A detachment rule for an operator tells you conditions under which the operator can be removed from that which it governs. So, to use a straightforward example, the detachment rule for the necessity operator is just the inference rule that allows us to infer p from p:
p |- p.
I’m interested in detachment rules for the probability operator. If the operator P is taken to mean “it is probable that”, then the analogue of the inference rule for the necessity operator is a disaster:
P(p) |- p.
The counterexamples should be obvious, but I’ll give one anyway. In a ten ticket lottery, this rule would allow you to conclude that the winner is one of the first 6 tickets sold.
There is pressure, however, to find some true detachment rule, even if the simple one is obviously false.
The pressure comes from the fact that statistical knowledge is possible. Sometimes, a sample teaches us that most North Dakotans do not have PhD’s, for example. So some version of the following has to be true: In certain circumstances a high enough probability for p allows one to detach the probability operator and conclude that p is true.
My interest is in what these circumstances have to be like in order for such a detachment rule to be acceptable. There are some true versions that I’m not interested in, however. For example, a body of evidence might confirm both the high probability of p and p itself. In such circumstances, the operator can be detached. But what I intend for the schema to require is that there are conditions on which a belief in p can be doxastically justified by being based on the probability claim itself, and not where the acceptability of premise and conclusion has merely a common cause.
One more caveat. It may be that there are perfectly fine instances of the schema for some ordinary kind of justification or rationality, but not for the kind of justification that fills the third slot in the typical account of knowledge. If that is true, it would be worth finding out, since if we are entitled to various beliefs that we don’t have sufficient evidence to know, that ought to tell us something about practical reason and appropriate assertion, I would expect.
The pressure to reject all instances of the schema above when discussing the kind of justification involved in an account of knowledge comes from lottery cases. The worry is that we’ll end up having to say that in lottery cases that you can know that your ticket will lose. But I’m also inclined to think it a mistake to deny that there is any such thing as purely statistical knowledge, so I’m trying to sort through all this. Before saying anything about what I’m thinking, though, I’ll stop here and see what others might think about detachment rules for probability operators like the one above.