We have shown the usefulness of counterfactuals in theories with a cartesian product structure, showing how to infer such counterfactuals and how to make inferences from them. We also consider more general tree-structured counterfactuals.
We do not claim there is a single set of true counterfactuals. Rather, a counterfactual can only be judged relative to a background theory. In the cartesian case, the theory involves a choice of a ``co-ordinate frame''.
These counterfactuals give information about how the world behaves, so that in future situations the reasoner can better predict what will happen.
To be useful, a counterfactual needs to be imbedded in a theory that includes goals or a notion of utility.
The theories inhabited by counterfactuals are usually approximate theories of the world and sometimes involve concepts and objects that are not fully defined. Approximate theories and approximate objects and their relationships are discussed in [McCarthy, 2000].