 For our purposes it is important to point out that conditional probability is always a lower bound for the probability of the material conditional. In other words, \[p(H|E)\leq p(E\rightarrow H),\] whenever \(p(E)&gt;0\). We can see this as a quantitative version of the qualitative principle of Conditionalization we discussed in Section 2.1.1. However confident a Bayesian agent becomes in \(H\) after updating on \(E,\) she must have been at least as confident that \(H\) is a material consequence of \(E\). Popper and Miller (1983) took this observation to be “completely devastating to the inductive interpration of the calculus of probability.” For the history of the Popper-Miller debate see Chapter 4 in Earman (1992). A similar property can be demonstrated for Jeffrey conditioning (Genin 2017, Other Internet Resources).