This example has all the essential elements of a cartesian counterfactual. We have a cartesian space, whose points are the models of the approximate theory and whose structure is given by the co-ordinates. Each of these co-ordinates in this example is the value of a variable. To find the truth value of a counterfactual we change the co-ordinate as specified by the left hand side and evaluate the right hand side.
The co-ordinate frame we chose here, using the variables x,y,z, is not the only possible frame. A transformation of those variables would give a similar theory but with different counterfactuals.. One possibility is,
Our initial state has the following values.
Given this co-ordinate system, the counterfactual is true, whereas it was false in the previous frame.
In this example, s was uniquely determined by the values of the co-ordinates. We can imagine that another value r, is not uniquely determined--we only know that it is smaller than s. Thus we have,
Again, let us choose x,y,z to have the values (1,2,1). In this case is true, as we know that and .
Furthermore, we know that is false. However, when we cannot uniquely determine all other components in terms of the co-ordinates, some counterfactuals are indeterminate. For instance, we do not know the truth value of .
Thus different co-ordinate frame make different counterfactuals true. Thus three factors influence the truth of a counterfactual:
The above rectangular co-ordinate system example hasn't enough structure to prefer one theory over another. However, suppose it were specified that x, y and z were the co-ordinates along the walls and the height of a point starting from the corner of a room. Then there would be some reason for preferring the x-y-z theory and its associated counterfactuals to the x'-y-z theory and its associated counterfactuals. When there is a clear reason to prefer one theory, its counterfactuals can have a somewhat objective character. These are the most useful. Even so, this would be a useful counterfactual only imbedded in a larger theory that includes some goal.