So now we have a better idea of what it is that a model of belief should do. But which part of our model of belief is representing which part of the belief state? The first thing to say is that \(P(X)\) is not an adequate representation of the belief in \(X\). That is, one of the values of the credal set approach is that it can capture certain kinds of non-logical relationships between propositions that are lost when focusing on, say, the associated set of probability values. For example, consider tossing a coin of unknown bias. \(P(H)=P(T)=[0,1]\), but this fails to represent the important fact that \(p(H)=1-p(T)\) for all \(p\in P\). Or that getting a heads on the first toss is at least as likely as heads on two consecutive tosses. These facts that aren’t captured by the sets-of-values view can play an important role in reasoning and decision.