Exactly how, upcoming, you will i establish ‘actual causation utilizing the structural equations construction?

(8) A varying Y counterfactually hinges on a varying X in the good design in the event that and only if it is actually the instance one X = x and you will Y = y there can be found opinions x? ? x and y? ? y in a fashion that replacing brand new picture to possess X which have X = x? production Y = y?.

A variable Y (different from X and Z) are intermediate between X and you can Z in the event that and just when it belongs to particular channel ranging from X and you can Z

Of course, so far we just have something we are calling a ‘causal model, ?V, E?; we havent been told anything about how to extract causal information from it. As should be obvious by now, the basic recipe is going to be roughly as follows: the truth of ‘c causes e (or ‘c is an actual cause of e), where c and e are particular, token events, will be a matter of the counterfactual relationship, as encoded by the model, between two variables X and Y, where the occurrence of c is represented by a structural equation of the form X = x₁ and the occurrence of e is represented by a structural equation of the form Y = y₁. That would get us the truth of “Suzys throw caused her rock to hit the bottle” (ST = 1 and SH = 1, and, since SH = ST is a member of E, we know that if we replace ST = 1 with ST = 0, we get SH = 0). But it wont get us, for example, the truth of “Suzys throw caused the bottle to shatter”, since if we replace ST = 1 with ST = 0 and work through the equations we still end up with BS = 1.

Better make it by the given exactly how SEF deals with cases of late preemption such as the Suzy and Billy instance. Halpern and you may Pearl (2001, 2005), Hitchcock (2001), and Woodward (2003) all of the offer approximately the same treatment of late preemption. The secret to their treatment is the aid of a particular means of investigations the existence of an effective causal relatives. The procedure is to find an integrated techniques connecting the fresh putative cause-and-effect; suppress the newest influence of its non-built-in surroundings because of the ‘freezing those individuals land as they unquestionably are; then subject the new putative end up in to help you a beneficial counterfactual sample. Thus, like, to check on whether or not Suzys putting a stone was the cause of bottle to shatter, we would like to look at the method running off ST courtesy SH to BS; keep augment in the the actual really worth (that is, 0) the newest varying BH that’s extrinsic to this processes; and then step brand new changeable ST to see if they changes the worth of BS. The very last tips cover researching the fresh counterfactual “When the Suzy hadnt tossed a rock and you will Billys stone hadnt strike the newest container, the latest package do not have shattered”. It is easy to see that it counterfactual is valid. On the other hand, once we manage an equivalent procedure to check if or not Billys putting a stone was the cause of bottles in order to shatter,the audience is necessary to check out the counterfactual “When the Billy hadnt thrown their material and you will Suzys stone got strike new package, the fresh new container won’t smashed”. That it counterfactual is false. It will be the difference between your situation-viewpoints of these two counterfactuals which explains that it try Suzys material putting, rather than Billys, one to caused the container so you can shatter. (An equivalent concept try developed in Yablo 2002 and you can 2004 no matter if beyond the structural equations structure.)

Hitchcock (2001) presents a useful regimentation of this reasoning. He defines a route between two variables X and Z in the set V to be an ordered sequence of variables 1,…, Y_n, Z> such that each variable in the sequence is in V and is a parent of its successor in the sequence. Then he introduces the new concept of an active causal route: