Appropriate Causal Models and the Stability of Causation
  Causal models defined in terms of structural equations have proved to be
quite a powerful way of representing knowledge regarding causality. However, a
number of authors have given examples that seem to show that the Halpern-Pearl
(HP) definition of causality gives intuitively unreasonable answers. Here it is
shown that, for each of these examples, we can give two stories consistent with
the description in the example, such that intuitions regarding causality are
quite different for each story. By adding additional variables, we can
disambiguate the stories. Moreover, in the resulting causal models, the HP
definition of causality gives the intuitively correct answer. It is also shown
that, by adding extra variables, a modification to the original HP definition
made to deal with an example of Hopkins and Pearl may not be necessary. Given
how much can be done by adding extra variables, there might be a concern that
the notion of causality is somewhat unstable. Can adding extra variables in a
"conservative" way (i.e., maintaining all the relations between the variables
in the original model) cause the answer to the question "Is X=x a cause of Y=y"
to alternate between "yes" and "no"? It is shown that we can have such
alternation infinitely often, but if we take normality into consideration, we
cannot. Indeed, under appropriate normality assumptions. adding an extra
variable can change the answer from "yes" to "no", but after that, it cannot
cannot change back to "yes".
