Alan Hajek Probabilities of Conditionals - Revisited

Is a rational agent’s subjective probability assignment to a conditional always the same as the corresponding subjective conditional probability? The hypothesis that it is has come to be known as Stalnaker’s Hypothesis:’ (SW P(A -+ C) = P(C/A) whenever P(A) > 0 where + is a conditional connective, and P(C/A) = P(CA)/P(A). (SH) has been the subject of much debate. I hope to show in this paper that it cannot be right in general, and that it in fact fails in all situations in which it would have the most obvious applicability. My claim may have a familiar ring to it, and to be sure, a number of authors have stalked Stalnaker’s Hypothesis. The best known arguments against it are the triviality results due to David Lewis (1976 and 1986). However, these results, which concern classes of probability functions, rest on certain assumptions, notably: A 1. Each class of probability functions is closed under certain operations (such as conditionalization or Jeffrey conditionalization). A2. The proposition expressed by a conditional sentence is independent of the probability function defined on it.2 Given his assumptions, Lewis succeeds in showing that (SH) has unfortunate consequences, namely triviality of all the probability functions in each such class. Lewis himself has pointed out Al (1976, p. 303 and 1986, p. 588), and Stalnaker has noted A2 (1976, p. 302), dubbing this the assumption of ‘metaphysical realism’. Furthermore, these assumptions have met with some opposition. Van Fraassen, for example, argues against Al (1989) and against A2 (1976, p. 275). I will now propose an argument against (SH) that is free of these assumptions of Lewis’ in fact, practically free of any assumption about +. All that I will ask is that whenever A and C are propositions, A -+ C is also a proposition.