Exactly how, up coming, might i describe ‘actual causation by using the structural equations framework?

(8) An adjustable Y counterfactually relies on a changeable X inside the good design in the event the and simply if it is actually the circumstances that X = x and Y = y so there can be found viewpoints x? ? x and you may y? ? y in a fashion that replacing the fresh new equation to have X that have X = x? efficiency Y = y?.

A varying Y (distinctive from X and you can Z) is advanced between X and you may Z if and just when it falls under particular route anywhere between 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 lesbian hookup apps review 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 = x_{step 1} 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 arrive by considering just how SEF works closely with instances of later preemption including the Suzy and you will Billy instance. Halpern and you may Pearl (2001, 2005), Hitchcock (2001), and you will Woodward (2003) all offer more or less a similar treatment of later preemption. The key to the treatment is the use of a particular procedure for research the clear presence of an excellent causal relation. The procedure is to find an intrinsic techniques connecting the brand new putative cause-and-effect; suppress the brand new influence of the non-built-in surroundings because of the ‘freezing people landscape while they unquestionably are; right after which topic the fresh putative bring about so you’re able to a good counterfactual test. So, such as, to test whether Suzys throwing a stone was the cause of package so you’re able to shatter, we wish to look at the process powering off ST owing to SH so you’re able to BS; keep augment at their actual worthy of (that’s, 0) the latest variable BH which is extrinsic compared to that procedure; and move the adjustable ST to see if they transform the value of BS. The final procedures include researching brand new counterfactual “If the Suzy hadnt tossed a rock and Billys stone hadnt hit the newest container, the latest bottles would not have shattered”. It is possible to notice that it counterfactual is valid. On the other hand, whenever we perform a similar process to check if or not Billys throwing a rock was the cause of package so you can shatter,we’re necessary to consider the counterfactual “If the Billy hadnt thrown his rock and you may Suzys material got hit the latest container, brand new bottle wouldn’t shattered”. So it counterfactual try false. It is the difference in the fact-viewpoints of these two counterfactuals which explains that they try Suzys rock putting, rather than Billys, one to was the cause of bottle so you can shatter. (A comparable principle is actually designed in Yablo 2002 and you may 2004 regardless of if beyond the structural equations construction.)

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 <X, Y₁,…, 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: