next up previous
Next: The Almost Crash-Elaborated Up: Detailed Examples Previous: Detailed Examples

Rectangular co-ordinates



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 tex2html_wrap_inline1400 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 tex2html_wrap_inline1412 is true, as we know that tex2html_wrap_inline1414 and tex2html_wrap_inline1416 .

Furthermore, we know that tex2html_wrap_inline1418 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 tex2html_wrap_inline1420 .

Thus different co-ordinate frame make different counterfactuals true. Thus three factors influence the truth of a counterfactual:

  1. The space of possible states.
  2. The co-ordinate frame on the space of possible states.
  3. 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.

next up previous
Next: The Almost Crash-Elaborated Up: Detailed Examples Previous: Detailed Examples

John McCarthy
Wed Jul 12 14:10:43 PDT 2000