Dutch Book against Lewis

According to the PCCP thesis, the probability of a conditional A → C is the conditional probability P(C|A). This claim is undermined by Lewis’ triviality results, which purport to show that apart from trivial cases, PCCP is not true. In the present article we show that the only rational, “Dutch Book-resistant” extension of the agent’s beliefs concerning non-conditional sentences A and C to the conditional A → C is by assuming that P(A → C) = P(C|A) (i.e., in accord with PCCP). In other cases a diachronic Dutch Book against the agent can be constructed. There is a tension between our findings and Lewis’ results, which needs to be explained. Therefore, we present a probability space which corresponds in a natural way to the diachronic Dutch Book—and which allows the conditional A → C to be interpreted as an event in a mathematically sound way. It also allows to formalize the notion of conditionalizing A → C on ¬C which plays a crucial role in Lewis’ proof. Our conclusion is that Lewis’ proof is circular, so it cannot be considered to be a sound argument against PCCP.