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Cartesian Counterfactuals via State Vectors

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 tex2html_wrap_inline1464 is tex2html_wrap_inline1466 , while the state vector that is like tex2html_wrap_inline1464 , save that x has been assigned the value v, is tex2html_wrap_inline1474 . We can axiomatize a and c as follows.


The numerical example of subsection 3.1 is expressed as follows. Let tex2html_wrap_inline1480 represent the actual state of the world. We have


We are interested in the function


The counterfactual


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.

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