To go beyond cartesian counterfactuals, it is useful to consider trees of possibilities. A node in the tree corresponds to fixing some of the aspects of the entity being considered and regarding the others as variable.
To fix the ideas, let the entity be the world and its history--but there are others.
We consider trees with finite branching and finite depth. We may regard the leaves as possible worlds.
A theory of counterfactuals based on the trees can have more structure than theories based solely on the possible worlds.
Returning to the adventures of Junior, we consider the node A from which the possibilities branch according to whether Junior went surfing, went skiing or stayed home. If Junior went skiing, it makes sense to ask whether he took cheap skiing lessons, expensive skiing lessons or none. These are edges leading from the node leading from the edge in which he went skiing. If he went surfing, there are no edges leading from that node corresponding to the different skiing lessons. [Therefore, we don't have a cartesian product structure, but we could force one by putting edges for the kinds of ski lessons leading from the node in which he went surfing. We don't do that.]
Now let us suppose that Junior had lunch on the given day, regardless of whether he skied or surfed. If he went skiing or surfing, we suppose that at lunch that he may have had either a hamburger or a hot dog or a pizza. If he stayed home he had chicken soup. In fact suppose that he had a hamburger. Now consider the counterfactual, ``If Junior had had a hot dog, he would have had indigestion.'' This counterfactual may be stated either about the skiing node or about the surfing node. Let's put in another branch on whether Junior telephoned Deborah or Sheila or neither.
We can imagine the tree of possibilities to have been constructed in two ways. in one the split on what Junior had for lunch precedes in the tree the split on whom he telephoned. We say ``precedes in the tree'', because we are not concerned with the temporal order of these events, ignoring the fact that if he telephoned Deborah he had to do it before lunch whereas telephoning Sheila would have been done after lunch.
Let the variable x have the value 1, 2 or 3 according to whether Junior went skiing, surfing or stayed home. Let y have one of these values according to whether he had expensive lessons, cheap lessons or none. Let z have values according to what he had for lunch and w have values according to whom he telephoned.
We can label the edges of the tree. The edges from node A are labeled, x=1, x=2, and x=3. We can use a notation for this reminiscent of the notation used for state vectors and write a(x,1,A) or more explicitly . However, in the tree case, we don't have all the nice properties of the state vector case. The edges leading from the node a(x,1,A) are labelled y=1, y=2 and y=3. However, the edges leading from a(x,2,A) and a(x,3,A) have no y-labels nor do any of their successors in the tree, i.e. a(y,1,a(x,2,A)) is undefined. If Junior didn't go skiing, there is no branch according to what kind of lessons he took.
Here are the trees. Tree 1 puts skiing, surfing and home on the edges leading from A, and Tree 2 puts whom he telephoned on these edges.
The z and w situations are simpler. Provided the value of x is 1 or 2, the tree may be arranged to put the edges labelled with z before or after the edges labelled with w. We can write, for example,
This is a local cartesian product case. We can say that the variables z and w commute. The relation between z and x is more complex, because telephoning and having hot dogs for lunch arise only when x is 1 or 2. Also when x=1, z and w may jointly commute with y.
We say that a pair of co-ordinates is locally cartesian under a condition , when
This notion make sense for sets of co-ordinates. A set of co-ordinates are locally cartesian under a condition , when all orders of assignment agree.
We can now write
The branching tree of propositional possibilities is isomorphic to the structure of compound conditional expressions discussed in [McCarthy, 1963].