Introduction

 (1): ``If another car had come over the hill when you passed that car, there would have been a head-on collision.''

is a useful counterfactual conditional sentence. The driver will estimate how long passing took and how much time he would have had if another car had come over the hill. In this he uses his observation of the distance to the top of the hill when he got back in the right hand lane and his estimate of how long it would have taken a car suddenly coming over the hill to cover that distance.

If he regards the counterfactual (1) as true, he will be more cautious about passing in the future. If he regards it as false, it will be because he thinks he had plenty of time to complete the maneuver and get back in the right lane. If he regards it merely as a material implication with a false antecedent, he'll believe it but won't take it as a reason to change his driving habits.

We discuss such useful counterfactuals for two reasons. (1) We expect them to be useful in AI systems. (2) We have found counterfactuals inferred from experience and having behavioral consequences to admit a richer theory than counterfactuals cooked up to serve as examples of the semantics.

In general, a counterfactual conditional sentence ( counterfactual for short ) is a sentence

in which the antecedent p is false. If implies is taken to be the ordinary mathematical logical implication , then all counterfactuals are true. However, in ordinary life, it is often useful to consider certain counterfactuals, as in the above example (1), as being possibly false. Also (1) may be a useful counterfactual because it permits the driver to learn from an experience he didn't quite have.

We will use to represent counterfactual implication, so that (1) may be written

Another car was coming when you passed there was a head-on collision.

Q. What can the driver learn if he believes (1)?

A. Whatever he could learn from ``Another car came when you passed, and there was a head-on collision.''

Three points.

1. We ignore the possibility that the collision might interfere with the driver's ability to learn.

2. What can be learned depends on the driver's mechanisms for learning. However, we propose that he applies the same learning mechanism to the almost experience as he would apply to the real experience.

3. The hypothetical experience is not a full possible world, e.g. the name of the driver of the car that came over the hill isn't determined. It is a theoretical entity in an approximate theory. [McCarthy, 2000] treats approximate theories in some detail, but the simplest approximate theories for discussing counterfactuals are the theories of cartesian counterfactuals of the present article.

One straightforward inference from (1) is (2)

Another car comes when you pass under sufficiently similar circumstances there is a head-on collision.

Thus a counterfactual conditional is used to infer a corresponding ordinary conditional, univerally quantified over ``similar circumstances''. Believing the counterfactual (1) lets us make the same generalization from the counterfactual conditional that we would make had the collision occurred.

There is some resemblance between this inference and those used to infer a scientific law, but there are two important differences. (1) ``similar circumstances'' is often not spelled out in language. It can be just circumstances that look the same. (2) The inference is from a single experience; more is needed to infer a scientific law.

While counterfactuals are used in daily life, they have been most studied in philosophy. The main philosophical goal has been to assign meaning to these sentences. The Lewis/Stalnaker [Stalnaker, 1968] [Stalnaker and Thomason, 1970] approach is the leading one. David Lewis [Lewis, 1973] defines the truth of a counterfactual using the notion of possible worlds. He posits that possible worlds are ordered by comparative possibility. He then regards as true provided q is true in the nearest possible worlds where p is true. He gives no way of judging the closeness of worlds save that of considering the counterfactuals. Thus it seems difficult to evaluate the truth of any particular counterfactual based on other evidence.

Moreover, he Lewis/Stalnaker approach offers no way of inferring non-counterfactual sentences from counterfactuals.

The truth of a counterfactual does not just depend on the state of the world the way a direct observation does. There is always a theory--in the above example, an implicit theory of driving shared by the passenger and the driver. The theory is based on their experience and what they have learned and been taught about driving.