable \(V \in {\mathcal {U}}\cup {\mathcal {V}}\) recursively as follows:
Graphically, a variable’s rank is the largest number of edges lying between that variable and an exogenous variable along a directed path. Let ‘\({\mathbf {Rank}}(i)\)’ denote the set of all variables of rank \(i\), and let ‘\({\mathbf {Rank}}(i, j, \ldots , k)\)’ denote the union \({\mathbf {Rank}}(i) \cup {\mathbf {Rank}}(j) \cup \dots \cup {\mathbf {Rank}}(k)\).
Lemma 1
Given \(({\mathcal {V}}1)\), \(({\mathcal {U}}1)\), and \((f1)\), for all \(V \in {\mathcal {V}}\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\), where \({\mathbf {PA}}(V)_{{\mathbf {X}} = {\mathbf {x}}}\) assigns the values to \({\mathbf {PA}}(V)\) determined by the structural equations in \({\mathcal {E}}- \bigcup _i (\phi _{X_i})\), for every endogenous \(X_i \in \mathbf {X}\), and \(({\mathcal {U}}- {\mathbf {X}})_w \cup \mathbf {x}\).
Proof
By induction on the rank of the variables in \({\mathcal {V}}\). \(\square \)
Base case
For all \(V \in {\mathbf {Rank}}(1)\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{{\mathbf {X}} = \mathbf {x}}\).
Proof
If \(rank(V) = 1\), then every \(P \in {\mathbf {PA}}(V)\) is exogenous. Without loss of generality, consider one \(P \in {\mathbf {PA}}(V)\). If \(P \in \mathbf {X}\), then \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_{\mathbf {X}=\mathbf {x}}\) (the value assigned to \(P\) by \(\mathbf {x}\)), by \((f1)\). If \(P \notin \mathbf {X}\), then \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_w\), by \(({\mathcal {U}}1)\). In either case, \(P\) takes on the value assigned to it by \({\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\). \(\square \)
Inductive step
If for all \(V \in {\mathbf {Rank}}(1, 2, \dots , k)\), it is true that, for all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\), then for all \(V \in {\mathbf {Rank}}(k+1)\), it will be true that, for all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\).
Proof
Without loss of generality, consider one \(V \in {\mathbf {Rank}}(k+1)\), one \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), one \(\mathbf {x}\), and one \(P \in {\mathbf {PA}}(V)\). Either \(P \in \mathbf {X}\) or \(P \notin \mathbf {X}\). Suppose that \(P \in \mathbf {X}\). Then, \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_{\mathbf {X}=\mathbf {x}}\), by \((f1)\). If \(P \notin \mathbf {X}\), then, since \(rank(P) \leqslant k, {\mathbf {PA}}(P)_{w'} = {\mathbf {PA}}(P)_{\mathbf {X} =\mathbf {x}}\), for all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\), by the inductive hypothesis (since \(P \notin {\mathbf {X}}, {\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- P)\)). Then, \(({\mathcal {V}}1)\) guarantees that \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = \phi _P({\mathbf {PA}}(P)_{\mathbf {X} = \mathbf {x}}\)). So, whether \(P \in \mathbf {X}\) or \(P \notin {\mathbf {X}}, P\) takes on the value \(P_{\mathbf {X}=\mathbf {x}}\) at every \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\). Since \(P, V, \mathbf {X}\), and \(\mathbf {x}\) were arbitrary, for all \(V \in {\mathbf {Rank}}(k+1)\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), and all \({\mathbf {x}}, {\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\) for every \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\). \(\square\)
Lemma 2
Given \(({\mathcal {V}}3)\), \(({\mathcal {U}}1)\), and \((f1)\), for all \(V \in {\mathcal {V}}\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\), where \({\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\) assigns the values to \({\mathbf {PA}}(V)\) determined by the assignment of values \(({\mathcal {U}}- {\mathbf {X}})_w \cup \mathbf {x}\) and the structural equations in \({\mathcal {E}}- \bigcup _i (\phi _{X_i})\), for every endogenous \(X_i \in \mathbf {X}\).
Proof
By induction on the rank of the variables in \({\mathcal {V}}\). \(\square \)
Base case
For all \(V \in {\mathbf {Rank}}(1)\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\).
Proof
Consider, without loss of generality, a variable \(V \in {\mathbf {Rank}}(1)\). Since \(V\)’s rank is 1, every \(P \in {\mathbf {PA}}(V)\) is exogenous. If \(P \in \mathbf {X}\), then \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_{\mathbf {X}=\mathbf {x}}\) (the value assigned to \(X\) by \(\mathbf {x}\)), by \((f1)\). If \(P \notin \mathbf {X}\), then \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_w\), by \(({\mathcal {U}}1)\). In either case, \(P\) takes on the value assigned to it by \({\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\). \(\square \)
Inductive step
If for all \(V \in {\mathbf {Rank}}(1, 2, \ldots , k)\), it is true that, for all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\), then for all \(V \in {\mathbf {Rank}}(k+1)\), it will be true that, for all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), all \(\mathbf {x}\), and all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w),\,{\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\).
Proof
Without loss of generality, consider one \(V \in {\mathbf {Rank}}(k+1)\), one \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), one \(\mathbf {x}\), and one \(P \in {\mathbf {PA}}(V)\). Either \(P \in \mathbf {X}\) or \(P \notin \mathbf {X}\). Suppose that \(P \in \mathbf {X}\). Then, \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = P_{\mathbf {X}=\mathbf {x}}\), by \((f1)\). If \(P \notin \mathbf {X}\), then, since \(rank(P) \leqslant k, {\mathbf {PA}}(P)_{w'} = {\mathbf {PA}}(P)_{\mathbf {X} = \mathbf {x}}\), for all \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\), by the inductive hypothesis (since \(P \notin \mathbf {X}, \mathbf {X} \subseteq {\mathcal {U}}\cup ( {\mathcal {V}}- P)\)). Then, \(({\mathcal {V}}3)\) and \(({\mathfrak {F}}1)\) guarantee that \(f({\mathbf {X}} = {\mathbf {x}}, w) \subset {\mathfrak {F}}_P\) and \({\mathfrak {F}}_P \,\models \, P = \phi _P({\mathbf {PA}}(P))\). So \(f({\mathbf {X}} = {\mathbf {x}}, w) \,\models \, P = \phi _P({\mathbf {PA}}(P)_{\mathbf {X} = \mathbf {x}}).\) So, whether \(P \in \mathbf {X}\) or \(P \notin {\mathbf {X}}, P\) takes on the value \(P_{\mathbf {X}=\mathbf {x}}\) at every \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\). Since \(P, V, \mathbf {X}\), and \(\mathbf {x}\) were arbitrary, for all \(V \in {\mathbf {Rank}}(k+1)\), all \(\mathbf {X} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), and all \({\mathbf {x}}, {\mathbf {PA}}(V)_{w'} = {\mathbf {PA}}(V)_{\mathbf {X} = \mathbf {x}}\) for every \(w' \in f(\mathbf {X} = \mathbf {x}, w)\). \(\square\)
Theorem 1
Given (f1), in a causal model \({\mathcal {M}}= ( {\mathcal {U}}, {\mathcal {V}}, {\mathcal {E}})\), if \({\mathcal {U}}\) satisfies \(({\mathcal {U}}1)\), then \({\mathcal {V}}\) satisfies \(({\mathcal {V}}3)\) iff \({\mathcal {V}}\) satisfies \(({\mathcal {V}}1)\).
Proof
First assume that \({\mathcal {V}}\) satisfies \(({\mathcal {V}}1)\). Then, we know that for all \(V \in {\mathcal {V}}\), all \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), and all assignments \(\mathbf {x}\) to \(\mathbf {X}\),
By Lemma 1, it then follows that
So
So, if for every \(V\), every \(\mathbf {X} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), and every \(\mathbf {x}\), we include every \(w' \in f({\mathbf {X}} = {\mathbf {x}}, w)\) in \({\mathfrak {F}}_V\), then we will have a set \({\mathfrak {F}}_V\) which satisfies \(({\mathfrak {F}}1)\) and which entails that \(V = \phi _V( {\mathbf {PA}}(V))\). So every \(V \in {\mathcal {V}}\) will satisfy \(({\mathcal {V}}3)\).
To establish the other direction, assume that \({\mathcal {V}}\) satisfies \(({\mathcal {V}}3)\). Then, for every \(V \in {\mathcal {V}}\), every \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ({\mathcal {V}}- V)\), and every \({\mathbf {x}}, f({\mathbf {X}} ={\mathbf {x}}, w) \in {\mathfrak {F}}_V\) and \({\mathfrak {F}}_V \,\models \, V = \phi _V({\mathbf {PA}}(V))\). By Lemma 2, it then follows that, for all \(V, \mathbf {X}\), and \(\mathbf {x}\),
So \({\mathcal {V}}\) must satisfy \(({\mathcal {V}}1)\) as well. \(\square\)
Theorem 2
Given (f3), if \(\phi _V\) belongs to a causal model satisfying \(({\mathcal {V}}4)\) at a world \(w_0\), then \(\phi _V\) will continue to belong to a causal model satisfying \(({\mathcal {V}}4)\) after any number of consecutive hypothetical interventions to set the values of any \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ( {\mathcal {V}}- V)\).
Proof
By induction on the number of interventions. \(\square \)
Inductive step
If \(\phi _V\) belongs to a causal model satisfying \(({\mathcal {V}}4)\) at world \(w_k\) after \(k\) interventions to set the values of any \({\mathbf {X}} \subseteq {\mathcal {U}}\cup ( {\mathcal {V}}- V)\), then \(\phi _V\) will belong to a causal model satisfying \(({\mathcal {V}}4)\) at the world \(w_{k+1}\) where there is a \(k+1^{st}\) intervention to set the values of any \(\mathbf {X} \subseteq {\mathcal {U}}\cup ( {\mathcal {V}}- V)\).
Proof
By the inductive hypothesis, \(\phi _V\) belongs to a causal model satisfying \(({\mathcal {V}}4)\) at \(w_k\). This means that there must exist a set of worlds \({\mathfrak {F}}_V\) which satisfies \(({\mathfrak {F}}2)\) and which contains \(w_k\). By \((f3)\), an intervention setting the value of some \(\mathbf {X} \subseteq {\mathcal {U}}\cup ( {\mathcal {V}}- V)\) to \(\mathbf {x}\) must take us to a world \(w_{k+1} \in f({\mathbf {X}} = {\mathbf {x}}, w_k)\). Since \(w_k \in {\mathfrak {F}}_V, ({\mathfrak {F}}2)\) guarantees that \(f({\mathbf {X}} = {\mathbf {x}}, w_k) \subseteq {\mathfrak {F}}_V\), so \(w_{k+1} \in {\mathfrak {F}}_V\) as well. And, by assumption, \({\mathfrak {F}}_V \,\models \, V = \phi _V({\mathbf {PA}}(V))\). So there is a \({\mathfrak {F}}_V \ni w_{k+1}\) such that \({\mathfrak {F}}_V\) satisfies \(({\mathfrak {F}}2)\) and \({\mathfrak {F}}_V \,\models \, V = \phi _V({\mathbf {PA}}(V))\). As \(\phi _V\) was arbitrary, the same holds for every \(V' \notin \mathbf {X}\). So, \(({\mathcal {V}}4)\) will hold at \(w_{k+1}\). So \(\phi _V\) will belong to a causal model satisfying \(({\mathcal {V}}4)\) at \(w_{k+1}\).
Setting \(k = 0\) in the proof of the inductive step establishes the base case. \({\square}\)
Theorem 3
Given (f1), if a causal model \({\mathcal {M}}= ( {\mathcal {U}}, {\mathcal {V}}, {\mathcal {E}})\) is correct according to \(({\mathcal {M}}2)\), then \({\mathfrak {F}}_{\mathcal {M}}\) \(\mathop {=}\limits ^\mathrm{{{def}}}\) \(\bigcap _{V \in {\mathcal {V}}} {\mathfrak {F}}_{V}\) contains all and only allowed assignment of values to the variables \(V \in {\mathcal {U}}\cup {\mathcal {V}}\), where an assignment is allowed just in case it is a solution to the equations in \({\mathcal {E}}\).
Proof
The proof proceeds by induction on the rank of the variables in \({\mathcal {V}}\). \(\square \)
Base case
\({\mathfrak {F}}_{\mathcal {M}}\) contains all and only allowed assignment of values to the variables in \({\mathbf {Rank}}(0)\).
Proof
For every \(V \in {\mathcal {V}}, {\mathfrak {F}}_V\) contains \(f({\mathcal {U}}= {\mathbf {u}}, w)\), for every assignment \(\mathbf {u}\) to \({\mathcal {U}}\), and every \(w \in {\mathfrak {F}}_V\). So \({\mathfrak {F}}_{\mathcal {M}}\) contains \(f({\mathcal {U}}= {\mathbf {u}}, w)\), for every assignment \(\mathbf {u}\) to \({\mathcal {U}}\) and every \(w \in {\mathfrak {F}}_{\mathcal {M}}\). If \(U \in {\mathbf {Rank}}(0)\), then \(U\) is exogenous, \(U \in {\mathcal {U}}\). Every assignment of values to the exogenous variables is allowed. So \({\mathfrak {F}}_{\mathcal {M}}\) contains all and only allowed assignments to the variables in \({\mathbf {Rank}}(0)\). \(\square \)
Inductive step
If \({\mathfrak {F}}_{\mathcal {M}}\) contains all and only allowed assignment of values to the variables in \({\mathbf {Rank}}(0, 1, \ldots , k)\), then it contains all and only allowed assignment of values to the variables in \({\mathbf {Rank}}(0, 1, \ldots , k, k+1)\).
Proof
Take an arbitrary \(V \in {\mathbf {Rank}}(k+1)\). Since \({\mathbf {PA}}(V)\subseteq {\mathbf {Rank}}(0, 1, \dots , k)\), the inductive hypothesis gets us that every and only the allowed assignment of values to \({\mathbf {PA}}(V)\) are realized in \({\mathfrak {F}}_{\mathcal {M}}\). And because \({\mathcal {M}}\) is correct, \(V_w = \phi ({\mathbf {PA}}(V)_w)\), for every \(w \in {\mathfrak {F}}_{V}\). Since \({\mathfrak {F}}_{\mathcal {M}}\subseteq {\mathfrak {F}}_{V}\), this means that \(V_w = \phi ({\mathbf {PA}}(V)_w)\) for every \(w \in {\mathfrak {F}}_{\mathcal {M}}\) as well. So \({\mathfrak {F}}_{\mathcal {M}}\) contains all and only the allowed values of \(V\). Since \(V\) was arbitrary, the above holds for every \(V \in {\mathbf {Rank}}(k+1)\). \(\square\)
Theorem 4
Given \(({\mathcal {M}}2)\) and (f3), for any \(\mathbf {V} \,\subsetneq \,{\mathcal {V}}\),
is non-empty and contains every assignment of values to the variables in \({\mathbf{V}} \cup {\mathcal {U}}\).
Proof
Take an arbitrary \(\mathbf {V} \,\subsetneq\, {\mathcal {V}}\), an arbitrary assignment of values \(\mathbf {v}\) to \(\mathbf {V}\), an arbitrary \(W \notin \mathbf {V}\), and an arbitrary assignment \(\mathbf {u}\) to \({\mathcal {U}}\). Then, \({\mathfrak {F}}_W\) contains worlds at which \(\mathbf {V} \cup {\mathcal {U}}\) is set to \(\mathbf {v} \cup \mathbf {u}\) by an intervention, by \(({\mathfrak {F}}2)\) and \((f3)\). These worlds are not in \(\bigcup _{V \in \mathbf {V}} {\mathfrak {F}}_V\), by the definition of an intervention. Since \(W, \mathbf {V}, \mathbf {v}, \mathbf {u}\), and \(V\) were arbitrary, for every \(W \notin \mathbf {V}\), every \(\mathbf {u}\), and every \(V \in \mathbf {V}\), there are worlds in \({\mathfrak {F}}_W\) which are not in \({\mathfrak {F}}_V\) and at which the value of \(V \cup {\mathcal {U}}\) is set to any value \(\mathbf {v} \cup \mathbf {u}\). Thus, \(\bigcap _{W \notin \mathbf {V}} {\mathfrak {F}}_W - \bigcup _{V \in \mathbf {V}} {\mathfrak {F}}_V\) is non-empty and contains every assignment of values to the variables in \(\mathbf {V} \cup {\mathcal {U}}\). \(\square\)
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Gallow, J.D. A theory of structural determination. Philos Stud 173, 159–186 (2016). https://doi.org/10.1007/s11098-015-0474-5
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DOI: https://doi.org/10.1007/s11098-015-0474-5