In the previous section we claimed that certain counterfactuals were true, given some co-ordinate frames. We now give a preliminary axiomatization in terms of state vectors that allows us to formally prove these statements.
We can define cartesian counterfactuals in terms of state vectors [McCarthy, 1962]. The value of a variable x in a state vector is , while the state vector that is like , save that x has been assigned the value v, is . We can axiomatize a and c as follows.
The numerical example of subsection 3.1 is expressed as follows. Let represent the actual state of the world. We have
We are interested in the function
takes the form
It is obviously false.
Notice that while cartesian counterfactuals give a meaning to ``if x were 7'', they do not give a meaning to ``if x + y were 7. This is a feature, not a bug, because in ordinary common sense, counterfactuals easily constructed from meaningful counterfactuals are often without meaning.
In the above the ``variables'' x, y, and z are logically constants. When we need to quantify over such variables, we need new variables with an appropriate typographical distinction.