Appropriate causal models and the stability of causation (Q2804474)

A \textit{causal model} is a pair \((\mathcal{S},\mathcal{F})\) where \(\mathcal S\), the model's \textit{signature}, lists endogenous and exogenous variables together with their possible values, and \(\mathcal F\) is a set of \textit{modifiable structural equations} that relate the values of the variables. The focus is on \textit{acyclic} models for which there exists a strict ordering \(\prec\) on the set of endogenous variables such that if \(X\prec Y\) then the value of \(X\) may affect the value of \(Y\), but not the other way around; so given a \textit{context}, that is, an assignment of values to the exogenous variables, there is a unique solution for all equations, obtained by solving the equations for each variable in the order given by \(\prec\) (starting with the \(\prec\)-minimal endogenous variables, whose values are immediately determined by the values of the exogenous variables). Six examples are discussed in the paper. The first one, also to be found in [\textit{J. Y. Halpern} and \textit{J. Pearl}, Br. J. Philos. Sci. 56, No. 4, 843--887 (2005; Zbl 1092.03003)], is the following.NEWLINENEWLINE``Suzy and Billy both pick up rocks and throw them at a bottle. Suzy's rock gets there first, shattering the bottle.''NEWLINENEWLINEA suitable model should capture the fact that Suzy's throw is a cause of the bottle shattering, while Billy's is not. Halpern and Pearl proposed a model (the HP model) based on counterfactuals, the idea being that \(A\) is a cause of \(B\) if, if \(A\) hadn't occurred (although it did), then \(B\) would not have occurred -- still the model is more complicated as this definition is too naive to work. A simplistic model has three endogenous variables \(\mathrm{ST}\), \(\mathrm{BT}\), and \(\mathrm{BS}\) for ``Suzy throws'', ``Billy throws'', and ``bottle shatters'', respectively, all taking the value 0 or 1 (False or True). There is the equation: \(\mathrm{BS} = \mathrm{ST}\vee\mathrm{BT}\), plus equations that, thanks to one endogenous variable, determine the values of \(\mathrm{ST} \) and \(\mathrm{BT}\). According to the HP model, both \(\mathrm{ST}=1\) and \(\mathrm{BT}=1\) are causes of \(\mathrm{BS}=1\). The way around illustrates one of the themes of the paper, namely, that of the introduction of additional variables: adding \(\mathrm{SH}\) and \(\mathrm{BH}\) for ``Suzy hits'' and ``Billy hits'', respectively, changing the equation \(\mathrm{BS} = \mathrm{ST} \vee \mathrm{BT}\) to \(\mathrm{BS} = \mathrm{SH}\vee\mathrm{BH}\), and adding the equations \(\mathrm{SH} = \mathrm{ST}\) and \(\mathrm{BH} = \mathrm{BT}\wedge\neg\mathrm{SH}\), results in only \(\mathrm{ST}=1\) being the cause of \(\mathrm{BS}=1\), on the basis of a technically involved definition of \(\vec{X}=\vec{x}\) (with \(\vec{X}\) a vector of endogenous variables) being an actual cause of a Boolean combination of formulas of the form \(X = x\) (with \(X\) an endogenous variable). The paper also refines the HP model, considering changing the condition NEWLINE\[NEWLINE(M,\vec{u})\models[\vec{X}\leftarrow\vec{x},\vec{W}\leftarrow\vec{w},\vec{Z'}\leftarrow\vec{z}]\phi\text{ for all subsets }\vec{Z'}\text{ of }\vec{Z}NEWLINE\]NEWLINE to NEWLINE\[NEWLINE(M,\vec{u})\models[\vec{X}\leftarrow\vec{x},\vec{W'}\leftarrow\vec{w},\vec{Z'}\leftarrow\vec{z}]\phi\text{ for all subsets }\vec{Z'}\text{ of }\vec{Z}\text{ and all subsets }\vec{W'}\text{ of }\vec{W}NEWLINE\]NEWLINE where {\parindent=0.7cm\begin{itemize}\item[--] \((M,\vec{u})\) is \(M\) with the exogenous variables set to \(\vec{u}\), \item[--] \([Y_1\leftarrow y_1,\dots,Y_k\leftarrow y_k]\phi\) expresses that \(\phi\) would hold if \(Y_i\) were set to \(y_i\) for \(i = 1,\dots,k\), \item[--] \(\vec{Z}\) is a vector of endogenous variables that make up the ``causal path'' from \(\vec{X}\) to \(\phi\), because changing the value of some variable in \(\vec{X}\) results in changing the values of some variables in \(\vec{Z}\), which results in changing the values of other variables in \(\vec{Z}\), \dots, which finally results in changing the value of \(\phi\), while \item[--] \(\vec{W}\) is a vector of the remaining endogenous variables, that may still have an indirect effect on what happens. NEWLINENEWLINE\end{itemize}} The key result of the paper is that this change can be made unnecessary by adding variables. The paper also discusses the notion of model \textit{normality} and causality \textit{stability}, and show how restriction to the former can imply the latter.