Kaufmann on the Probabilities of Conditionals

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different probabilities to a single conditional at the same time. According to the Thesis, the probability of an indicative conditional (henceforth, simply “conditional”) equals the probability of its consequent conditional on the antecedent. While the Thesis is beset by various problems, it is widely thought to accord quite well with our pretheoretical verdicts. However, Kaufmann [2004] argues that under certain specifiable circumstances the pretheoretically “right” probability of a conditional departs systematically, and sometimes rather starkly, from the probability it ought to be assigned according to the Thesis. He also presents another rule for calculating the probability of a conditional—yielding the conditional’s local probability, as he calls it—that, he thinks, does better in this respect. Below, I aim to show that this rule is inconsistent in that it may lead one to assign different (local) probabilities to a single conditional at the same time. I further argue that the intuitions that appear to underlie Kaufmann’s proposal are to be explained away rather than explained. Kaufmann starts by considering a scenario in which we are supposed to pick a ball from one of two bags, X and Y, whose contents are as specified in the table below. We do not know which of the two bags we are facing, but we do know that it is three times as likely that we are facing Y as that we are facing X; we also know the contents of each bag. Pr(Bag X) = 1/4 Pr(Bag Y) = 3/4 10 red balls, 10 red balls, 9 of them with a black spot; 1 of them with a black spot; 2 white balls 50 white balls He next asks whether in this situation the strength of our belief in the following conditional would be best characterized as being high, about fifty–fifty, or low: If I pick a red ball, it will have a black spot. (1)