ifc-0219

16.2.1 From relations to mechanisms

Suppose observations support a relation between \(X\) and \(Y\). Prediction, intervention, and explanation remain distinct achievements:

\[ \begin{aligned} \text{prediction:}\quad & P(Y\mid X)\text{ is stable on held-out observations},\\[2pt] \text{intervention:}\quad & P(Y\mid \operatorname {do}(X=x))\text{ is predicted after}\\[-1pt]& \qquad \text{a declared mechanism change},\\[2pt] \text{explanation:}\quad & \text{one typed mechanism survives discriminating tests.} \end{aligned} \]

The first does not imply the second, and the second does not by itself establish the third [ Pearl , 2009 ] . A simulator is useful because it makes the gap executable.

Let \(\mathbb S\) be a scientific sketch and \(\mathcal E\) the typed category of simulator procedures. A realization

\[ R:\mathbb S\longrightarrow \mathcal E \]

maps scientific objects to state, parameter, or observable types and maps mechanisms to executable procedures with matching interfaces. An equation \(p=q\) in a strict sketch model must satisfy \(R(p)=R(q)\). Before admission, an observer may measure a residual between the two executable paths; a nontrivial comparison cell is permitted only when the realization has explicitly been declared lax or approximate. If a proposed arrow has no realization, or a declared equation fails its registered observer, the defect may lie in the theory, the simulator interface, the realization doctrine, or their interpretation.

. A simulator grounds a theory by making its consequences executable. It does not turn executable consequences into facts about nature.