The Other Option

Hanti Lin and Kevin Kelly [LK12] introduce a way of relating probability functions and binary beliefs, referred to as odds-threshold method, which is based on comparing the probabilities of the elementary outcomes. Outcome o1 is preferred to outcome o2 if o1 is sufficiently more likely than o2; the agent’s belief-set is the set of mostly preferred outcomes. As main contribution, we give a probability-independent definition of the preferential orders generated by the odds-threshold method. This allows for a logical characterization of the belief revision properties induced on binary beliefs by Jeffrey conditioning (JC) on a piece of evidence. Specifically, we show that if JC is assumed to be successful – namely, the evidence is implied by the belief-set induced by the posterior probability – then the corresponding belief revision on binary beliefs satisfies all the rules of the nonmonotonic system P together with Disjunctive Rationality [Fre93] and a new rule named Plausible Monotonicity. As JC does not require the evidence to be certain, we also considered rules for iterated belief revision. In the case of successful JC, the induced iterated belief revision satisfies the rules (C1), (C2) and (C3), proposed by Darwiche and Pearl [DP97]. In addition, if JC is also assumed to be confirmatory – i.e. the posterior probability of the evidence is strictly greater than its prior – then the rule (C4) is satisfied as well.