Causality and Karl Popper
I learned today of some absolutely fascinating interactions between Popperian critical rationalism and the theory of causality. The counterfactual theory of causality is to scientists probably the most useful of the causal theories. The way in which Popperian philosophy enters is in the meaning of counterfactuals and whether they are related by a so-called Structural Equation Model (SEM).
To illustrate, suppose we observe and are interested in three variables, . We know that comes before and chronologically, and comes before . Let’s talk about counterfactuals first. We observe , but suppose we now ask ourselves what would have happened to if, instead of letting nature calculate , we intervened to set it to some value such as . We denote this new, imagined variable by . We might also be interested in the counterfactual variable , which is the new variable that “would result” if we could intervene to set to 0. Counterfactual theories assume that variables such as “exist” and are “available to nature”, and that if actually happens, then nature responds by producing this version of . If, instead, occurs, then nature produces the entirely different variable .
Let us try to reconcile the existence of such counterfactual variables with common views of determinism. If we tried to imagine a process by which nature “calculates” the variables , we’d be tempted to come up with something like this: . Here are three independent random variables (often called “noise”). So nature first calculates from some noise variable using a secret process (just a function, really, but unknown to us). Then it calculates from the previously calculated value of and the noise variable using the function , and finally calculates from using .
Such SEMs are very useful. They can be represented graphically and analyzed mathematically to answer a great many questions in causality, leading to a rich theory developed by Judea Pearl, among many others. An additional bonus is these models account for counterfactual variables in a very natural way. Suppose, for example, that we are interested in the counterfactual , that is the version of that results if someone intervened and set the variable to . (The mechanism by which this might be done is not relevant here.) Nature then simply computes the function . Another example is the counterfactual , which means that is first computed as , and then . The counterfactual is naturally defined as . Thus counterfactuals have very natural semantics in the SEM setting.
The SEM is also a very natural model to humans. Much of what we perceive as Newtonian physics and determinism works this way. Event happens, this influences event (but doesn’t determine it perfectly, which is why is necessary), and so on. Indeed, when I ask myself what other models of the universe might exist (and this question excludes quantum uncertainty and other similar weirdness), I am unable to conceive of a process which doesn’t boil down to an SEM of the type shown above. Most people, when asked to imagine a process by which the universe “creates” events, will probably come up with an SEM.
Perhaps unsurprisingly, this kind of model is inadmissible in the Popperian view, since it uses an underlying justificiation. Intuitively, the structural equations justify the counterfactuals. Delving into things a little more deeply, it turns out that this simple model makes a large number of assumptions that are not verifiable, and so are inadmissible in the Popperian view. But we need a specific example.
A Weird Counterfactual
Specifically, consider the “cross-world counterfactual” . Mathematically, there is no problem with this definition; the function is evaluated at which are themselves well-defined constants or random variables. The problem is with interpretation and observability. Any of the previous counterfactuals could be obtained in a natural way. To observe , set to 0, let nature determine and observe the resulting . To observe , set both to and to and observe the corresponding . But the new quantity is fundamentally different; it involves both (the first argument of ) and needed to observe . This means what we’re trying to do is the following complicated procedure: first intervene and set , obtain the random variable , and then turn back the clock, and now intervene to set as well as set , using the observed before turning back the clock.
Why Critical Rationalism Excludes SEMs
But of course, we can’t really turn back clocks. Thus the weird counterfactual above is unobservable; there should be no way for us to evaluate it under Popperian critical rationalism. If we can evaluate it, it means we are using a rich justification with implications that cannot be verified experimentally. Now, under the SEM it can be shown that
Since all of the quantities on the right hand side are based on observed variables, it means this cross-world counterfactual, which we “couldn’t possibly identify” without turning back the clock, is actually identifiable — a paradox!
What’s the resolution to this paradox? Simply that the SEM makes so many hidden assumptions that we can actually identify it. This explains why SEMs don’t conform to Popperian thought, and why a lot of work focuses on causality and counterfactuals without structural equations.