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 space of possible states.
- The co-ordinate frame on the space of possible states.
- The current state.

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.

Wed Jul 12 14:10:43 PDT 2000