So far we have considered cartesian counterfactuals. In this section we consider how we can go beyond this basic case. The essential restriction that characterizes cartesian counterfactuals is that every point in the product space is a meaningful state of affairs. Thus, if we have a frame with co-ordinates and , we can choose any values for and , say , and , and the theory that we get when we add to our approximate theory correctly predicts the truth of counterfactuals with premises asserting the values of , etc.
Sometimes there are assignments of co-ordinates that are not meaningful. This can be because the co-ordinates having those values breaks some rule, or is impossible. In other cases, though those values of the co-ordinates are not excluded by our approximate theory, it may be that our approximate theory does not correctly predict the outcome for those values. Finally, so co-ordinates are not meaningful given when other co-ordinates have certain values. For instance, a choice of whether to bend ones knees or not does not make sense unless you are skiing.
This is made clearer by some examples.
The first example shows a case where some values of the co-ordinates are excluded by the approximate theory. In this theory x is the height of the bottom of a spring, and y the height of the top of the spring, s is then the height of the spring, and f the force outwards on each endpoint. l is the length of the spring at rest, and e the elasticity. In this case, Hooke's Law tells us ``ut tensio sic vis'', i.e. the force is proportional to the extension of the spring, where e is the proportion, and further, it is in the opposite direction.
Consider the following approximate theory,
In the current world x = 3, y = 5, so the spring is at its nominal length.
Then, if we have as our co-ordinate frame, we have that if , further we have that , that is, the force on the bottom of the string is 2 units upwards.
What would happen if we moved the bottom of the string to position 6. In the real world, this would be either impossible, (as we would try to push something through itself), or possibly we might end up with a reversed situation. In either case, the approximate theory is no longer the case. We can tell this because the theory makes the following counterfactual true .
However, sometimes we will go outside the range of meaningfulness of our approximate theory without reaching a contradiction inside it. Consider our spring theory above. Suppose that the spring breaks when it is stretched to length 6.
Our theory tells us that y = 10 > f = - 10. This is incorrect, but we cannot tell this from our approximate theory alone. In the previous case we could detect a problem because of inconsistency. In this case, we cannot detect the problem in that way.
In both these cases, out theory was cartesian in a certain region, but outside that region, that simple structure broke down, and the cartesian structure no longer provided meaningful answers.