2005 Volume 13 Issue 2 Pages 105-126
Bayesianism has the following two principles. (B1) Any rational belief state can be represented by a probability function. (B2) Any rational belief change can be represented by conditionalisation. Besides (B1) and (B2), Bayesian confirmation theory has the following principle. (B3) Evidence E confirms a theory T ⇔ the probability of T under the condition E is greater than the probability of T. Glymour argues about what he believes is a counterexample to (B3). When we represent by P the belief state of Einstein at a certain time in November 1915 and consider E to be old evidence for the general theory of relativity (GTR), we obtain P (E)=1. Then we obtain (1) P (GTR|E)=P (GTR). On the other hand, according to (B3), we obtain (2) P (GTR|E)>P (GTR). (2) contradicts (1). This is called the old evidence problem. Howson handles it as follows. The belief state of each agent at each time is relativised to the stock of background information which he has at the time. The reason why the old evidence problem arises is that we relativise the belief state of Einstein to the unsuitable stock of background information. Let K be the stock of background information which Einstein had at that time and let P be the probability function which represents the belief state relativised to the stock. When we choose this P in applying (B3), the old evidence problem arises. Let K_??_{(E)} be the result of deleting from K everything in K dependent on E and let P' be the probability function which represents the belief state relativised to K_??_{(E)}. Because P' (E)≠1, the old evidence problem does not arise. But as Chihara criticises, it is not clear what K_??_{(E)} and P' are like. I handle Chihara's criticism in terms of a probabilistic version of AGM theory. AGM theory can describe such types of belief changes as expansion, contraction and revision. I handle the old evidence problem by means of retaining (B1) and (B3) and relaxing (B2) so as to admit the type of belief change which can be represented by the change from P to P', that is, contraction. Following this line of thought, I show that we can consider P' to be the contraction of P with respect to E. Relying mainly on the writings of Gärdenfors and Spohn, I show that we can construct a probabilistic contraction function which generates the contraction of P with respect to E.
