We now look at an axiomatization that formalizes some examples of counterfactuals in a skiing domain. We sketch a formalization of skiing, which includes both some facts about lessons, and some facts about what happens when you ski. The formalization uses the situation calculus discussed in [McCarthy and Hayes, 1969].
The analysis is designed to highlight several points.
We consider four counterfactuals. The first two are from our previously mentioned two ski instructors. Before we consider the counterfactuals, we consider the following story.
Junior had 25 dollars. He went to a ski slope and spent his money on one cheap lesson, that cost 25 dollars. Cheap lessons teach you one skill each lesson, either bending your knees on bumps, or if you have learned that, then placing your weight on your downhill ski on turns. Expensive lessons cost twice as much, but teach two skills per lesson, in the same order. Thus, Junior learns to bend his knees. He then goes skiing, at a time picked out by situation 1, but when he comes to Slope4 he falls.
Slope4 is a turn, and according to our theory, unless you place your weight on your downhill ski on turns you fall. We also believe that unless you bend your knees on a bump you fall, and that these behaviors happen only if you have learned these skills.
Our first counterfactual states that h``if he had actually bent his knees in situation 1 then he would not have fallen in the next situation''. This will be represented by the following counterfactual.
The other states that h``if he had actually put his weight on the downhill ski in situation 1 then he would not have fallen in the next situation''. In our logical formalization this will be written.
The difference can be resolved by finding out whether Slope4 has a bump or is a turn. As Slope4 is a turn, the first is false and the second is true. Here, the counterfactuals are evaluated in a frame where what ly happens is a component. Thus, we can change the fact that he bent his knees, without changing any other co-ordinate, and thus infer that he does not fall.
Both our instructors can agree that h``if he had chosen more expensive lessons, then he would not have fallen''.
Here they choose to keep the number of lessons fixed, but make the lessons better.
Our hero does not assent to the above conditional. He knows that ``if he chose the expensive lessons, he would not have had any lessons'', as he cannot afford it. Thus he believes that he would fall if he chose expensive lessons,
Thus he still would not have learned the requisite skills. Expensive lessons cost more, so he would not have been able to take as many lessons.
The reason that the ski instructors differ from our hero on this conditional is that they allow the amount of money to vary, while our hero allows the number of lessons to vary. They use different frames and thus get different results.
We give an axiomatization of part of this domain in Section 5, and now present a set of frames that gives the results for the first two counterfactuals.