Dutch Book against Lewis

According to the PCCP thesis, the probability of a conditional A -> C is the conditional probability P(C vertical bar 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 vertical bar 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 not sign inverted left perpendicularC 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.

Automated summary

According to the PCCP thesis, the probability of a conditional A -> C is the conditional probability P(C | A). However, Lewis' triviality results show that PCCP is not true except in trivial cases. The only rational extension of beliefs concerning non-conditional sentences A and C to the conditional A -> C is by assuming that P(A -> C) = P(C | A), as other cases can lead to a diachronic Dutch Book against the agent.