With the axiomatization of the above story given below we can now consider the first two counterfactuals. In each case we take an approximate theory, which will be some subset of the consequences of the axioms in Section 5 , and a co-ordinate frame. From the theory and the frame we judge the truth of the counterfactual. We show that choosing different approximate theories, or choosing different frames can lead to different choices.
We give the co-ordinate frame that the two instructors use. They
choose what situations were actual, and what fluents held at the
situation 
, and the type of Slope4 as their frame. Here tr
is a truth value. The co-ordinates in the frame are the following
terms, (two of these are propositions, and so take on the values true
or false).
As their core approximate theory they take the axioms about the
effects of various ski moves, 15, the axiom
relating 
and 
, 16, and the unique names and domain closure axioms
and the frame axioms 18.
As the current world, they choose the values,
as they agree that Junior did fall on a slope with a bump, while skiing Slope4.
With this frame the following counterfactuals are false and true, respectively.
In contrast the counterfactual
is true. We prove the first two claims formally after introducing the necessary machinery in the next section.